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Theorem poimirlem7 36085
Description: Lemma for poimir 36111, similar to poimirlem6 36084, but for vertices after the opposite vertex. (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}))))}
poimirlem9.1 (𝜑𝑇𝑆)
poimirlem9.2 (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))
poimirlem7.3 (𝜑𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
Assertion
Ref Expression
poimirlem7 (𝜑 → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑀))
Distinct variable groups:   𝑓,𝑗,𝑛,𝑡,𝑦   𝜑,𝑗,𝑛,𝑦   𝑗,𝐹,𝑛,𝑦   𝑗,𝑀,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑛,𝑡   𝑓,𝑀,𝑡   𝑓,𝑁,𝑡   𝑇,𝑓   𝑓,𝐹,𝑡   𝑡,𝑇   𝑆,𝑗,𝑛,𝑡,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem7
StepHypRef Expression
1 poimirlem9.1 . . . . . . . 8 (𝜑𝑇𝑆)
2 elrabi 3639 . . . . . . . . 9 (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)))
3 poimirlem22.s . . . . . . . . 9 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}))))}
42, 3eleq2s 2856 . . . . . . . 8 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
51, 4syl 17 . . . . . . 7 (𝜑𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
6 xp1st 7953 . . . . . . 7 (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)}))
75, 6syl 17 . . . . . 6 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
8 xp2nd 7954 . . . . . 6 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
97, 8syl 17 . . . . 5 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
10 fvex 6855 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
11 f1oeq1 6772 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3630 . . . . 5 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
139, 12sylib 217 . . . 4 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
14 f1of 6784 . . . 4 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
1513, 14syl 17 . . 3 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
16 poimirlem9.2 . . . . . . . . 9 (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))
17 elfznn 13470 . . . . . . . . 9 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
1816, 17syl 17 . . . . . . . 8 (𝜑 → (2nd𝑇) ∈ ℕ)
1918peano2nnd 12170 . . . . . . 7 (𝜑 → ((2nd𝑇) + 1) ∈ ℕ)
2019peano2nnd 12170 . . . . . 6 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℕ)
21 nnuz 12806 . . . . . 6 ℕ = (ℤ‘1)
2220, 21eleqtrdi 2848 . . . . 5 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ (ℤ‘1))
23 fzss1 13480 . . . . 5 ((((2nd𝑇) + 1) + 1) ∈ (ℤ‘1) → ((((2nd𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁))
2422, 23syl 17 . . . 4 (𝜑 → ((((2nd𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁))
25 poimirlem7.3 . . . 4 (𝜑𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
2624, 25sseldd 3945 . . 3 (𝜑𝑀 ∈ (1...𝑁))
2715, 26ffvelcdmd 7036 . 2 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁))
28 xp1st 7953 . . . . . . . . . . . . 13 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
297, 28syl 17 . . . . . . . . . . . 12 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
30 elmapfn 8803 . . . . . . . . . . . 12 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
3129, 30syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
3231adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
33 1ex 11151 . . . . . . . . . . . . . . 15 1 ∈ V
34 fnconstg 6730 . . . . . . . . . . . . . . 15 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))))
3533, 34ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1)))
36 c0ex 11149 . . . . . . . . . . . . . . 15 0 ∈ V
37 fnconstg 6730 . . . . . . . . . . . . . . 15 (0 ∈ V → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
3836, 37ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))
3935, 38pm3.2i 471 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
40 dff1o3 6790 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
4140simprbi 497 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
4213, 41syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun (2nd ‘(1st𝑇)))
43 imain 6586 . . . . . . . . . . . . . . 15 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
4442, 43syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
4525elfzelzd 13442 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℤ)
4645zred 12607 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℝ)
4746ltm1d 12087 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 1) < 𝑀)
48 fzdisj 13468 . . . . . . . . . . . . . . . . 17 ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
4947, 48syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
5049imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
51 ima0 6029 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
5250, 51eqtrdi 2792 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
5344, 52eqtr3d 2778 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = ∅)
54 fnun 6614 . . . . . . . . . . . . 13 ((((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
5539, 53, 54sylancr 587 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
5645zcnd 12608 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℂ)
57 npcan1 11580 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
5856, 57syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
59 1red 11156 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 1 ∈ ℝ)
6020nnred 12168 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℝ)
6119nnred 12168 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
6219nnge1d 12201 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ≤ ((2nd𝑇) + 1))
