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Theorem poimirlem20 35077
Description: Lemma for poimir 35090 establishing existence for poimirlem21 35078. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
poimirlem22.2 (𝜑𝑇𝑆)
poimirlem22.3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
poimirlem21.4 (𝜑 → (2nd𝑇) = 𝑁)
Assertion
Ref Expression
poimirlem20 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧   𝜑,𝑗,𝑛,𝑦   𝑗,𝐹,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝜑,𝑝,𝑡   𝑓,𝐾,𝑗,𝑛,𝑝,𝑡   𝑓,𝑁,𝑝,𝑡   𝑇,𝑓,𝑝   𝜑,𝑧   𝑓,𝐹,𝑝,𝑡,𝑧   𝑧,𝐾   𝑧,𝑁   𝑡,𝑇,𝑧   𝑆,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem20
StepHypRef Expression
1 oveq2 7143 . . . . . . . . 9 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) − 1) = (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
21eleq1d 2874 . . . . . . . 8 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((1st ‘(1st𝑇))‘𝑛) − 1) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾)))
3 oveq2 7143 . . . . . . . . 9 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) − 0) = (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
43eleq1d 2874 . . . . . . . 8 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((1st ‘(1st𝑇))‘𝑛) − 0) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾)))
5 fveq2 6645 . . . . . . . . . . . 12 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
65oveq1d 7150 . . . . . . . . . . 11 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((1st ‘(1st𝑇))‘𝑛) − 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1))
76adantl 485 . . . . . . . . . 10 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1))
8 poimirlem22.2 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑇𝑆)
9 elrabi 3623 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
10 poimirlem22.s . . . . . . . . . . . . . . . . . . . . 21 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
119, 10eleq2s 2908 . . . . . . . . . . . . . . . . . . . 20 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
128, 11syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
13 xp1st 7703 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
1412, 13syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
15 xp1st 7703 . . . . . . . . . . . . . . . . . 18 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
1614, 15syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
17 elmapi 8411 . . . . . . . . . . . . . . . . 17 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
1816, 17syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
19 xp2nd 7704 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
2014, 19syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
21 fvex 6658 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘(1st𝑇)) ∈ V
22 f1oeq1 6579 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
2321, 22elab 3615 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
2420, 23sylib 221 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
25 f1of 6590 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
2624, 25syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
27 poimir.0 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℕ)
28 elfz1end 12932 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
2927, 28sylib 221 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (1...𝑁))
3026, 29ffvelrnd 6829 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁))
3118, 30ffvelrnd 6829 . . . . . . . . . . . . . . 15 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾))
32 elfzonn0 13077 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0)
3331, 32syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0)
34 fvex 6658 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇))‘𝑁) ∈ V
35 eleq1 2877 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (𝑛 ∈ (1...𝑁) ↔ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)))
3635anbi2d 631 . . . . . . . . . . . . . . . . . 18 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((𝜑𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁))))
37 fveq2 6645 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (𝑝𝑛) = (𝑝‘((2nd ‘(1st𝑇))‘𝑁)))
3837neeq1d 3046 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((𝑝𝑛) ≠ 0 ↔ (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
3938rexbidv 3256 . . . . . . . . . . . . . . . . . 18 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
4036, 39imbi12d 348 . . . . . . . . . . . . . . . . 17 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0) ↔ ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)))
41 poimirlem22.3 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
4234, 40, 41vtocl 3507 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)
4330, 42mpdan 686 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)
44 fveq1 6644 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)))
4518ffnd 6488 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
4645adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
47 1ex 10626 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ V
48 fnconstg 6541 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)))
4947, 48ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦))
50 c0ex 10624 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ V
51 fnconstg 6541 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))
5349, 52pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
54 dff1o3 6596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
5554simprbi 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
5624, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → Fun (2nd ‘(1st𝑇)))
57 imain 6409 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
5856, 57syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
59 elfznn0 12995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
6059nn0red 11944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
