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Theorem poimirlem28 37043
Description: Lemma for poimir 37048, a variant of Sperner's lemma. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (πœ‘ β†’ 𝑁 ∈ β„•)
poimirlem28.1 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ 𝐡 = 𝐢)
poimirlem28.2 ((πœ‘ ∧ 𝑝:(1...𝑁)⟢(0...𝐾)) β†’ 𝐡 ∈ (0...𝑁))
poimirlem28.3 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) β†’ 𝐡 < 𝑛)
poimirlem28.4 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 𝐾)) β†’ 𝐡 β‰  (𝑛 βˆ’ 1))
poimirlem28.5 (πœ‘ β†’ 𝐾 ∈ β„•)
Assertion
Ref Expression
poimirlem28 (πœ‘ β†’ βˆƒπ‘  ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢)
Distinct variable groups:   𝑓,𝑖,𝑗,𝑛,𝑝,𝑠   πœ‘,𝑗,𝑛   𝑗,𝑁,𝑛   πœ‘,𝑖,𝑝,𝑠   𝐡,𝑓,𝑖,𝑗,𝑛,𝑠   𝑓,𝐾,𝑖,𝑗,𝑛,𝑝,𝑠   𝑓,𝑁,𝑖,𝑝,𝑠   𝐢,𝑖,𝑛,𝑝
Allowed substitution hints:   πœ‘(𝑓)   𝐡(𝑝)   𝐢(𝑓,𝑗,𝑠)
Proof of Theorem poimirlem28
Dummy variables π‘˜ π‘š π‘ž 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ β„•)
21nnnn0d 12548 . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„•0)
31nnred 12243 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ ℝ)
43leidd 11796 . . . . 5 (πœ‘ β†’ 𝑁 ≀ 𝑁)
52, 2, 43jca 1126 . . . 4 (πœ‘ β†’ (𝑁 ∈ β„•0 ∧ 𝑁 ∈ β„•0 ∧ 𝑁 ≀ 𝑁))
6 oveq2 7422 . . . . . . . . . . . . . . . 16 (π‘˜ = 0 β†’ (1...π‘˜) = (1...0))
7 fz10 13540 . . . . . . . . . . . . . . . 16 (1...0) = βˆ…
86, 7eqtrdi 2783 . . . . . . . . . . . . . . 15 (π‘˜ = 0 β†’ (1...π‘˜) = βˆ…)
98oveq2d 7430 . . . . . . . . . . . . . 14 (π‘˜ = 0 β†’ ((0..^𝐾) ↑m (1...π‘˜)) = ((0..^𝐾) ↑m βˆ…))
10 fzofi 13957 . . . . . . . . . . . . . . . 16 (0..^𝐾) ∈ Fin
11 map0e 8890 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ∈ Fin β†’ ((0..^𝐾) ↑m βˆ…) = 1o)
1210, 11ax-mp 5 . . . . . . . . . . . . . . 15 ((0..^𝐾) ↑m βˆ…) = 1o
13 df1o2 8485 . . . . . . . . . . . . . . 15 1o = {βˆ…}
1412, 13eqtri 2755 . . . . . . . . . . . . . 14 ((0..^𝐾) ↑m βˆ…) = {βˆ…}
159, 14eqtrdi 2783 . . . . . . . . . . . . 13 (π‘˜ = 0 β†’ ((0..^𝐾) ↑m (1...π‘˜)) = {βˆ…})
16 eqidd 2728 . . . . . . . . . . . . . . . . 17 (π‘˜ = 0 β†’ 𝑓 = 𝑓)
1716, 8, 8f1oeq123d 6827 . . . . . . . . . . . . . . . 16 (π‘˜ = 0 β†’ (𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜) ↔ 𝑓:βˆ…β€“1-1-ontoβ†’βˆ…))
18 eqid 2727 . . . . . . . . . . . . . . . . 17 βˆ… = βˆ…
19 f1o00 6868 . . . . . . . . . . . . . . . . 17 (𝑓:βˆ…β€“1-1-ontoβ†’βˆ… ↔ (𝑓 = βˆ… ∧ βˆ… = βˆ…))
2018, 19mpbiran2 709 . . . . . . . . . . . . . . . 16 (𝑓:βˆ…β€“1-1-ontoβ†’βˆ… ↔ 𝑓 = βˆ…)
2117, 20bitrdi 287 . . . . . . . . . . . . . . 15 (π‘˜ = 0 β†’ (𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜) ↔ 𝑓 = βˆ…))
2221abbidv 2796 . . . . . . . . . . . . . 14 (π‘˜ = 0 β†’ {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)} = {𝑓 ∣ 𝑓 = βˆ…})
23 df-sn 4625 . . . . . . . . . . . . . 14 {βˆ…} = {𝑓 ∣ 𝑓 = βˆ…}
2422, 23eqtr4di 2785 . . . . . . . . . . . . 13 (π‘˜ = 0 β†’ {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)} = {βˆ…})
2515, 24xpeq12d 5703 . . . . . . . . . . . 12 (π‘˜ = 0 β†’ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) = ({βˆ…} Γ— {βˆ…}))
26 0ex 5301 . . . . . . . . . . . . 13 βˆ… ∈ V
2726, 26xpsn 7144 . . . . . . . . . . . 12 ({βˆ…} Γ— {βˆ…}) = {βŸ¨βˆ…, βˆ…βŸ©}
2825, 27eqtr2di 2784 . . . . . . . . . . 11 (π‘˜ = 0 β†’ {βŸ¨βˆ…, βˆ…βŸ©} = (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}))
29 elsni 4641 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ 𝑠 = βŸ¨βˆ…, βˆ…βŸ©)
3026, 26op1std 7995 . . . . . . . . . . . . . . . . . . 19 (𝑠 = βŸ¨βˆ…, βˆ…βŸ© β†’ (1st β€˜π‘ ) = βˆ…)
3129, 30syl 17 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ (1st β€˜π‘ ) = βˆ…)
3231oveq1d 7429 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ ((1st β€˜π‘ ) ∘f + βˆ…) = (βˆ… ∘f + βˆ…))
33 f0 6772 . . . . . . . . . . . . . . . . . . . 20 βˆ…:βˆ…βŸΆβˆ…
34 ffn 6716 . . . . . . . . . . . . . . . . . . . 20 (βˆ…:βˆ…βŸΆβˆ… β†’ βˆ… Fn βˆ…)
3533, 34mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ βˆ… Fn βˆ…)
3626a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ βˆ… ∈ V)
37 inidm 4214 . . . . . . . . . . . . . . . . . . 19 (βˆ… ∩ βˆ…) = βˆ…
38 0fv 6935 . . . . . . . . . . . . . . . . . . . 20 (βˆ…β€˜π‘›) = βˆ…
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∧ 𝑛 ∈ βˆ…) β†’ (βˆ…β€˜π‘›) = βˆ…)
4035, 35, 36, 36, 37, 39, 39offval 7686 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ (βˆ… ∘f + βˆ…) = (𝑛 ∈ βˆ… ↦ (βˆ… + βˆ…)))
41 mpt0 6691 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ βˆ… ↦ (βˆ… + βˆ…)) = βˆ…
4240, 41eqtrdi 2783 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ (βˆ… ∘f + βˆ…) = βˆ…)
4332, 42eqtrd 2767 . . . . . . . . . . . . . . . 16 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ ((1st β€˜π‘ ) ∘f + βˆ…) = βˆ…)
4443uneq1d 4158 . . . . . . . . . . . . . . 15 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ (((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) = (βˆ… βˆͺ ((1...𝑁) Γ— {0})))
45 uncom 4149 . . . . . . . . . . . . . . . 16 (βˆ… βˆͺ ((1...𝑁) Γ— {0})) = (((1...𝑁) Γ— {0}) βˆͺ βˆ…)
46 un0 4386 . . . . . . . . . . . . . . . 16 (((1...𝑁) Γ— {0}) βˆͺ βˆ…) = ((1...𝑁) Γ— {0})
4745, 46eqtri 2755 . . . . . . . . . . . . . . 15 (βˆ… βˆͺ ((1...𝑁) Γ— {0})) = ((1...𝑁) Γ— {0})
4844, 47eqtr2di 2784 . . . . . . . . . . . . . 14 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ ((1...𝑁) Γ— {0}) = (((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})))
4948csbeq1d 3893 . . . . . . . . . . . . 13 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅)
5049eqeq2d 2738 . . . . . . . . . . . 12 (𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} β†’ (0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ ↔ 0 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅))
51 oveq2 7422 . . . . . . . . . . . . . . 15 (π‘˜ = 0 β†’ (0...π‘˜) = (0...0))
52 0z 12585 . . . . . . . . . . . . . . . 16 0 ∈ β„€
53 fzsn 13561 . . . . . . . . . . . . . . . 16 (0 ∈ β„€ β†’ (0...0) = {0})
5452, 53ax-mp 5 . . . . . . . . . . . . . . 15 (0...0) = {0}
5551, 54eqtrdi 2783 . . . . . . . . . . . . . 14 (π‘˜ = 0 β†’ (0...π‘˜) = {0})
56 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘˜ = 0 β†’ ((𝑗 + 1)...π‘˜) = ((𝑗 + 1)...0))
5756imaeq2d 6057 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ = 0 β†’ ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) = ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)))
5857xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 0 β†’ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}) = (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))
5958uneq2d 4159 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 0 β†’ ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0})) = ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0})))
6059oveq2d 7430 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 0 β†’ ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))))
61 oveq1 7421 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ = 0 β†’ (π‘˜ + 1) = (0 + 1))
62 0p1e1 12350 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6361, 62eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 0 β†’ (π‘˜ + 1) = 1)
6463oveq1d 7429 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 0 β†’ ((π‘˜ + 1)...𝑁) = (1...𝑁))
6564xpeq1d 5701 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 0 β†’ (((π‘˜ + 1)...𝑁) Γ— {0}) = ((1...𝑁) Γ— {0}))
6660, 65uneq12d 4160 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 0 β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) = (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})))
6766csbeq1d 3893 . . . . . . . . . . . . . . . . 17 (π‘˜ = 0 β†’ ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅)
6867eqeq2d 2738 . . . . . . . . . . . . . . . 16 (π‘˜ = 0 β†’ (𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅))
6955, 68rexeqbidv 3338 . . . . . . . . . . . . . . 15 (π‘˜ = 0 β†’ (βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆƒπ‘— ∈ {0}𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅))
70 c0ex 11224 . . . . . . . . . . . . . . . 16 0 ∈ V
71 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 β†’ (1...𝑗) = (1...0))
7271, 7eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 β†’ (1...𝑗) = βˆ…)
7372imaeq2d 6057 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 β†’ ((2nd β€˜π‘ ) β€œ (1...𝑗)) = ((2nd β€˜π‘ ) β€œ βˆ…))
74 ima0 6074 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd β€˜π‘ ) β€œ βˆ…) = βˆ…
7573, 74eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 β†’ ((2nd β€˜π‘ ) β€œ (1...𝑗)) = βˆ…)
7675xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 β†’ (((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) = (βˆ… Γ— {1}))
77 0xp 5770 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ… Γ— {1}) = βˆ…
7876, 77eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 β†’ (((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) = βˆ…)
79 oveq1 7421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 0 β†’ (𝑗 + 1) = (0 + 1))
8079, 62eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 0 β†’ (𝑗 + 1) = 1)
8180oveq1d 7429 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 β†’ ((𝑗 + 1)...0) = (1...0))
8281, 7eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 β†’ ((𝑗 + 1)...0) = βˆ…)
8382imaeq2d 6057 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 β†’ ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) = ((2nd β€˜π‘ ) β€œ βˆ…))
8483, 74eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 β†’ ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) = βˆ…)
8584xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 β†’ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}) = (βˆ… Γ— {0}))