6361ltp1d 12085 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1))
6459, 61, 60, 62, 63lelttrd 11313 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 1 < (((2nd𝑇) + 1) + 1))
65 elfzle1 13444 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ 𝑀)
6625, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((2nd𝑇) + 1) + 1) ≤ 𝑀)
6759, 60, 46, 64, 66ltletrd 11315 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 1 < 𝑀)
6859, 46, 67ltled 11303 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ≤ 𝑀)
69 elnnz1 12529 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤ 𝑀))
7045, 68, 69sylanbrc 583 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℕ)
7170, 21eleqtrdi 2848 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘1))
7258, 71eqeltrd 2838 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘1))
73 peano2zm 12546 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
7445, 73syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 − 1) ∈ ℤ)
75 uzid 12778 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
76 peano2uz 12826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
7774, 75, 763syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
7858, 77eqeltrrd 2839 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ‘(𝑀 − 1)))
79 uzss 12786 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘(𝑀 − 1)) → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
8078, 79syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
81 elfzuz3 13438 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → 𝑁 ∈ (ℤ𝑀))
8225, 81syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ𝑀))
8380, 82sseldd 3945 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘(𝑀 − 1)))
84 fzsplit2 13466 . . . . . . . . . . . . . . . . . 18 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
8572, 83, 84syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
8658oveq1d 7372 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
8786uneq2d 4123 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
8885, 87eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
8988imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
90 imaundi 6102 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
9189, 90eqtrdi 2792 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
92 f1ofo 6791 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
9313, 92syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
94 foima 6761 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9593, 94syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9691, 95eqtr3d 2778 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = (1...𝑁))
9796fneq2d 6596 . . . . . . . . . . . 12 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)))
9855, 97mpbid 231 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
9998adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
100 ovexd 7392 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1...𝑁) ∈ V)
101 inidm 4178 . . . . . . . . . 10 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
102 eqidd 2737 . . . . . . . . . 10 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
103 imaundi 6102 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
104 fzpred 13489 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
10582, 104syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
106105imaeq2d 6013 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))))
107 f1ofn 6785 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
10813, 107syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
109 fnsnfv 6920 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
110108, 26, 109syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
111110uneq1d 4122 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
112103, 106, 1113eqtr4a 2802 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
113112xpeq1d 5662 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}))
114 xpundir 5701 . . . . . . . . . . . . . . . 16 (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
115113, 114eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
116115uneq2d 4123 . . . . . . . . . . . . . 14 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
117 un12 4127 . . . . . . . . . . . . . 14 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
118116, 117eqtrdi 2792 . . . . . . . . . . . . 13 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
119118fveq1d 6844 . . . . . . . . . . . 12 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
120119ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
121 fnconstg 6730 . . . . . . . . . . . . . . . . 17 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
12236, 121ax-mp 5 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))
12335, 122pm3.2i 471 . . . . . . . . . . . . . . 15 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
124 imain 6586 . . . . . . . . . . . . . . . . 17 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
12542, 124syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
12674zred 12607 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 − 1) ∈ ℝ)
127 peano2re 11328 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
12846, 127syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 + 1) ∈ ℝ)
12946ltp1d 12085 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 < (𝑀 + 1))
130126, 46, 128, 47, 129lttrd 11316 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 − 1) < (𝑀 + 1))
131 fzdisj 13468 . . . . . . . . . . . . . . . . . . 19 ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