6160ltp1d 11559 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
62 fzdisj 12929 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
6361, 62syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
6463imaeq2d 5896 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
65 ima0 5912 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2nd ‘(1st𝑇)) “ ∅) = ∅
6664, 65eqtrdi 2849 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
6758, 66sylan9req 2854 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅)
68 fnun 6434 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
6953, 67, 68sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
70 imaundi 5975 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
71 nn0p1nn 11924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
72 nnuz 12269 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ℕ = (ℤ‘1)
7371, 72eleqtrdi 2900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ (ℤ‘1))
7459, 73syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
7574adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ‘1))
7627nncnd 11641 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝑁 ∈ ℂ)
77 npcan1 11054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
7876, 77syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
7978adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
80 elfzuz3 12899 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
81 peano2uz 12289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
8280, 81syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
8382adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
8479, 83eqeltrrd 2891 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ𝑦))
85 fzsplit2 12927 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑦 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
8675, 84, 85syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
8786imaeq2d 5896 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
88 f1ofo 6597 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
89 foima 6570 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9024, 88, 893syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9190adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9287, 91eqtr3d 2835 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
9370, 92syl5eqr 2847 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
9493fneq2d 6417 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
9569, 94mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
96 ovex 7168 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1...𝑁) ∈ V
9796a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
98 inidm 4145 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
99 eqidd 2799 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
100 f1ofn 6591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
10124, 100syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
102101adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
103 fzss1 12941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
10474, 103syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
105104adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
106 eluzp1p1 12258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
107 uzss 12253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)) → (ℤ‘((𝑁 − 1) + 1)) ⊆ (ℤ‘(𝑦 + 1)))
10880, 106, 1073syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → (ℤ‘((𝑁 − 1) + 1)) ⊆ (ℤ‘(𝑦 + 1)))
109108adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (ℤ‘((𝑁 − 1) + 1)) ⊆ (ℤ‘(𝑦 + 1)))
11027nnzd 12074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝑁 ∈ ℤ)
111 uzid 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
112110, 111syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑𝑁 ∈ (ℤ𝑁))
11378fveq2d 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → (ℤ‘((𝑁 − 1) + 1)) = (ℤ𝑁))
114112, 113eleqtrrd 2893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑁 ∈ (ℤ‘((𝑁 − 1) + 1)))
115114adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘((𝑁 − 1) + 1)))
116109, 115sseldd 3916 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
117 eluzfz2 12910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
118116, 117syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
119 fnfvima 6973 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
120102, 105, 118, 119syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
121 fvun2 6730 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
12249, 52, 121mp3an12 1448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
12367, 120, 122syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
12450fvconst2 6943 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
125120, 124syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
126123, 125eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
127126adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
12846, 95, 97, 97, 98, 99, 127ofval 7398 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0))
12930, 128mpidan 688 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0))
13033nn0cnd 11945 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℂ)
131130addid1d 10829 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
132131adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
133129, 132eqtrd 2833 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
13444, 133sylan9eqr 2855 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
135134adantllr 718 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑝 ∈ ran 𝐹) ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
136 fveq2 6645 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
137136breq2d 5042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
138137ifbid 4447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
139138csbeq1d 3832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
140 2fveq3 6650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