86 0xp 5770 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ… Γ— {0}) = βˆ…
8785, 86eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 β†’ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}) = βˆ…)
8878, 87uneq12d 4160 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 β†’ ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0})) = (βˆ… βˆͺ βˆ…))
89 un0 4386 . . . . . . . . . . . . . . . . . . . . 21 (βˆ… βˆͺ βˆ…) = βˆ…
9088, 89eqtrdi 2783 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 β†’ ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0})) = βˆ…)
9190oveq2d 7430 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 β†’ ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) = ((1st β€˜π‘ ) ∘f + βˆ…))
9291uneq1d 4158 . . . . . . . . . . . . . . . . . 18 (𝑗 = 0 β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})) = (((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})))
9392csbeq1d 3893 . . . . . . . . . . . . . . . . 17 (𝑗 = 0 β†’ ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅ = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅)
9493eqeq2d 2738 . . . . . . . . . . . . . . . 16 (𝑗 = 0 β†’ (𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅))
9570, 94rexsn 4682 . . . . . . . . . . . . . . 15 (βˆƒπ‘— ∈ {0}𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...0)) Γ— {0}))) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅)
9669, 95bitrdi 287 . . . . . . . . . . . . . 14 (π‘˜ = 0 β†’ (βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅))
9755, 96raleqbidv 3337 . . . . . . . . . . . . 13 (π‘˜ = 0 β†’ (βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ {0}𝑖 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅))
98 eqeq1 2731 . . . . . . . . . . . . . 14 (𝑖 = 0 β†’ (𝑖 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 0 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅))
9970, 98ralsn 4681 . . . . . . . . . . . . 13 (βˆ€π‘– ∈ {0}𝑖 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 0 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅)
10097, 99bitr2di 288 . . . . . . . . . . . 12 (π‘˜ = 0 β†’ (0 = ⦋(((1st β€˜π‘ ) ∘f + βˆ…) βˆͺ ((1...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
10150, 100sylan9bbr 510 . . . . . . . . . . 11 ((π‘˜ = 0 ∧ 𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©}) β†’ (0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
10228, 101rabeqbidva 3443 . . . . . . . . . 10 (π‘˜ = 0 β†’ {𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅} = {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})
103102eqcomd 2733 . . . . . . . . 9 (π‘˜ = 0 β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} = {𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅})
104103fveq2d 6895 . . . . . . . 8 (π‘˜ = 0 β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅}))
105104breq2d 5154 . . . . . . 7 (π‘˜ = 0 β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅})))
106105notbid 318 . . . . . 6 (π‘˜ = 0 β†’ (Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅})))
107106imbi2d 340 . . . . 5 (π‘˜ = 0 β†’ ((πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) ↔ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅}))))
108 oveq2 7422 . . . . . . . . . . . 12 (π‘˜ = π‘š β†’ (1...π‘˜) = (1...π‘š))
109108oveq2d 7430 . . . . . . . . . . 11 (π‘˜ = π‘š β†’ ((0..^𝐾) ↑m (1...π‘˜)) = ((0..^𝐾) ↑m (1...π‘š)))
110 eqidd 2728 . . . . . . . . . . . . 13 (π‘˜ = π‘š β†’ 𝑓 = 𝑓)
111110, 108, 108f1oeq123d 6827 . . . . . . . . . . . 12 (π‘˜ = π‘š β†’ (𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜) ↔ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)))
112111abbidv 2796 . . . . . . . . . . 11 (π‘˜ = π‘š β†’ {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)} = {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)})
113109, 112xpeq12d 5703 . . . . . . . . . 10 (π‘˜ = π‘š β†’ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) = (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}))
114 oveq2 7422 . . . . . . . . . . 11 (π‘˜ = π‘š β†’ (0...π‘˜) = (0...π‘š))
115 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = π‘š β†’ ((𝑗 + 1)...π‘˜) = ((𝑗 + 1)...π‘š))
116115imaeq2d 6057 . . . . . . . . . . . . . . . . . 18 (π‘˜ = π‘š β†’ ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) = ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)))
117116xpeq1d 5701 . . . . . . . . . . . . . . . . 17 (π‘˜ = π‘š β†’ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}) = (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))
118117uneq2d 4159 . . . . . . . . . . . . . . . 16 (π‘˜ = π‘š β†’ ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0})) = ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0})))
119118oveq2d 7430 . . . . . . . . . . . . . . 15 (π‘˜ = π‘š β†’ ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))))
120 oveq1 7421 . . . . . . . . . . . . . . . . 17 (π‘˜ = π‘š β†’ (π‘˜ + 1) = (π‘š + 1))
121120oveq1d 7429 . . . . . . . . . . . . . . . 16 (π‘˜ = π‘š β†’ ((π‘˜ + 1)...𝑁) = ((π‘š + 1)...𝑁))
122121xpeq1d 5701 . . . . . . . . . . . . . . 15 (π‘˜ = π‘š β†’ (((π‘˜ + 1)...𝑁) Γ— {0}) = (((π‘š + 1)...𝑁) Γ— {0}))
123119, 122uneq12d 4160 . . . . . . . . . . . . . 14 (π‘˜ = π‘š β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) = (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})))
124123csbeq1d 3893 . . . . . . . . . . . . 13 (π‘˜ = π‘š β†’ ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅)
125124eqeq2d 2738 . . . . . . . . . . . 12 (π‘˜ = π‘š β†’ (𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
126114, 125rexeqbidv 3338 . . . . . . . . . . 11 (π‘˜ = π‘š β†’ (βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
127114, 126raleqbidv 3337 . . . . . . . . . 10 (π‘˜ = π‘š β†’ (βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
128113, 127rabeqbidv 3444 . . . . . . . . 9 (π‘˜ = π‘š β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} = {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})
129128fveq2d 6895 . . . . . . . 8 (π‘˜ = π‘š β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))
130129breq2d 5154 . . . . . . 7 (π‘˜ = π‘š β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
131130notbid 318 . . . . . 6 (π‘˜ = π‘š β†’ (Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
132131imbi2d 340 . . . . 5 (π‘˜ = π‘š β†’ ((πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) ↔ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
133 oveq2 7422 . . . . . . . . . . . 12 (π‘˜ = (π‘š + 1) β†’ (1...π‘˜) = (1...(π‘š + 1)))
134133oveq2d 7430 . . . . . . . . . . 11 (π‘˜ = (π‘š + 1) β†’ ((0..^𝐾) ↑m (1...π‘˜)) = ((0..^𝐾) ↑m (1...(π‘š + 1))))
135 eqidd 2728 . . . . . . . . . . . . 13 (π‘˜ = (π‘š + 1) β†’ 𝑓 = 𝑓)
136135, 133, 133f1oeq123d 6827 . . . . . . . . . . . 12 (π‘˜ = (π‘š + 1) β†’ (𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜) ↔ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))))
137136abbidv 2796 . . . . . . . . . . 11 (π‘˜ = (π‘š + 1) β†’ {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)} = {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))})
138134, 137xpeq12d 5703 . . . . . . . . . 10 (π‘˜ = (π‘š + 1) β†’ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) = (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}))
139 oveq2 7422 . . . . . . . . . . 11 (π‘˜ = (π‘š + 1) β†’ (0...π‘˜) = (0...(π‘š + 1)))
140 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = (π‘š + 1) β†’ ((𝑗 + 1)...π‘˜) = ((𝑗 + 1)...(π‘š + 1)))
141140imaeq2d 6057 . . . . . . . . . . . . . . . . . 18 (π‘˜ = (π‘š + 1) β†’ ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) = ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))))
142141xpeq1d 5701 . . . . . . . . . . . . . . . . 17 (π‘˜ = (π‘š + 1) β†’ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}) = (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))
143142uneq2d 4159 . . . . . . . . . . . . . . . 16 (π‘˜ = (π‘š + 1) β†’ ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0})) = ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0})))
144143oveq2d 7430 . . . . . . . . . . . . . . 15 (π‘˜ = (π‘š + 1) β†’ ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))))
145 oveq1 7421 . . . . . . . . . . . . . . . . 17 (π‘˜ = (π‘š + 1) β†’ (π‘˜ + 1) = ((π‘š + 1) + 1))
146145oveq1d 7429 . . . . . . . . . . . . . . . 16 (π‘˜ = (π‘š + 1) β†’ ((π‘˜ + 1)...𝑁) = (((π‘š + 1) + 1)...𝑁))
147146xpeq1d 5701 . . . . . . . . . . . . . . 15 (π‘˜ = (π‘š + 1) β†’ (((π‘˜ + 1)...𝑁) Γ— {0}) = ((((π‘š + 1) + 1)...𝑁) Γ— {0}))
148144, 147uneq12d 4160 . . . . . . . . . . . . . 14 (π‘˜ = (π‘š + 1) β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) = (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})))
149148csbeq1d 3893 . . . . . . . . . . . . 13 (π‘˜ = (π‘š + 1) β†’ ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅)
150149eqeq2d 2738 . . . . . . . . . . . 12 (π‘˜ = (π‘š + 1) β†’ (𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
151139, 150rexeqbidv 3338 . . . . . . . . . . 11 (π‘˜ = (π‘š + 1) β†’ (βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
152139, 151raleqbidv 3337 . . . . . . . . . 10 (π‘˜ = (π‘š + 1) β†’ (βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
153138, 152rabeqbidv 3444 . . . . . . . . 9 (π‘˜ = (π‘š + 1) β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} = {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})
154153fveq2d 6895 . . . . . . . 8 (π‘˜ = (π‘š + 1) β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))
155154breq2d 5154 . . . . . . 7 (π‘˜ = (π‘š + 1) β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