132130, 131syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
133132imaeq2d 6013 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
134133, 51eqtrdi 2792 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅)
135125, 134eqtr3d 2778 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
136 fnun 6614 . . . . . . . . . . . . . . 15 ((((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
137123, 135, 136sylancr 587 . . . . . . . . . . . . . 14 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
138 imaundi 6102 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
139 imadif 6585 . . . . . . . . . . . . . . . . . 18 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
14042, 139syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
141 fzsplit 13467 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
14226, 141syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
143142difeq1d 4081 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}))
144 difundir 4240 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀}))
145 fzsplit2 13466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
14672, 78, 145syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
14758oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀))
148 fzsn 13483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
14945, 148syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑀...𝑀) = {𝑀})
150147, 149eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀})
151150uneq2d 4123 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀}))
152146, 151eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀}))
153152difeq1d 4081 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}))
154 difun2 4440 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀})
155126, 46ltnled 11302 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1)))
15647, 155mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1))
157 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1))
158156, 157nsyl 140 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1)))
159 difsn 4758 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
160158, 159syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
161154, 160eqtrid 2788 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1)))
162153, 161eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1)))
16346, 128ltnled 11302 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
164129, 163mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
165 elfzle1 13444 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀)
166164, 165nsyl 140 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁))
167 difsn 4758 . . . . . . . . . . . . . . . . . . . . . 22 𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
168166, 167syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
169162, 168uneq12d 4124 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
170144, 169eqtrid 2788 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
171143, 170eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
172171imaeq2d 6013 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))))
173110eqcomd 2742 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑀}) = {((2nd ‘(1st𝑇))‘𝑀)})
17495, 173difeq12d 4083 . . . . . . . . . . . . . . . . 17 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
175140, 172, 1743eqtr3d 2784 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
176138, 175eqtr3id 2790 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
177176fneq2d 6596 . . . . . . . . . . . . . 14 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})))
178137, 177mpbid 231 . . . . . . . . . . . . 13 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
179 eldifsn 4747 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)))
180179biimpri 227 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
181180ancoms 459 . . . . . . . . . . . . 13 ((𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
182 disjdif 4431 . . . . . . . . . . . . . 14 ({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅
183 fnconstg 6730 . . . . . . . . . . . . . . . 16 (0 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
18436, 183ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
185 fvun2 6933 . . . . . . . . . . . . . . 15 ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)} ∧ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ∧ (({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
186184, 185mp3an1 1448 . . . . . . . . . . . . . 14 ((((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ∧ (({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
187182, 186mpanr1 701 . . . . . . . . . . . . 13 ((((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
188178, 181, 187syl2an 596 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
189188anassrs 468 . . . . . . . . . . 11 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
190120, 189eqtrd 2776 . . . . . . . . . 10 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
19132, 99, 100, 100, 101, 102, 190ofval 7628 . . . . . . . . 9 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)))
192 fnconstg 6730 . . . . . . . . . . . . . . 15 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)))
19333, 192ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀))
194193, 122pm3.2i 471 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
195 imain 6586 . . . . . . . . . . . . . . 15 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
19642, 195syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