141 2fveq3 6650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
142141imaeq1d 5895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
143142xpeq1d 5548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
144141imaeq1d 5895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
145144xpeq1d 5548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
146143, 145uneq12d 4091 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
147140, 146oveq12d 7153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑡 = 𝑇 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
148147csbeq2dv 3835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑡 = 𝑇if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
149139, 148eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
150149mpteq2dv 5126 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
151150eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
152151, 10elrab2 3631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑇𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
153152simprbi 500 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑇𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
1548, 153syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
15560adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
156 peano2zm 12013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
157110, 156syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → (𝑁 − 1) ∈ ℤ)
158157zred 12075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝑁 − 1) ∈ ℝ)
159158adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
16027nnred 11640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑁 ∈ ℝ)
161160adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
162 elfzle2 12906 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
163162adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
164160ltm1d 11561 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝑁 − 1) < 𝑁)
165164adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
166155, 159, 161, 163, 165lelttrd 10787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
167 poimirlem21.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (2nd𝑇) = 𝑁)
168167adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 𝑁)
169166, 168breqtrrd 5058 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd𝑇))
170169iftrued 4433 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
171170csbeq1d 3832 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑦 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
172 vex 3444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
173 oveq2 7143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
174173imaeq2d 5896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
175174xpeq1d 5548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}))
176 oveq1 7142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
177176oveq1d 7150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
178177imaeq2d 5896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
179178xpeq1d 5548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
180175, 179uneq12d 4091 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
181180oveq2d 7151 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑦 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
182172, 181csbie 3863 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
183171, 182eqtrdi 2849 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
184183mpteq2dva 5125 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
185154, 184eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
186185rneqd 5772 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ran 𝐹 = ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
187186eleq2d 2875 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑝 ∈ ran 𝐹𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))))
188 eqid 2798 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
189 ovex 7168 . . . . . . . . . . . . . . . . . . . . . 22 ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) ∈ V
190188, 189elrnmpti 5796 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
191187, 190syl6bb 290 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
192191biimpa 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ ran 𝐹) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
193135, 192r19.29a 3248 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ ran 𝐹) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
194193neeq1d 3046 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ ran 𝐹) → ((𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0 ↔ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
195194biimpd 232 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ ran 𝐹) → ((𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
196195rexlimdva 3243 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
19743, 196mpd 15 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)
198 elnnne0 11899 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0 ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
19933, 197, 198sylanbrc 586 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ)
200 nnm1nn0 11926 . . . . . . . . . . . . 13 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℕ0)
201199, 200syl 17 . . . . . . . . . . . 12 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℕ0)
202 elfzo0 13073 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾))
20331, 202sylib 221 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾))
204203simp2d 1140 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℕ)
205201nn0red 11944 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℝ)
20633nn0red 11944 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℝ)
207204nnred 11640 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ ℝ)