156155notbid 318 . . . . . 6 (π‘˜ = (π‘š + 1) β†’ (Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
157156imbi2d 340 . . . . 5 (π‘˜ = (π‘š + 1) β†’ ((πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) ↔ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
158 oveq2 7422 . . . . . . . . . . . 12 (π‘˜ = 𝑁 β†’ (1...π‘˜) = (1...𝑁))
159158oveq2d 7430 . . . . . . . . . . 11 (π‘˜ = 𝑁 β†’ ((0..^𝐾) ↑m (1...π‘˜)) = ((0..^𝐾) ↑m (1...𝑁)))
160 eqidd 2728 . . . . . . . . . . . . 13 (π‘˜ = 𝑁 β†’ 𝑓 = 𝑓)
161160, 158, 158f1oeq123d 6827 . . . . . . . . . . . 12 (π‘˜ = 𝑁 β†’ (𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜) ↔ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)))
162161abbidv 2796 . . . . . . . . . . 11 (π‘˜ = 𝑁 β†’ {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)} = {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
163159, 162xpeq12d 5703 . . . . . . . . . 10 (π‘˜ = 𝑁 β†’ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) = (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
164 oveq2 7422 . . . . . . . . . . 11 (π‘˜ = 𝑁 β†’ (0...π‘˜) = (0...𝑁))
165 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑁 β†’ ((𝑗 + 1)...π‘˜) = ((𝑗 + 1)...𝑁))
166165imaeq2d 6057 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 𝑁 β†’ ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) = ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)))
167166xpeq1d 5701 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑁 β†’ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}) = (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))
168167uneq2d 4159 . . . . . . . . . . . . . . . 16 (π‘˜ = 𝑁 β†’ ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0})) = ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))
169168oveq2d 7430 . . . . . . . . . . . . . . 15 (π‘˜ = 𝑁 β†’ ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
170 oveq1 7421 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑁 β†’ (π‘˜ + 1) = (𝑁 + 1))
171170oveq1d 7429 . . . . . . . . . . . . . . . 16 (π‘˜ = 𝑁 β†’ ((π‘˜ + 1)...𝑁) = ((𝑁 + 1)...𝑁))
172171xpeq1d 5701 . . . . . . . . . . . . . . 15 (π‘˜ = 𝑁 β†’ (((π‘˜ + 1)...𝑁) Γ— {0}) = (((𝑁 + 1)...𝑁) Γ— {0}))
173169, 172uneq12d 4160 . . . . . . . . . . . . . 14 (π‘˜ = 𝑁 β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) = (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})))
174173csbeq1d 3893 . . . . . . . . . . . . 13 (π‘˜ = 𝑁 β†’ ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅)
175174eqeq2d 2738 . . . . . . . . . . . 12 (π‘˜ = 𝑁 β†’ (𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
176164, 175rexeqbidv 3338 . . . . . . . . . . 11 (π‘˜ = 𝑁 β†’ (βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
177164, 176raleqbidv 3337 . . . . . . . . . 10 (π‘˜ = 𝑁 β†’ (βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
178163, 177rabeqbidv 3444 . . . . . . . . 9 (π‘˜ = 𝑁 β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})
179178fveq2d 6895 . . . . . . . 8 (π‘˜ = 𝑁 β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))
180179breq2d 5154 . . . . . . 7 (π‘˜ = 𝑁 β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
181180notbid 318 . . . . . 6 (π‘˜ = 𝑁 β†’ (Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
182181imbi2d 340 . . . . 5 (π‘˜ = 𝑁 β†’ ((πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘˜)) Γ— {𝑓 ∣ 𝑓:(1...π‘˜)–1-1-ontoβ†’(1...π‘˜)}) ∣ βˆ€π‘– ∈ (0...π‘˜)βˆƒπ‘— ∈ (0...π‘˜)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘˜)) Γ— {0}))) βˆͺ (((π‘˜ + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) ↔ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
183 n2dvds1 16330 . . . . . . 7 Β¬ 2 βˆ₯ 1
184 opex 5460 . . . . . . . . . 10 βŸ¨βˆ…, βˆ…βŸ© ∈ V
185 hashsng 14346 . . . . . . . . . 10 (βŸ¨βˆ…, βˆ…βŸ© ∈ V β†’ (β™―β€˜{βŸ¨βˆ…, βˆ…βŸ©}) = 1)
186184, 185ax-mp 5 . . . . . . . . 9 (β™―β€˜{βŸ¨βˆ…, βˆ…βŸ©}) = 1
187 nnuz 12881 . . . . . . . . . . . . . . . . 17 β„• = (β„€β‰₯β€˜1)
1881, 187eleqtrdi 2838 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜1))
189 eluzfz1 13526 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (β„€β‰₯β€˜1) β†’ 1 ∈ (1...𝑁))
190188, 189syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 1 ∈ (1...𝑁))
191 poimirlem28.5 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐾 ∈ β„•)
192191nnnn0d 12548 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐾 ∈ β„•0)
193 0elfz 13616 . . . . . . . . . . . . . . . 16 (𝐾 ∈ β„•0 β†’ 0 ∈ (0...𝐾))
194 fconst6g 6780 . . . . . . . . . . . . . . . 16 (0 ∈ (0...𝐾) β†’ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾))
195192, 193, 1943syl 18 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾))
19670fvconst2 7210 . . . . . . . . . . . . . . . 16 (1 ∈ (1...𝑁) β†’ (((1...𝑁) Γ— {0})β€˜1) = 0)
197190, 196syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (((1...𝑁) Γ— {0})β€˜1) = 0)
198190, 195, 1973jca 1126 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1 ∈ (1...𝑁) ∧ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾) ∧ (((1...𝑁) Γ— {0})β€˜1) = 0))
199 nfv 1910 . . . . . . . . . . . . . . . 16 Ⅎ𝑝(πœ‘ ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾) ∧ (((1...𝑁) Γ— {0})β€˜1) = 0))
200 nfcsb1v 3914 . . . . . . . . . . . . . . . . 17 Ⅎ𝑝⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅
201200nfeq1 2913 . . . . . . . . . . . . . . . 16 Ⅎ𝑝⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ = 0
202199, 201nfim 1892 . . . . . . . . . . . . . . 15 Ⅎ𝑝((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾) ∧ (((1...𝑁) Γ— {0})β€˜1) = 0)) β†’ ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ = 0)
203 ovex 7447 . . . . . . . . . . . . . . . 16 (1...𝑁) ∈ V
204 snex 5427 . . . . . . . . . . . . . . . 16 {0} ∈ V
205203, 204xpex 7747 . . . . . . . . . . . . . . 15 ((1...𝑁) Γ— {0}) ∈ V
206 feq1 6697 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) Γ— {0}) β†’ (𝑝:(1...𝑁)⟢(0...𝐾) ↔ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾)))
207 fveq1 6890 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ((1...𝑁) Γ— {0}) β†’ (π‘β€˜1) = (((1...𝑁) Γ— {0})β€˜1))
208207eqeq1d 2729 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) Γ— {0}) β†’ ((π‘β€˜1) = 0 ↔ (((1...𝑁) Γ— {0})β€˜1) = 0))
209206, 2083anbi23d 1436 . . . . . . . . . . . . . . . . 17 (𝑝 = ((1...𝑁) Γ— {0}) β†’ ((1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0) ↔ (1 ∈ (1...𝑁) ∧ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾) ∧ (((1...𝑁) Γ— {0})β€˜1) = 0)))
210209anbi2d 628 . . . . . . . . . . . . . . . 16 (𝑝 = ((1...𝑁) Γ— {0}) β†’ ((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0)) ↔ (πœ‘ ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾) ∧ (((1...𝑁) Γ— {0})β€˜1) = 0))))
211 csbeq1a 3903 . . . . . . . . . . . . . . . . 17 (𝑝 = ((1...𝑁) Γ— {0}) β†’ 𝐡 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅)
212211eqeq1d 2729 . . . . . . . . . . . . . . . 16 (𝑝 = ((1...𝑁) Γ— {0}) β†’ (𝐡 = 0 ↔ ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ = 0))
213210, 212imbi12d 344 . . . . . . . . . . . . . . 15 (𝑝 = ((1...𝑁) Γ— {0}) β†’ (((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0)) β†’ 𝐡 = 0) ↔ ((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾) ∧ (((1...𝑁) Γ— {0})β€˜1) = 0)) β†’ ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ = 0)))
214 1ex 11226 . . . . . . . . . . . . . . . . 17 1 ∈ V
215 eleq1 2816 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 β†’ (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
216 fveqeq2 6900 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 β†’ ((π‘β€˜π‘›) = 0 ↔ (π‘β€˜1) = 0))
217215, 2163anbi13d 1435 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 β†’ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0) ↔ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0)))
218217anbi2d 628 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 β†’ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) ↔ (πœ‘ ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0))))
219 breq2 5146 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 β†’ (𝐡 < 𝑛 ↔ 𝐡 < 1))
220218, 219imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 β†’ (((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) β†’ 𝐡 < 𝑛) ↔ ((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0)) β†’ 𝐡 < 1)))
221 poimirlem28.3 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) β†’ 𝐡 < 𝑛)
222214, 220, 221vtocl 3541 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0)) β†’ 𝐡 < 1)
223 poimirlem28.2 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑝:(1...𝑁)⟢(0...𝐾)) β†’ 𝐡 ∈ (0...𝑁))
224 elfznn0 13612 . . . . . . . . . . . . . . . . . 18 (𝐡 ∈ (0...𝑁) β†’ 𝐡 ∈ β„•0)
225 nn0lt10b 12640 . . . . . . . . . . . . . . . . . 18 (𝐡 ∈ β„•0 β†’ (𝐡 < 1 ↔ 𝐡 = 0))
226223, 224, 2253syl 18 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑝:(1...𝑁)⟢(0...𝐾)) β†’ (𝐡 < 1 ↔ 𝐡 = 0))
2272263ad2antr2 1187 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0)) β†’ (𝐡 < 1 ↔ 𝐡 = 0))
228222, 227mpbid 231 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜1) = 0)) β†’ 𝐡 = 0)
229202, 205, 213, 228vtoclf 3547 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) Γ— {0}):(1...𝑁)⟢(0...𝐾) ∧ (((1...𝑁) Γ— {0})β€˜1) = 0)) β†’ ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ = 0)
230198, 229mpdan 686 . . . . . . . . . . . . 13 (πœ‘ β†’ ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅ = 0)
231230eqcomd 2733 . . . . . . . . . . . 12 (πœ‘ β†’ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅)
232231ralrimivw 3145 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘  ∈ {βŸ¨βˆ…, βˆ…βŸ©}0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅)