197 fzdisj 13468 . . . . . . . . . . . . . . . . 17 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
198129, 197syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
199198imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
200199, 51eqtrdi 2792 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
201196, 200eqtr3d 2778 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
202 fnun 6614 . . . . . . . . . . . . 13 ((((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
203194, 201, 202sylancr 587 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
204142imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
205 imaundi 6102 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
206204, 205eqtrdi 2792 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
207206, 95eqtr3d 2778 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
208207fneq2d 6596 . . . . . . . . . . . 12 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
209203, 208mpbid 231 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
210209adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
211 imaundi 6102 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀}))
212152imaeq2d 6013 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})))
213110uneq2d 4123 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀})))
214211, 212, 2133eqtr4a 2802 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}))
215214xpeq1d 5662 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}))
216 xpundir 5701 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1}))
217215, 216eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1})))
218217uneq1d 4122 . . . . . . . . . . . . . 14 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
219 un23 4128 . . . . . . . . . . . . . . 15 (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1}))
220219equncomi 4115 . . . . . . . . . . . . . 14 (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
221218, 220eqtrdi 2792 . . . . . . . . . . . . 13 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
222221fveq1d 6844 . . . . . . . . . . . 12 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
223222ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
224 fnconstg 6730 . . . . . . . . . . . . . . . 16 (1 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
22533, 224ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
226 fvun2 6933 . . . . . . . . . . . . . . 15 ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)} ∧ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ∧ (({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
227225, 226mp3an1 1448 . . . . . . . . . . . . . 14 ((((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ∧ (({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
228182, 227mpanr1 701 . . . . . . . . . . . . 13 ((((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
229178, 181, 228syl2an 596 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
230229anassrs 468 . . . . . . . . . . 11 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2nd ‘(1st𝑇))‘𝑀)} × {1}) ∪ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
231223, 230eqtrd 2776 . . . . . . . . . 10 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
23232, 210, 100, 100, 101, 102, 231ofval 7628 . . . . . . . . 9 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)))
233191, 232eqtr4d 2779 . . . . . . . 8 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
234233an32s 650 . . . . . . 7 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
235234anasss 467 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
236 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
237236breq2d 5117 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
238237ifbid 4509 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
239238csbeq1d 3859 . . . . . . . . . . . . . . 15 (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}))))
240 2fveq3 6847 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
241 2fveq3 6847 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
242241imaeq1d 6012 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
243242xpeq1d 5662 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
244241imaeq1d 6012 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
245244xpeq1d 5662 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
246243, 245uneq12d 4124 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
247240, 246oveq12d 7375 . . . . . . . . . . . . . . . 16 (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}))))
248247csbeq2dv 3862 . . . . . . . . . . . . . . 15 (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}))))
249239, 248eqtrd 2776 . . . . . . . . . . . . . 14 (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}))))
250249mpteq2dv 5207 . . . . . . . . . . . . 13 (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})))))
251250eqeq2d 2747 . . . . . . . . . . . 12 (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}))))))
252251, 3elrab2 3648 . . . . . . . . . . 11 (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}))))))
253252simprbi 497 . . . . . . . . . 10 (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})))))
2541, 253syl 17 . . . . . . . . 9 (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})))))
255 breq1 5108 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 2) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 2) < (2nd𝑇)))
256255adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 2)) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 2) < (2nd𝑇)))