208206ltm1d 11561 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) < ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
209 elfzolt2 13042 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾)
21031, 209syl 17 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾)
211205, 206, 207, 208, 210lttrd 10790 . . . . . . . . . . . 12 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) < 𝐾)
212 elfzo0 13073 . . . . . . . . . . . 12 ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ (0..^𝐾) ↔ ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℕ0𝐾 ∈ ℕ ∧ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) < 𝐾))
213201, 204, 211, 212syl3anbrc 1340 . . . . . . . . . . 11 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ (0..^𝐾))
214213adantr 484 . . . . . . . . . 10 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ (0..^𝐾))
2157, 214eqeltrd 2890 . . . . . . . . 9 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 1) ∈ (0..^𝐾))
216215adantlr 714 . . . . . . . 8 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 1) ∈ (0..^𝐾))
21718ffvelrnda 6828 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
218 elfzonn0 13077 . . . . . . . . . . . . 13 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
219217, 218syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
220219nn0cnd 11945 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
221220subid1d 10975 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 0) = ((1st ‘(1st𝑇))‘𝑛))
222221, 217eqeltrd 2890 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 0) ∈ (0..^𝐾))
223222adantr 484 . . . . . . . 8 (((𝜑𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 0) ∈ (0..^𝐾))
2242, 4, 216, 223ifbothda 4462 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾))
225224fmpttd 6856 . . . . . 6 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))):(1...𝑁)⟶(0..^𝐾))
226 ovex 7168 . . . . . . 7 (0..^𝐾) ∈ V
227226, 96elmap 8418 . . . . . 6 ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑m (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))):(1...𝑁)⟶(0..^𝐾))
228225, 227sylibr 237 . . . . 5 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑m (1...𝑁)))
229 simpr 488 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁))
230 1z 12000 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
231 peano2z 12011 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
232230, 231ax-mp 5 . . . . . . . . . . . . . . 15 (1 + 1) ∈ ℤ
233110, 232jctil 523 . . . . . . . . . . . . . 14 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
234 elfzelz 12902 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ)
235234, 230jctir 524 . . . . . . . . . . . . . 14 (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
236 fzsubel 12938 . . . . . . . . . . . . . 14 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
237233, 235, 236syl2an 598 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
238229, 237mpbid 235 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
239 ax-1cn 10584 . . . . . . . . . . . . . 14 1 ∈ ℂ
240239, 239pncan3oi 10891 . . . . . . . . . . . . 13 ((1 + 1) − 1) = 1
241240oveq1i 7145 . . . . . . . . . . . 12 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
242238, 241eleqtrdi 2900 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1)))
243242ralrimiva 3149 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)))
244 simpr 488 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1)))
245157, 230jctil 523 . . . . . . . . . . . . . . 15 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
246 elfzelz 12902 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ)
247246, 230jctir 524 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
248 fzaddel 12936 . . . . . . . . . . . . . . 15 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
249245, 247, 248syl2an 598 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
250244, 249mpbid 235 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
25178oveq2d 7151 . . . . . . . . . . . . . 14 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
252251adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
253250, 252eleqtrd 2892 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁))
254234zcnd 12076 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ)
255246zcnd 12076 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ)
256 subadd2 10879 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
257239, 256mp3an2 1446 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
258 eqcom 2805 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦)
259 eqcom 2805 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛)
260257, 258, 2593bitr4g 317 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
261254, 255, 260syl2anr 599 . . . . . . . . . . . . . 14 ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
262261ralrimiva 3149 . . . . . . . . . . . . 13 (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
263262adantl 485 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
264 reu6i 3667 . . . . . . . . . . . 12 (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
265253, 263, 264syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
266265ralrimiva 3149 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
267 eqid 2798 . . . . . . . . . . 11 (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
268267f1ompt 6852 . . . . . . . . . 10 ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ↔ (∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)))
269243, 266, 268sylanbrc 586 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)))
270 f1osng 6630 . . . . . . . . . 10 ((1 ∈ V ∧ 𝑁 ∈ ℕ) → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
27147, 27, 270sylancr 590 . . . . . . . . 9 (𝜑 → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
272158, 160ltnled 10776 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
273164, 272mpbid 235 . . . . . . . . . . 11 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