233 rabid2 3459 . . . . . . . . . . 11 ({βŸ¨βˆ…, βˆ…βŸ©} = {𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅} ↔ βˆ€π‘  ∈ {βŸ¨βˆ…, βˆ…βŸ©}0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅)
234232, 233sylibr 233 . . . . . . . . . 10 (πœ‘ β†’ {βŸ¨βˆ…, βˆ…βŸ©} = {𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅})
235234fveq2d 6895 . . . . . . . . 9 (πœ‘ β†’ (β™―β€˜{βŸ¨βˆ…, βˆ…βŸ©}) = (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅}))
236186, 235eqtr3id 2781 . . . . . . . 8 (πœ‘ β†’ 1 = (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅}))
237236breq2d 5154 . . . . . . 7 (πœ‘ β†’ (2 βˆ₯ 1 ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅})))
238183, 237mtbii 326 . . . . . 6 (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅}))
239238a1i 11 . . . . 5 (𝑁 ∈ β„•0 β†’ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ {βŸ¨βˆ…, βˆ…βŸ©} ∣ 0 = ⦋((1...𝑁) Γ— {0}) / π‘β¦Œπ΅})))
240 2z 12610 . . . . . . . . . . . . 13 2 ∈ β„€
241 fzfi 13955 . . . . . . . . . . . . . . . . 17 (1...(π‘š + 1)) ∈ Fin
242 mapfi 9362 . . . . . . . . . . . . . . . . 17 (((0..^𝐾) ∈ Fin ∧ (1...(π‘š + 1)) ∈ Fin) β†’ ((0..^𝐾) ↑m (1...(π‘š + 1))) ∈ Fin)
24310, 241, 242mp2an 691 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ↑m (1...(π‘š + 1))) ∈ Fin
244 ovex 7447 . . . . . . . . . . . . . . . . . . 19 (1...(π‘š + 1)) ∈ V
245244, 244mapval 8846 . . . . . . . . . . . . . . . . . 18 ((1...(π‘š + 1)) ↑m (1...(π‘š + 1))) = {𝑓 ∣ 𝑓:(1...(π‘š + 1))⟢(1...(π‘š + 1))}
246 mapfi 9362 . . . . . . . . . . . . . . . . . . 19 (((1...(π‘š + 1)) ∈ Fin ∧ (1...(π‘š + 1)) ∈ Fin) β†’ ((1...(π‘š + 1)) ↑m (1...(π‘š + 1))) ∈ Fin)
247241, 241, 246mp2an 691 . . . . . . . . . . . . . . . . . 18 ((1...(π‘š + 1)) ↑m (1...(π‘š + 1))) ∈ Fin
248245, 247eqeltrri 2825 . . . . . . . . . . . . . . . . 17 {𝑓 ∣ 𝑓:(1...(π‘š + 1))⟢(1...(π‘š + 1))} ∈ Fin
249 f1of 6833 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1)) β†’ 𝑓:(1...(π‘š + 1))⟢(1...(π‘š + 1)))
250249ss2abi 4059 . . . . . . . . . . . . . . . . 17 {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))} βŠ† {𝑓 ∣ 𝑓:(1...(π‘š + 1))⟢(1...(π‘š + 1))}
251 ssfi 9187 . . . . . . . . . . . . . . . . 17 (({𝑓 ∣ 𝑓:(1...(π‘š + 1))⟢(1...(π‘š + 1))} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))} βŠ† {𝑓 ∣ 𝑓:(1...(π‘š + 1))⟢(1...(π‘š + 1))}) β†’ {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))} ∈ Fin)
252248, 250, 251mp2an 691 . . . . . . . . . . . . . . . 16 {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))} ∈ Fin
253 xpfi 9331 . . . . . . . . . . . . . . . 16 ((((0..^𝐾) ↑m (1...(π‘š + 1))) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))} ∈ Fin) β†’ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∈ Fin)
254243, 252, 253mp2an 691 . . . . . . . . . . . . . . 15 (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∈ Fin
255 rabfi 9283 . . . . . . . . . . . . . . 15 ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∈ Fin β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} ∈ Fin)
256 hashcl 14333 . . . . . . . . . . . . . . 15 ({𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} ∈ Fin β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„•0)
257254, 255, 256mp2b 10 . . . . . . . . . . . . . 14 (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„•0
258257nn0zi 12603 . . . . . . . . . . . . 13 (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„€
259 rabfi 9283 . . . . . . . . . . . . . . 15 ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∈ Fin β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))} ∈ Fin)
260 hashcl 14333 . . . . . . . . . . . . . . 15 ({𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))} ∈ Fin β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„•0)
261254, 259, 260mp2b 10 . . . . . . . . . . . . . 14 (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„•0
262261nn0zi 12603 . . . . . . . . . . . . 13 (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„€
263240, 258, 2623pm3.2i 1337 . . . . . . . . . . . 12 (2 ∈ β„€ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„€ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„€)
264 nn0p1nn 12527 . . . . . . . . . . . . . . . 16 (π‘š ∈ β„•0 β†’ (π‘š + 1) ∈ β„•)
265264ad2antrl 727 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (π‘š + 1) ∈ β„•)
266 uneq1 4152 . . . . . . . . . . . . . . . 16 (π‘ž = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) = (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})))
267266csbeq1d 3893 . . . . . . . . . . . . . . 15 (π‘ž = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅)
26870fconst 6777 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘š + 1) + 1)...𝑁) Γ— {0}):(((π‘š + 1) + 1)...𝑁)⟢{0}
269268jctr 524 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘ž:(1...(π‘š + 1))⟢(0...𝐾) β†’ (π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ ((((π‘š + 1) + 1)...𝑁) Γ— {0}):(((π‘š + 1) + 1)...𝑁)⟢{0}))
270264nnred 12243 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘š ∈ β„•0 β†’ (π‘š + 1) ∈ ℝ)
271270ltp1d 12160 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘š ∈ β„•0 β†’ (π‘š + 1) < ((π‘š + 1) + 1))
272 fzdisj 13546 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘š + 1) < ((π‘š + 1) + 1) β†’ ((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ…)
273271, 272syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š ∈ β„•0 β†’ ((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ…)
274 fun 6753 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ ((((π‘š + 1) + 1)...𝑁) Γ— {0}):(((π‘š + 1) + 1)...𝑁)⟢{0}) ∧ ((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ…) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁))⟢((0...𝐾) βˆͺ {0}))
275269, 273, 274syl2anr 596 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘š ∈ β„•0 ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁))⟢((0...𝐾) βˆͺ {0}))
276275adantlr 714 . . . . . . . . . . . . . . . . . . . . 21 (((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁))⟢((0...𝐾) βˆͺ {0}))
277276adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁))⟢((0...𝐾) βˆͺ {0}))
278264peano2nnd 12245 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘š ∈ β„•0 β†’ ((π‘š + 1) + 1) ∈ β„•)
279278, 187eleqtrdi 2838 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘š ∈ β„•0 β†’ ((π‘š + 1) + 1) ∈ (β„€β‰₯β€˜1))
280279ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ ((π‘š + 1) + 1) ∈ (β„€β‰₯β€˜1))
281 nn0z 12599 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘š ∈ β„•0 β†’ π‘š ∈ β„€)
2821nnzd 12601 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (πœ‘ β†’ 𝑁 ∈ β„€)
283 zltp1le 12628 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘š ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (π‘š < 𝑁 ↔ (π‘š + 1) ≀ 𝑁))
284281, 282, 283syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘š < 𝑁 ↔ (π‘š + 1) ≀ 𝑁))
285284biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘š < 𝑁) β†’ (π‘š + 1) ≀ 𝑁)
286285anasss 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (π‘š + 1) ≀ 𝑁)
287281peano2zd 12685 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘š ∈ β„•0 β†’ (π‘š + 1) ∈ β„€)
288287adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) β†’ (π‘š + 1) ∈ β„€)
289 eluz 12852 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘š + 1) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑁 ∈ (β„€β‰₯β€˜(π‘š + 1)) ↔ (π‘š + 1) ≀ 𝑁))
290288, 282, 289syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (𝑁 ∈ (β„€β‰₯β€˜(π‘š + 1)) ↔ (π‘š + 1) ≀ 𝑁))
291286, 290mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ 𝑁 ∈ (β„€β‰₯β€˜(π‘š + 1)))
292 fzsplit2 13544 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘š + 1) + 1) ∈ (β„€β‰₯β€˜1) ∧ 𝑁 ∈ (β„€β‰₯β€˜(π‘š + 1))) β†’ (1...𝑁) = ((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁)))
293280, 291, 292syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (1...𝑁) = ((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁)))
294293eqcomd 2733 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ ((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁)) = (1...𝑁))
295192, 193syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ 0 ∈ (0...𝐾))
296295snssd 4808 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ {0} βŠ† (0...𝐾))
297 ssequn2 4179 . . . . . . . . . . . . . . . . . . . . . . . 24 ({0} βŠ† (0...𝐾) ↔ ((0...𝐾) βˆͺ {0}) = (0...𝐾))
298296, 297sylib 217 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ ((0...𝐾) βˆͺ {0}) = (0...𝐾))
299298adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ ((0...𝐾) βˆͺ {0}) = (0...𝐾))
300294, 299feq23d 6711 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁))⟢((0...𝐾) βˆͺ {0}) ↔ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾)))
301300adantrr 716 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):((1...(π‘š + 1)) βˆͺ (((π‘š + 1) + 1)...𝑁))⟢((0...𝐾) βˆͺ {0}) ↔ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾)))
302277, 301mpbid 231 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
303 nfv 1910 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑝(πœ‘ ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
304 nfcsb1v 3914 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑝⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅
305304nfel1 2914 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑝⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁)
306303, 305nfim 1892 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑝((πœ‘ ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁))
307 vex 3473 . . . . . . . . . . . . . . . . . . . . 21 π‘ž ∈ V
308 ovex 7447 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘š + 1) + 1)...𝑁) ∈ V