257 oveq1 7364 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 2) → (𝑦 + 1) = ((𝑀 − 2) + 1))
258 sub1m1 12405 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℂ → ((𝑀 − 1) − 1) = (𝑀 − 2))
25956, 258syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑀 − 1) − 1) = (𝑀 − 2))
260259oveq1d 7372 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑀 − 1) − 1) + 1) = ((𝑀 − 2) + 1))
26174zcnd 12608 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 − 1) ∈ ℂ)
262 npcan1 11580 . . . . . . . . . . . . . . . 16 ((𝑀 − 1) ∈ ℂ → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1))
263261, 262syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1))
264260, 263eqtr3d 2778 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 − 2) + 1) = (𝑀 − 1))
265257, 264sylan9eqr 2798 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 2)) → (𝑦 + 1) = (𝑀 − 1))
266256, 265ifbieq2d 4512 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 2) < (2nd𝑇), 𝑦, (𝑀 − 1)))
26718nncnd 12169 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℂ)
268 add1p1 12404 . . . . . . . . . . . . . . . . 17 ((2nd𝑇) ∈ ℂ → (((2nd𝑇) + 1) + 1) = ((2nd𝑇) + 2))
269267, 268syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) + 1) = ((2nd𝑇) + 2))
270269, 66eqbrtrrd 5129 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) + 2) ≤ 𝑀)
27118nnred 12168 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℝ)
272 2re 12227 . . . . . . . . . . . . . . . . . 18 2 ∈ ℝ
273 leaddsub 11631 . . . . . . . . . . . . . . . . . 18 (((2nd𝑇) ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 2) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 2)))
274272, 273mp3an2 1449 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 2) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 2)))
275271, 46, 274syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 2) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 2)))
27659, 46posdifd 11742 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1 < 𝑀 ↔ 0 < (𝑀 − 1)))
27767, 276mpbid 231 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 0 < (𝑀 − 1))
278 elnnz 12509 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 − 1) ∈ ℕ ↔ ((𝑀 − 1) ∈ ℤ ∧ 0 < (𝑀 − 1)))
27974, 277, 278sylanbrc 583 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 − 1) ∈ ℕ)
280 nnm1nn0 12454 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 − 1) ∈ ℕ → ((𝑀 − 1) − 1) ∈ ℕ0)
281279, 280syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑀 − 1) − 1) ∈ ℕ0)
282259, 281eqeltrrd 2839 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 − 2) ∈ ℕ0)
283282nn0red 12474 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 2) ∈ ℝ)
284271, 283lenltd 11301 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) ≤ (𝑀 − 2) ↔ ¬ (𝑀 − 2) < (2nd𝑇)))
285275, 284bitrd 278 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 2) ≤ 𝑀 ↔ ¬ (𝑀 − 2) < (2nd𝑇)))
286270, 285mpbid 231 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑀 − 2) < (2nd𝑇))
287286iffalsed 4497 . . . . . . . . . . . . 13 (𝜑 → if((𝑀 − 2) < (2nd𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1))
288287adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 2)) → if((𝑀 − 2) < (2nd𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1))
289266, 288eqtrd 2776 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1))
290289csbeq1d 3859 . . . . . . . . . 10 ((𝜑𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑀 − 1) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
291 oveq2 7365 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
292291imaeq2d 6013 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 − 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))))
293292xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 − 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
294293adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
295 oveq1 7364 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
296295, 58sylan9eqr 2798 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
297296oveq1d 7372 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
298297imaeq2d 6013 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = (𝑀 − 1)) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
299298xpeq1d 5662 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))
300294, 299uneq12d 4124 . . . . . . . . . . . . 13 ((𝜑𝑗 = (𝑀 − 1)) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))
301300oveq2d 7373 . . . . . . . . . . . 12 ((𝜑𝑗 = (𝑀 − 1)) → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
30274, 301csbied 3893 . . . . . . . . . . 11 (𝜑(𝑀 − 1) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
303302adantr 481 . . . . . . . . . 10 ((𝜑𝑦 = (𝑀 − 2)) → (𝑀 − 1) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
304290, 303eqtrd 2776 . . . . . . . . 9 ((𝜑𝑦 = (𝑀 − 2)) → 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...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
305 poimir.0 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ)
306 nnm1nn0 12454 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
307305, 306syl 17 . . . . . . . . . 10 (𝜑 → (𝑁 − 1) ∈ ℕ0)
308305nnred 12168 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℝ)
30946lem1d 12088 . . . . . . . . . . . . 13 (𝜑 → (𝑀 − 1) ≤ 𝑀)
310 elfzle2 13445 . . . . . . . . . . . . . 14 (𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → 𝑀𝑁)
31125, 310syl 17 . . . . . . . . . . . . 13 (𝜑𝑀𝑁)
312126, 46, 308, 309, 311letrd 11312 . . . . . . . . . . . 12 (𝜑 → (𝑀 − 1) ≤ 𝑁)