274 elfzle2 12906 . . . . . . . . . . 11 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
275273, 274nsyl 142 . . . . . . . . . 10 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
276 disjsn 4607 . . . . . . . . . 10 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
277275, 276sylibr 237 . . . . . . . . 9 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
278 1re 10630 . . . . . . . . . . . . . 14 1 ∈ ℝ
279278ltp1i 11533 . . . . . . . . . . . . 13 1 < (1 + 1)
280232zrei 11975 . . . . . . . . . . . . . 14 (1 + 1) ∈ ℝ
281278, 280ltnlei 10750 . . . . . . . . . . . . 13 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
282279, 281mpbi 233 . . . . . . . . . . . 12 ¬ (1 + 1) ≤ 1
283 elfzle1 12905 . . . . . . . . . . . 12 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
284282, 283mto 200 . . . . . . . . . . 11 ¬ 1 ∈ ((1 + 1)...𝑁)
285 disjsn 4607 . . . . . . . . . . 11 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
286284, 285mpbir 234 . . . . . . . . . 10 (((1 + 1)...𝑁) ∩ {1}) = ∅
287 f1oun 6609 . . . . . . . . . 10 ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
288286, 287mpanr1 702 . . . . . . . . 9 ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
289269, 271, 277, 288syl21anc 836 . . . . . . . 8 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
290 eleq1 2877 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁)))
291284, 290mtbiri 330 . . . . . . . . . . . . . 14 (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁))
292291necon2ai 3016 . . . . . . . . . . . . 13 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1)
293 ifnefalse 4437 . . . . . . . . . . . . 13 (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
294292, 293syl 17 . . . . . . . . . . . 12 (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
295294mpteq2ia 5121 . . . . . . . . . . 11 (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
296295uneq1i 4086 . . . . . . . . . 10 ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩})
29747a1i 11 . . . . . . . . . . 11 (𝜑 → 1 ∈ V)
298 ssv 3939 . . . . . . . . . . . 12 ℕ ⊆ V
299298, 27sseldi 3913 . . . . . . . . . . 11 (𝜑𝑁 ∈ V)
30027, 72eleqtrdi 2900 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ (ℤ‘1))
301 fzpred 12950 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
302300, 301syl 17 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
303 uncom 4080 . . . . . . . . . . . 12 ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
304302, 303eqtr2di 2850 . . . . . . . . . . 11 (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁))
305 iftrue 4431 . . . . . . . . . . . 12 (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
306305adantl 485 . . . . . . . . . . 11 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
307297, 299, 304, 306fmptapd 6910 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
308296, 307syl5eqr 2847 . . . . . . . . 9 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
30978, 300eqeltrd 2890 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
310 uzid 12246 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
311 peano2uz 12289 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
312157, 310, 3113syl 18 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
31378, 312eqeltrrd 2891 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
314 fzsplit2 12927 . . . . . . . . . . 11 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
315309, 313, 314syl2anc 587 . . . . . . . . . 10 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
31678oveq1d 7150 . . . . . . . . . . . 12 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
317 fzsn 12944 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
318110, 317syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑁...𝑁) = {𝑁})
319316, 318eqtrd 2833 . . . . . . . . . . 11 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
320319uneq2d 4090 . . . . . . . . . 10 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
321315, 320eqtr2d 2834 . . . . . . . . 9 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
322308, 304, 321f1oeq123d 6585 . . . . . . . 8 (𝜑 → (((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)))
323289, 322mpbid 235 . . . . . . 7 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))
324 f1oco 6612 . . . . . . 7 (((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
32524, 323, 324syl2anc 587 . . . . . 6 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
32696mptex 6963 . . . . . . . 8 (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∈ V
32721, 326coex 7617 . . . . . . 7 ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ V
328 f1oeq1 6579 . . . . . . 7 (𝑓 = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁)))
329327, 328elab 3615 . . . . . 6 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
330325, 329sylibr 237 . . . . 5 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
331 opelxpi 5556 . . . . 5 (((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑m (1...𝑁)) ∧ ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
332228, 330, 331syl2anc 587 . . . 4 (𝜑 → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
33327nnnn0d 11943 . . . . 5 (𝜑𝑁 ∈ ℕ0)
334 0elfz 12999 . . . . 5 (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
335333, 334syl 17 . . . 4 (𝜑 → 0 ∈ (0...𝑁))
336 opelxpi 5556 . . . 4 ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 0 ∈ (0...𝑁)) → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
337332, 335, 336syl2anc 587 . . 3 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
338 poimirlem22.1 . . . . 5 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
33927, 10, 338, 8, 41, 167poimirlem19 35076 . . . 4 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
340 elfzle1 12905 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
341 0re 10632 . . . . . . . . . 10 0 ∈ ℝ