309308, 204xpex 7747 . . . . . . . . . . . . . . . . . . . . 21 ((((π‘š + 1) + 1)...𝑁) Γ— {0}) ∈ V
310307, 309unex 7740 . . . . . . . . . . . . . . . . . . . 20 (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) ∈ V
311 feq1 6697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (𝑝:(1...𝑁)⟢(0...𝐾) ↔ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾)))
312311anbi2d 628 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((πœ‘ ∧ 𝑝:(1...𝑁)⟢(0...𝐾)) ↔ (πœ‘ ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))))
313 csbeq1a 3903 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ 𝐡 = ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅)
314313eleq1d 2813 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (𝐡 ∈ (0...𝑁) ↔ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁)))
315312, 314imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (((πœ‘ ∧ 𝑝:(1...𝑁)⟢(0...𝐾)) β†’ 𝐡 ∈ (0...𝑁)) ↔ ((πœ‘ ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁))))
316306, 310, 315, 223vtoclf 3547 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁))
317302, 316syldan 590 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁))
318317anassrs 467 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁))
319 elfznn0 13612 . . . . . . . . . . . . . . . . 17 (⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ β„•0)
320318, 319syl 17 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ β„•0)
321264nnnn0d 12548 . . . . . . . . . . . . . . . . . 18 (π‘š ∈ β„•0 β†’ (π‘š + 1) ∈ β„•0)
322321adantr 480 . . . . . . . . . . . . . . . . 17 ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) β†’ (π‘š + 1) ∈ β„•0)
323322ad2antlr 726 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ (π‘š + 1) ∈ β„•0)
324 leloe 11316 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘š + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ ((π‘š + 1) ≀ 𝑁 ↔ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)))
325270, 3, 324syl2anr 596 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ ((π‘š + 1) ≀ 𝑁 ↔ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)))
326284, 325bitrd 279 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘š < 𝑁 ↔ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)))
327326biimpd 228 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘š < 𝑁 β†’ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)))
328327imdistani 568 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘š < 𝑁) β†’ ((πœ‘ ∧ π‘š ∈ β„•0) ∧ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)))
329328anasss 466 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ ((πœ‘ ∧ π‘š ∈ β„•0) ∧ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)))
330 simplll 774 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ πœ‘)
331278nnge1d 12276 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘š ∈ β„•0 β†’ 1 ≀ ((π‘š + 1) + 1))
332331ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ 1 ≀ ((π‘š + 1) + 1))
333 zltp1le 12628 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((π‘š + 1) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((π‘š + 1) < 𝑁 ↔ ((π‘š + 1) + 1) ≀ 𝑁))
334287, 282, 333syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ ((π‘š + 1) < 𝑁 ↔ ((π‘š + 1) + 1) ≀ 𝑁))
335334biimpa 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ ((π‘š + 1) + 1) ≀ 𝑁)
336287peano2zd 12685 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘š ∈ β„•0 β†’ ((π‘š + 1) + 1) ∈ β„€)
337 1z 12608 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 ∈ β„€
338 elfz 13508 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘š + 1) + 1) ∈ β„€ ∧ 1 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (((π‘š + 1) + 1) ∈ (1...𝑁) ↔ (1 ≀ ((π‘š + 1) + 1) ∧ ((π‘š + 1) + 1) ≀ 𝑁)))
339337, 338mp3an2 1446 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((π‘š + 1) + 1) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (((π‘š + 1) + 1) ∈ (1...𝑁) ↔ (1 ≀ ((π‘š + 1) + 1) ∧ ((π‘š + 1) + 1) ≀ 𝑁)))
340336, 282, 339syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (((π‘š + 1) + 1) ∈ (1...𝑁) ↔ (1 ≀ ((π‘š + 1) + 1) ∧ ((π‘š + 1) + 1) ≀ 𝑁)))
341340adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ (((π‘š + 1) + 1) ∈ (1...𝑁) ↔ (1 ≀ ((π‘š + 1) + 1) ∧ ((π‘š + 1) + 1) ≀ 𝑁)))
342332, 335, 341mpbir2and 712 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ ((π‘š + 1) + 1) ∈ (1...𝑁))
343342adantlr 714 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ((π‘š + 1) + 1) ∈ (1...𝑁))
344 nn0re 12497 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘š ∈ β„•0 β†’ π‘š ∈ ℝ)
345344ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ π‘š ∈ ℝ)
346270ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ (π‘š + 1) ∈ ℝ)
3473ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ 𝑁 ∈ ℝ)
348344ltp1d 12160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘š ∈ β„•0 β†’ π‘š < (π‘š + 1))
349348ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ π‘š < (π‘š + 1))
350 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ (π‘š + 1) < 𝑁)
351345, 346, 347, 349, 350lttrd 11391 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ π‘š < 𝑁)
352351adantlr 714 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ π‘š < 𝑁)
353 anass 468 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘š < 𝑁) ↔ (πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)))
354302anassrs 467 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
355353, 354sylanb 580 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘š < 𝑁) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
356355an32s 651 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ π‘š < 𝑁) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
357352, 356syldan 590 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
358 ffn 6716 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘ž:(1...(π‘š + 1))⟢(0...𝐾) β†’ π‘ž Fn (1...(π‘š + 1)))
359358ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ π‘ž Fn (1...(π‘š + 1)))
360273ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ…)
361 eluz 12852 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((π‘š + 1) + 1) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑁 ∈ (β„€β‰₯β€˜((π‘š + 1) + 1)) ↔ ((π‘š + 1) + 1) ≀ 𝑁))
362336, 282, 361syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (𝑁 ∈ (β„€β‰₯β€˜((π‘š + 1) + 1)) ↔ ((π‘š + 1) + 1) ≀ 𝑁))
363362adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ (𝑁 ∈ (β„€β‰₯β€˜((π‘š + 1) + 1)) ↔ ((π‘š + 1) + 1) ≀ 𝑁))
364335, 363mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ 𝑁 ∈ (β„€β‰₯β€˜((π‘š + 1) + 1)))
365 eluzfz1 13526 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (β„€β‰₯β€˜((π‘š + 1) + 1)) β†’ ((π‘š + 1) + 1) ∈ (((π‘š + 1) + 1)...𝑁))
366364, 365syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (π‘š + 1) < 𝑁) β†’ ((π‘š + 1) + 1) ∈ (((π‘š + 1) + 1)...𝑁))
367366adantlr 714 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ((π‘š + 1) + 1) ∈ (((π‘š + 1) + 1)...𝑁))
368 fnconstg 6779 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ V β†’ ((((π‘š + 1) + 1)...𝑁) Γ— {0}) Fn (((π‘š + 1) + 1)...𝑁))
36970, 368ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘š + 1) + 1)...𝑁) Γ— {0}) Fn (((π‘š + 1) + 1)...𝑁)
370 fvun2 6984 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘ž Fn (1...(π‘š + 1)) ∧ ((((π‘š + 1) + 1)...𝑁) Γ— {0}) Fn (((π‘š + 1) + 1)...𝑁) ∧ (((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ… ∧ ((π‘š + 1) + 1) ∈ (((π‘š + 1) + 1)...𝑁))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = (((((π‘š + 1) + 1)...𝑁) Γ— {0})β€˜((π‘š + 1) + 1)))
371369, 370mp3an2 1446 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘ž Fn (1...(π‘š + 1)) ∧ (((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ… ∧ ((π‘š + 1) + 1) ∈ (((π‘š + 1) + 1)...𝑁))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = (((((π‘š + 1) + 1)...𝑁) Γ— {0})β€˜((π‘š + 1) + 1)))
372359, 360, 367, 371syl12anc 836 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = (((((π‘š + 1) + 1)...𝑁) Γ— {0})β€˜((π‘š + 1) + 1)))
37370fvconst2 7210 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘š + 1) + 1) ∈ (((π‘š + 1) + 1)...𝑁) β†’ (((((π‘š + 1) + 1)...𝑁) Γ— {0})β€˜((π‘š + 1) + 1)) = 0)
374367, 373syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ (((((π‘š + 1) + 1)...𝑁) Γ— {0})β€˜((π‘š + 1) + 1)) = 0)
375372, 374eqtrd 2767 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0)
376 nfv 1910 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑝(πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0))
377 nfcv 2898 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑝 <
378 nfcv 2898 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑝((π‘š + 1) + 1)
379304, 377, 378nfbr 5189 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑝⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1)
380376, 379nfim 1892 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑝((πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1))
381 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (π‘β€˜((π‘š + 1) + 1)) = ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)))
382381eqeq1d 2729 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((π‘β€˜((π‘š + 1) + 1)) = 0 ↔ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0))
383311, 3823anbi23d 1436 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((((π‘š + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜((π‘š + 1) + 1)) = 0) ↔ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0)))
384383anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜((π‘š + 1) + 1)) = 0)) ↔ (πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0))))
385313breq1d 5152 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (𝐡 < ((π‘š + 1) + 1) ↔ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1)))
386384, 385imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (((πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜((π‘š + 1) + 1)) = 0)) β†’ 𝐡 < ((π‘š + 1) + 1)) ↔ ((πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1))))
387 ovex 7447 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘š + 1) + 1) ∈ V
388 eleq1 2816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((π‘š + 1) + 1) β†’ (𝑛 ∈ (1...𝑁) ↔ ((π‘š + 1) + 1) ∈ (1...𝑁)))
389 fveqeq2 6900 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((π‘š + 1) + 1) β†’ ((π‘β€˜π‘›) = 0 ↔ (π‘β€˜((π‘š + 1) + 1)) = 0))
390388, 3893anbi13d 1435 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = ((π‘š + 1) + 1) β†’ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0) ↔ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜((π‘š + 1) + 1)) = 0)))
391390anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = ((π‘š + 1) + 1) β†’ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) ↔ (πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜((π‘š + 1) + 1)) = 0))))
392 breq2 5146 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = ((π‘š + 1) + 1) β†’ (𝐡 < 𝑛 ↔ 𝐡 < ((π‘š + 1) + 1)))
393391, 392imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = ((π‘š + 1) + 1) β†’ (((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) β†’ 𝐡 < 𝑛) ↔ ((πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜((π‘š + 1) + 1)) = 0)) β†’ 𝐡 < ((π‘š + 1) + 1))))
394387, 393, 221vtocl 3541 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜((π‘š + 1) + 1)) = 0)) β†’ 𝐡 < ((π‘š + 1) + 1))
395380, 310, 386, 394vtoclf 3547 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ (((π‘š + 1) + 1) ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜((π‘š + 1) + 1)) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1))
396330, 343, 357, 375, 395syl13anc 1370 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1))
397353, 318sylanb 580 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘š < 𝑁) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁))
398397an32s 651 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ π‘š < 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁))
399398elfzelzd 13520 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ π‘š < 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ β„€)
400352, 399syldan 590 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ β„€)
401287ad3antlr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ (π‘š + 1) ∈ β„€)
402 zleltp1 12629 . . . . . . . . . . . . . . . . . . . . 21 ((⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ β„€ ∧ (π‘š + 1) ∈ β„€) β†’ (⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1) ↔ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1)))
403400, 401, 402syl2anc 583 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ (⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1) ↔ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < ((π‘š + 1) + 1)))
404396, 403mpbird 257 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) < 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1))
405348ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ π‘š < (π‘š + 1))
406 breq2 5146 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘š + 1) = 𝑁 β†’ (π‘š < (π‘š + 1) ↔ π‘š < 𝑁))
407406biimpac 478 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘š < (π‘š + 1) ∧ (π‘š + 1) = 𝑁) β†’ π‘š < 𝑁)
408405, 407sylan 579 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) = 𝑁) β†’ π‘š < 𝑁)
409 elfzle2 13523 . . . . . . . . . . . . . . . . . . . . . 22 (⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ 𝑁)
410398, 409syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ π‘š < 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ 𝑁)
411408, 410syldan 590 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) = 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ 𝑁)
412 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) = 𝑁) β†’ (π‘š + 1) = 𝑁)
413411, 412breqtrrd 5170 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ (π‘š + 1) = 𝑁) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1))
414404, 413jaodan 956 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) ∧ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1))
415414an32s 651 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘š ∈ β„•0) ∧ ((π‘š + 1) < 𝑁 ∨ (π‘š + 1) = 𝑁)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1))
416329, 415sylan 579 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1))
417 elfz2nn0 13610 . . . . . . . . . . . . . . . 16 (⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...(π‘š + 1)) ↔ (⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ β„•0 ∧ (π‘š + 1) ∈ β„•0 ∧ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ≀ (π‘š + 1)))
418320, 323, 416, 417syl3anbrc 1341 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∈ (0...(π‘š + 1)))
419 fzss2 13559 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (β„€β‰₯β€˜(π‘š + 1)) β†’ (1...(π‘š + 1)) βŠ† (1...𝑁))
420291, 419syl 17 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (1...(π‘š + 1)) βŠ† (1...𝑁))
421420sselda 3978 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ 𝑛 ∈ (1...(π‘š + 1))) β†’ 𝑛 ∈ (1...𝑁))
4224213ad2antr1 1186 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 0)) β†’ 𝑛 ∈ (1...𝑁))
4233543ad2antr2 1187 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 0)) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
424358ad2antll 728 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ π‘ž Fn (1...(π‘š + 1)))
425273ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ ((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ…)
426 simprl 770 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ 𝑛 ∈ (1...(π‘š + 1)))
427 fvun1 6983 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘ž Fn (1...(π‘š + 1)) ∧ ((((π‘š + 1) + 1)...𝑁) Γ— {0}) Fn (((π‘š + 1) + 1)...𝑁) ∧ (((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ… ∧ 𝑛 ∈ (1...(π‘š + 1)))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = (π‘žβ€˜π‘›))
428369, 427mp3an2 1446 . . . . . . . . . . . . . . . . . . . . 21 ((π‘ž Fn (1...(π‘š + 1)) ∧ (((1...(π‘š + 1)) ∩ (((π‘š + 1) + 1)...𝑁)) = βˆ… ∧ 𝑛 ∈ (1...(π‘š + 1)))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = (π‘žβ€˜π‘›))
429424, 425, 426, 428syl12anc 836 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = (π‘žβ€˜π‘›))
430429adantlrr 720 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = (π‘žβ€˜π‘›))
4314303adantr3 1169 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 0)) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = (π‘žβ€˜π‘›))
432 simpr3 1194 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 0)) β†’ (π‘žβ€˜π‘›) = 0)
433431, 432eqtrd 2767 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 0)) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0)
434422, 423, 4333jca 1126 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 0)) β†’ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0))
435 nfv 1910 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑝(πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0))
436 nfcv 2898 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑝𝑛
437304, 377, 436nfbr 5189 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑝⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < 𝑛
438435, 437nfim 1892 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑝((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < 𝑛)
439 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (π‘β€˜π‘›) = ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›))
440439eqeq1d 2729 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((π‘β€˜π‘›) = 0 ↔ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0))
441311, 4403anbi23d 1436 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0) ↔ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0)))
442441anbi2d 628 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) ↔ (πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0))))
443313breq1d 5152 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (𝐡 < 𝑛 ↔ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < 𝑛))
444442, 443imbi12d 344 . . . . . . . . . . . . . . . . . 18 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 0)) β†’ 𝐡 < 𝑛) ↔ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < 𝑛)))
445438, 310, 444, 221vtoclf 3547 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < 𝑛)
446445adantlr 714 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < 𝑛)
447434, 446syldan 590 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 0)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ < 𝑛)
448 simp1 1134 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾) β†’ 𝑛 ∈ (1...(π‘š + 1)))
449421anasss 466 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ 𝑛 ∈ (1...(π‘š + 1)))) β†’ 𝑛 ∈ (1...𝑁))
450448, 449sylanr2 682 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾))) β†’ 𝑛 ∈ (1...𝑁))
451 simp2 1135 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾) β†’ π‘ž:(1...(π‘š + 1))⟢(0...𝐾))
452451, 302sylanr2 682 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾))) β†’ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾))
4534293adantr3 1169 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾)) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = (π‘žβ€˜π‘›))
454 simpr3 1194 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾)) β†’ (π‘žβ€˜π‘›) = 𝐾)