342 lenlt 10708 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
343341, 60, 342sylancr 590 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 − 1)) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
344340, 343mpbid 235 . . . . . . . 8 (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 𝑦 < 0)
345344iffalsed 4436 . . . . . . 7 (𝑦 ∈ (0...(𝑁 − 1)) → if(𝑦 < 0, 𝑦, (𝑦 + 1)) = (𝑦 + 1))
346345csbeq1d 3832 . . . . . 6 (𝑦 ∈ (0...(𝑁 − 1)) → if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑦 + 1) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
347 ovex 7168 . . . . . . 7 (𝑦 + 1) ∈ V
348 oveq2 7143 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
349348imaeq2d 5896 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
350349xpeq1d 5548 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}))
351 oveq1 7142 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
352351oveq1d 7150 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
353352imaeq2d 5896 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
354353xpeq1d 5548 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
355350, 354uneq12d 4091 . . . . . . . 8 (𝑗 = (𝑦 + 1) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
356355oveq2d 7151 . . . . . . 7 (𝑗 = (𝑦 + 1) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
357347, 356csbie 3863 . . . . . 6 (𝑦 + 1) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
358346, 357eqtrdi 2849 . . . . 5 (𝑦 ∈ (0...(𝑁 − 1)) → if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
359358mpteq2ia 5121 . . . 4 (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
360339, 359eqtr4di 2851 . . 3 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))))
361 opex 5321 . . . . . . . . . . 11 ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ V
362361, 50op2ndd 7682 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2nd𝑡) = 0)
363362breq2d 5042 . . . . . . . . 9 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝑦 < (2nd𝑡) ↔ 𝑦 < 0))
364363ifbid 4447 . . . . . . . 8 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < 0, 𝑦, (𝑦 + 1)))
365364csbeq1d 3832 . . . . . . 7 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
366361, 50op1std 7681 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (1st𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩)
36796mptex 6963 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ V
368367, 327op1std 7681 . . . . . . . . . 10 ((1st𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ → (1st ‘(1st𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
369366, 368syl 17 . . . . . . . . 9 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (1st ‘(1st𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
370367, 327op2ndd 7682 . . . . . . . . . . . . 13 ((1st𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ → (2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
371366, 370syl 17 . . . . . . . . . . . 12 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
372371imaeq1d 5895 . . . . . . . . . . 11 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)))
373372xpeq1d 5548 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}))
374371imaeq1d 5895 . . . . . . . . . . 11 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)))
375374xpeq1d 5548 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))
376373, 375uneq12d 4091 . . . . . . . . 9 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))
377369, 376oveq12d 7153 . . . . . . . 8 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
378377csbeq2dv 3835 . . . . . . 7 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
379365, 378eqtrd 2833 . . . . . 6 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
380379mpteq2dv 5126 . . . . 5 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))))
381380eqeq2d 2809 . . . 4 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
382381, 10elrab2 3631 . . 3 (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆 ↔ (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
383337, 360, 382sylanbrc 586 . 2 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆)
384361, 50op2ndd 7682 . . . . . 6 (𝑇 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2nd𝑇) = 0)
385384eqcoms 2806 . . . . 5 (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ = 𝑇 → (2nd𝑇) = 0)
38627nnne0d 11675 . . . . . . 7 (𝜑𝑁 ≠ 0)
387386necomd 3042 . . . . . 6 (𝜑 → 0 ≠ 𝑁)
388 neeq1 3049 . . . . . 6 ((2nd𝑇) = 0 → ((2nd𝑇) ≠ 𝑁 ↔ 0 ≠ 𝑁))
389387, 388syl5ibrcom 250 . . . . 5 (𝜑 → ((2nd𝑇) = 0 → (2nd𝑇) ≠ 𝑁))
390385, 389syl5 34 . . . 4 (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ = 𝑇 → (2nd𝑇) ≠ 𝑁))
391390necon2d 3010 . . 3 (𝜑 → ((2nd𝑇) = 𝑁 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇))
392167, 391mpd 15 . 2 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇)
393 neeq1 3049 . . 3 (𝑧 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝑧𝑇 ↔ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇))
394393rspcev 3571 . 2 ((⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆 ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇) → ∃𝑧𝑆 𝑧𝑇)
395383, 392, 394syl2anc 587 1 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  {cab 2776  wne 2987  wral 3106  wrex 3107  ∃!wreu 3108  {crab 3110  Vcvv 3441  csb 3828  cun 3879  cin 3880  wss 3881  c0 4243  ifcif 4425  {csn 4525  cop 4531   class class class wbr 5030  cmpt 5110   × cxp 5517  ccnv 5518  ran crn 5520  cima 5522  ccom 5523  Fun wfun 6318   Fn wfn 6319  wf 6320  ontowfo 6322  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  f cof 7387  1st c1st 7669  2nd c2nd 7670  m cmap 8389  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  cle 10665  cmin 10859  cn 11625  0cn0 11885  cz 11969  cuz 12231  ...cfz 12885  ..^cfzo 13028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029
This theorem is referenced by:  poimirlem21  35078
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