455453, 454eqtrd 2767 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘š ∈ β„•0) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾)) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾)
456455anasss 466 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾)
457456adantrlr 722 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾))) β†’ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾)
458450, 452, 4573jca 1126 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾))) β†’ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾))
459 nfv 1910 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑝(πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾))
460 nfcv 2898 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑝(𝑛 βˆ’ 1)
461304, 460nfne 3038 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑝⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ β‰  (𝑛 βˆ’ 1)
462459, 461nfim 1892 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑝((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ β‰  (𝑛 βˆ’ 1))
463439eqeq1d 2729 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((π‘β€˜π‘›) = 𝐾 ↔ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾))
464311, 4633anbi23d 1436 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 𝐾) ↔ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾)))
465464anbi2d 628 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 𝐾)) ↔ (πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾))))
466313neeq1d 2995 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (𝐡 β‰  (𝑛 βˆ’ 1) ↔ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ β‰  (𝑛 βˆ’ 1)))
467465, 466imbi12d 344 . . . . . . . . . . . . . . . . . 18 (𝑝 = (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) β†’ (((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 𝐾)) β†’ 𝐡 β‰  (𝑛 βˆ’ 1)) ↔ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ β‰  (𝑛 βˆ’ 1))))
468 poimirlem28.4 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...𝐾) ∧ (π‘β€˜π‘›) = 𝐾)) β†’ 𝐡 β‰  (𝑛 βˆ’ 1))
469462, 310, 467, 468vtoclf 3547 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})):(1...𝑁)⟢(0...𝐾) ∧ ((π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0}))β€˜π‘›) = 𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ β‰  (𝑛 βˆ’ 1))
470458, 469syldan 590 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾))) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ β‰  (𝑛 βˆ’ 1))
471470anassrs 467 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) ∧ (𝑛 ∈ (1...(π‘š + 1)) ∧ π‘ž:(1...(π‘š + 1))⟢(0...𝐾) ∧ (π‘žβ€˜π‘›) = 𝐾)) β†’ ⦋(π‘ž βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ β‰  (𝑛 βˆ’ 1))
472265, 267, 418, 447, 471poimirlem27 37042 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ 2 βˆ₯ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})))
473265, 267, 418poimirlem26 37041 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ 2 βˆ₯ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
474 fzfi 13955 . . . . . . . . . . . . . . . . . . 19 (0...(π‘š + 1)) ∈ Fin
475 xpfi 9331 . . . . . . . . . . . . . . . . . . 19 (((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∈ Fin ∧ (0...(π‘š + 1)) ∈ Fin) β†’ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∈ Fin)
476254, 474, 475mp2an 691 . . . . . . . . . . . . . . . . . 18 ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∈ Fin
477 rabfi 9283 . . . . . . . . . . . . . . . . . 18 (((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∈ Fin β†’ {𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} ∈ Fin)
478 hashcl 14333 . . . . . . . . . . . . . . . . . 18 ({𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} ∈ Fin β†’ (β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„•0)
479476, 477, 478mp2b 10 . . . . . . . . . . . . . . . . 17 (β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„•0
480479nn0zi 12603 . . . . . . . . . . . . . . . 16 (β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„€
481 zsubcl 12620 . . . . . . . . . . . . . . . 16 (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„€ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„€) β†’ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) ∈ β„€)
482480, 262, 481mp2an 691 . . . . . . . . . . . . . . 15 ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) ∈ β„€
483 zsubcl 12620 . . . . . . . . . . . . . . . 16 (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„€ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„€) β†’ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) ∈ β„€)
484480, 258, 483mp2an 691 . . . . . . . . . . . . . . 15 ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) ∈ β„€
485 dvds2sub 16253 . . . . . . . . . . . . . . 15 ((2 ∈ β„€ ∧ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) ∈ β„€ ∧ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) ∈ β„€) β†’ ((2 βˆ₯ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) ∧ 2 βˆ₯ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))) β†’ 2 βˆ₯ (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) βˆ’ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))))
486240, 482, 484, 485mp3an 1458 . . . . . . . . . . . . . 14 ((2 βˆ₯ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) ∧ 2 βˆ₯ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))) β†’ 2 βˆ₯ (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) βˆ’ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
487472, 473, 486syl2anc 583 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ 2 βˆ₯ (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) βˆ’ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
488479nn0cni 12500 . . . . . . . . . . . . . 14 (β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„‚
489261nn0cni 12500 . . . . . . . . . . . . . 14 (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„‚
490257nn0cni 12500 . . . . . . . . . . . . . 14 (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„‚
491 nnncan1 11512 . . . . . . . . . . . . . 14 (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„‚ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„‚ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„‚) β†’ (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) βˆ’ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))) = ((β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})))
492488, 489, 490, 491mp3an 1458 . . . . . . . . . . . . 13 (((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})) βˆ’ ((β™―β€˜{𝑑 ∈ ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) Γ— (0...(π‘š + 1))) ∣ βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ ((0...(π‘š + 1)) βˆ– {(2nd β€˜π‘‘)})𝑖 = ⦋(1st β€˜π‘‘) / π‘ β¦Œβ¦‹(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))) = ((β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}))
493487, 492breqtrdi 5183 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ 2 βˆ₯ ((β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})))
494 dvdssub2 16263 . . . . . . . . . . . 12 (((2 ∈ β„€ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ∈ β„€ ∧ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ∈ β„€) ∧ 2 βˆ₯ ((β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) βˆ’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}))) β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})))
495263, 493, 494sylancr 586 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})))
496 nn0cn 12498 . . . . . . . . . . . . . . . . . . . 20 (π‘š ∈ β„•0 β†’ π‘š ∈ β„‚)
497 pncan1 11654 . . . . . . . . . . . . . . . . . . . 20 (π‘š ∈ β„‚ β†’ ((π‘š + 1) βˆ’ 1) = π‘š)
498496, 497syl 17 . . . . . . . . . . . . . . . . . . 19 (π‘š ∈ β„•0 β†’ ((π‘š + 1) βˆ’ 1) = π‘š)
499498oveq2d 7430 . . . . . . . . . . . . . . . . . 18 (π‘š ∈ β„•0 β†’ (0...((π‘š + 1) βˆ’ 1)) = (0...π‘š))
500499rexeqdv 3321 . . . . . . . . . . . . . . . . . 18 (π‘š ∈ β„•0 β†’ (βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
501499, 500raleqbidv 3337 . . . . . . . . . . . . . . . . 17 (π‘š ∈ β„•0 β†’ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅))
5025013anbi1d 1437 . . . . . . . . . . . . . . . 16 (π‘š ∈ β„•0 β†’ ((βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1)) ↔ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))))
503502rabbidv 3435 . . . . . . . . . . . . . . 15 (π‘š ∈ β„•0 β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))} = {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})
504503fveq2d 6895 . . . . . . . . . . . . . 14 (π‘š ∈ β„•0 β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}))
505504ad2antrl 727 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}))
5061adantr 480 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ 𝑁 ∈ β„•)
507191adantr 480 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ 𝐾 ∈ β„•)
508 simprl 770 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ π‘š ∈ β„•0)
509 simprr 772 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ π‘š < 𝑁)
510506, 507, 508, 509poimirlem4 37019 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} β‰ˆ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})
511 fzfi 13955 . . . . . . . . . . . . . . . . . 18 (1...π‘š) ∈ Fin
512 mapfi 9362 . . . . . . . . . . . . . . . . . 18 (((0..^𝐾) ∈ Fin ∧ (1...π‘š) ∈ Fin) β†’ ((0..^𝐾) ↑m (1...π‘š)) ∈ Fin)
51310, 511, 512mp2an 691 . . . . . . . . . . . . . . . . 17 ((0..^𝐾) ↑m (1...π‘š)) ∈ Fin
514 ovex 7447 . . . . . . . . . . . . . . . . . . . 20 (1...π‘š) ∈ V
515514, 514mapval 8846 . . . . . . . . . . . . . . . . . . 19 ((1...π‘š) ↑m (1...π‘š)) = {𝑓 ∣ 𝑓:(1...π‘š)⟢(1...π‘š)}
516 mapfi 9362 . . . . . . . . . . . . . . . . . . . 20 (((1...π‘š) ∈ Fin ∧ (1...π‘š) ∈ Fin) β†’ ((1...π‘š) ↑m (1...π‘š)) ∈ Fin)
517511, 511, 516mp2an 691 . . . . . . . . . . . . . . . . . . 19 ((1...π‘š) ↑m (1...π‘š)) ∈ Fin
518515, 517eqeltrri 2825 . . . . . . . . . . . . . . . . . 18 {𝑓 ∣ 𝑓:(1...π‘š)⟢(1...π‘š)} ∈ Fin
519 f1of 6833 . . . . . . . . . . . . . . . . . . 19 (𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š) β†’ 𝑓:(1...π‘š)⟢(1...π‘š))
520519ss2abi 4059 . . . . . . . . . . . . . . . . . 18 {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)} βŠ† {𝑓 ∣ 𝑓:(1...π‘š)⟢(1...π‘š)}
521 ssfi 9187 . . . . . . . . . . . . . . . . . 18 (({𝑓 ∣ 𝑓:(1...π‘š)⟢(1...π‘š)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)} βŠ† {𝑓 ∣ 𝑓:(1...π‘š)⟢(1...π‘š)}) β†’ {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)} ∈ Fin)
522518, 520, 521mp2an 691 . . . . . . . . . . . . . . . . 17 {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)} ∈ Fin
523 xpfi 9331 . . . . . . . . . . . . . . . . 17 ((((0..^𝐾) ↑m (1...π‘š)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)} ∈ Fin) β†’ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∈ Fin)
524513, 522, 523mp2an 691 . . . . . . . . . . . . . . . 16 (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∈ Fin
525 rabfi 9283 . . . . . . . . . . . . . . . 16 ((((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∈ Fin β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} ∈ Fin)
526524, 525ax-mp 5 . . . . . . . . . . . . . . 15 {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} ∈ Fin
527 rabfi 9283 . . . . . . . . . . . . . . . 16 ((((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∈ Fin β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))} ∈ Fin)
528254, 527ax-mp 5 . . . . . . . . . . . . . . 15 {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))} ∈ Fin
529 hashen 14324 . . . . . . . . . . . . . . 15 (({𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))} ∈ Fin) β†’ ((β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} β‰ˆ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}))
530526, 528, 529mp2an 691 . . . . . . . . . . . . . 14 ((β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} β‰ˆ {𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))})
531510, 530sylibr 233 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}))
532505, 531eqtr4d 2770 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))
533532breq2d 5154 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ (βˆ€π‘– ∈ (0...((π‘š + 1) βˆ’ 1))βˆƒπ‘— ∈ (0...((π‘š + 1) βˆ’ 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ∧ ((1st β€˜π‘ )β€˜(π‘š + 1)) = 0 ∧ ((2nd β€˜π‘ )β€˜(π‘š + 1)) = (π‘š + 1))}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
534495, 533bitrd 279 . . . . . . . . . 10 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
535534biimpd 228 . . . . . . . . 9 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) β†’ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
536535con3d 152 . . . . . . . 8 ((πœ‘ ∧ (π‘š ∈ β„•0 ∧ π‘š < 𝑁)) β†’ (Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
537536expcom 413 . . . . . . 7 ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) β†’ (πœ‘ β†’ (Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
538537a2d 29 . . . . . 6 ((π‘š ∈ β„•0 ∧ π‘š < 𝑁) β†’ ((πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) β†’ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
5395383adant1 1128 . . . . 5 ((𝑁 ∈ β„•0 ∧ π‘š ∈ β„•0 ∧ π‘š < 𝑁) β†’ ((πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...π‘š)) Γ— {𝑓 ∣ 𝑓:(1...π‘š)–1-1-ontoβ†’(1...π‘š)}) ∣ βˆ€π‘– ∈ (0...π‘š)βˆƒπ‘— ∈ (0...π‘š)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...π‘š)) Γ— {0}))) βˆͺ (((π‘š + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})) β†’ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...(π‘š + 1))) Γ— {𝑓 ∣ 𝑓:(1...(π‘š + 1))–1-1-ontoβ†’(1...(π‘š + 1))}) ∣ βˆ€π‘– ∈ (0...(π‘š + 1))βˆƒπ‘— ∈ (0...(π‘š + 1))𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...(π‘š + 1))) Γ— {0}))) βˆͺ ((((π‘š + 1) + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))))
540107, 132, 157, 182, 239, 539fnn0ind 12677 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝑁 ∈ β„•0 ∧ 𝑁 ≀ 𝑁) β†’ (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
5415, 540mpcom 38 . . 3 (πœ‘ β†’ Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}))
542 dvds0 16234 . . . . . . . 8 (2 ∈ β„€ β†’ 2 βˆ₯ 0)
543240, 542ax-mp 5 . . . . . . 7 2 βˆ₯ 0
544 hash0 14344 . . . . . . 7 (β™―β€˜βˆ…) = 0
545543, 544breqtrri 5169 . . . . . 6 2 βˆ₯ (β™―β€˜βˆ…)
546 fveq2 6891 . . . . . 6 ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢} = βˆ… β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢}) = (β™―β€˜βˆ…))
547545, 546breqtrrid 5180 . . . . 5 ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢} = βˆ… β†’ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢}))
5483ltp1d 12160 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝑁 < (𝑁 + 1))
549282peano2zd 12685 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (𝑁 + 1) ∈ β„€)
550 fzn 13535 . . . . . . . . . . . . . . . . . . 19 (((𝑁 + 1) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = βˆ…))
551549, 282, 550syl2anc 583 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = βˆ…))
552548, 551mpbid 231 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ((𝑁 + 1)...𝑁) = βˆ…)
553552xpeq1d 5701 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (((𝑁 + 1)...𝑁) Γ— {0}) = (βˆ… Γ— {0}))
554553, 86eqtrdi 2783 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (((𝑁 + 1)...𝑁) Γ— {0}) = βˆ…)
555554uneq2d 4159 . . . . . . . . . . . . . 14 (πœ‘ β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) = (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ βˆ…))
556 un0 4386 . . . . . . . . . . . . . 14 (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ βˆ…) = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))
557555, 556eqtrdi 2783 . . . . . . . . . . . . 13 (πœ‘ β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
558557csbeq1d 3893 . . . . . . . . . . . 12 (πœ‘ β†’ ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ = ⦋((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) / π‘β¦Œπ΅)
559 ovex 7447 . . . . . . . . . . . . 13 ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∈ V
560 poimirlem28.1 . . . . . . . . . . . . 13 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ 𝐡 = 𝐢)
561559, 560csbie 3925 . . . . . . . . . . . 12 ⦋((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) / π‘β¦Œπ΅ = 𝐢
562558, 561eqtrdi 2783 . . . . . . . . . . 11 (πœ‘ β†’ ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ = 𝐢)
563562eqeq2d 2738 . . . . . . . . . 10 (πœ‘ β†’ (𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ 𝑖 = 𝐢))
564563rexbidv 3173 . . . . . . . . 9 (πœ‘ β†’ (βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢))
565564ralbidv 3172 . . . . . . . 8 (πœ‘ β†’ (βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅ ↔ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢))
566565rabbidv 3435 . . . . . . 7 (πœ‘ β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅} = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢})
567566fveq2d 6895 . . . . . 6 (πœ‘ β†’ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) = (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢}))
568567breq2d 5154 . . . . 5 (πœ‘ β†’ (2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) ↔ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢})))
569547, 568imbitrrid 245 . . . 4 (πœ‘ β†’ ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢} = βˆ… β†’ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅})))
570569necon3bd 2949 . . 3 (πœ‘ β†’ (Β¬ 2 βˆ₯ (β™―β€˜{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = ⦋(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) βˆͺ (((𝑁 + 1)...𝑁) Γ— {0})) / π‘β¦Œπ΅}) β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢} β‰  βˆ…))
571541, 570mpd 15 . 2 (πœ‘ β†’ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢} β‰  βˆ…)
572 rabn0 4381 . 2 ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢} β‰  βˆ… ↔ βˆƒπ‘  ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢)
573571, 572sylib 217 1 (πœ‘ β†’ βˆƒπ‘  ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = 𝐢)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  {cab 2704   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  {crab 3427  Vcvv 3469  β¦‹csb 3889   βˆ– cdif 3941   βˆͺ cun 3942   ∩ cin 3943   βŠ† wss 3944  βˆ…c0 4318  {csn 4624  βŸ¨cop 4630   class class class wbr 5142   ↦ cmpt 5225   Γ— cxp 5670   β€œ cima 5675   Fn wfn 6537  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   ∘f cof 7675  1st c1st 7983  2nd c2nd 7984  1oc1o 8471   ↑m cmap 8834   β‰ˆ cen 8950  Fincfn 8953  β„‚cc 11122  β„cr 11123  0cc0 11124  1c1 11125   + caddc 11127   < clt 11264   ≀ cle 11265   βˆ’ cmin 11460  β„•cn 12228  2c2 12283  β„•0cn0 12488  β„€cz 12574  β„€β‰₯cuz 12838  ...cfz 13502  ..^cfzo 13645  β™―chash 14307   βˆ₯ cdvds 16216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-inf2 9650  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7677  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-er 8716  df-map 8836  df-pm 8837  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-oi 9519  df-dju 9910  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-rp 12993  df-fz 13503  df-fzo 13646  df-seq 13985  df-exp 14045  df-fac 14251  df-bc 14280  df-hash 14308  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201  df-clim 15450  df-sum 15651  df-dvds 16217
This theorem is referenced by:  poimirlem32  37047
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