313126, 308, 59, 312lesub1dd 11771 . . . . . . . . . . 11 (𝜑 → ((𝑀 − 1) − 1) ≤ (𝑁 − 1))
314259, 313eqbrtrrd 5129 . . . . . . . . . 10 (𝜑 → (𝑀 − 2) ≤ (𝑁 − 1))
315 elfz2nn0 13532 . . . . . . . . . 10 ((𝑀 − 2) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 2) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑀 − 2) ≤ (𝑁 − 1)))
316282, 307, 314, 315syl3anbrc 1343 . . . . . . . . 9 (𝜑 → (𝑀 − 2) ∈ (0...(𝑁 − 1)))
317 ovexd 7392 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V)
318254, 304, 316, 317fvmptd 6955 . . . . . . . 8 (𝜑 → (𝐹‘(𝑀 − 2)) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
319318fveq1d 6844 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
320319adantr 481 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
321 breq1 5108 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 1) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < (2nd𝑇)))
322321adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < (2nd𝑇)))
323 oveq1 7364 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1))
324323, 58sylan9eqr 2798 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀)
325322, 324ifbieq2d 4512 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2nd𝑇), 𝑦, 𝑀))
32661lep1d 12086 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) + 1) ≤ (((2nd𝑇) + 1) + 1))
32761, 60, 46, 326, 66letrd 11312 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) + 1) ≤ 𝑀)
328 1re 11155 . . . . . . . . . . . . . . . . . 18 1 ∈ ℝ
329 leaddsub 11631 . . . . . . . . . . . . . . . . . 18 (((2nd𝑇) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 1) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 1)))
330328, 329mp3an2 1449 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 1) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 1)))
331271, 46, 330syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 1)))
332271, 126lenltd 11301 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < (2nd𝑇)))
333331, 332bitrd 278 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 1) ≤ 𝑀 ↔ ¬ (𝑀 − 1) < (2nd𝑇)))
334327, 333mpbid 231 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑀 − 1) < (2nd𝑇))
335334iffalsed 4497 . . . . . . . . . . . . 13 (𝜑 → if((𝑀 − 1) < (2nd𝑇), 𝑦, 𝑀) = 𝑀)
336335adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < (2nd𝑇), 𝑦, 𝑀) = 𝑀)
337325, 336eqtrd 2776 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
338337csbeq1d 3859 . . . . . . . . . 10 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑀 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
339 oveq2 7365 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
340339imaeq2d 6013 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑀)))
341340xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}))
342 oveq1 7364 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
343342oveq1d 7372 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
344343imaeq2d 6013 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
345344xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
346341, 345uneq12d 4124 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
347346oveq2d 7373 . . . . . . . . . . . . 13 (𝑗 = 𝑀 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
348347adantl 482 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑀) → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
34925, 348csbied 3893 . . . . . . . . . . 11 (𝜑𝑀 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
350349adantr 481 . . . . . . . . . 10 ((𝜑𝑦 = (𝑀 − 1)) → 𝑀 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
351338, 350eqtrd 2776 . . . . . . . . 9 ((𝜑𝑦 = (𝑀 − 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...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
352279nnnn0d 12473 . . . . . . . . . 10 (𝜑 → (𝑀 − 1) ∈ ℕ0)
35346, 308, 59, 311lesub1dd 11771 . . . . . . . . . 10 (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1))
354 elfz2nn0 13532 . . . . . . . . . 10 ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑀 − 1) ≤ (𝑁 − 1)))
355352, 307, 353, 354syl3anbrc 1343 . . . . . . . . 9 (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
356 ovexd 7392 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
357254, 351, 355, 356fvmptd 6955 . . . . . . . 8 (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
358357fveq1d 6844 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
359358adantr 481 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
360235, 320, 3593eqtr4d 2786 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛))
361360expr 457 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛)))
362361necon1d 2965 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) → 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
363 elmapi 8787 . . . . . . . . . . 11 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
36429, 363syl 17 . . . . . . . . . 10 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
365364, 27ffvelcdmd 7036 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾))
366 elfzonn0 13617 . . . . . . . . 9 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
367365, 366syl 17 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
368367nn0red 12474 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℝ)
369368ltp1d 12085 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) < (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
370368, 369ltned 11291 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
371318fveq1d 6844 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)))
372 ovexd 7392 . . . . . . . . 9 (𝜑 → (1...𝑁) ∈ V)
373 eqidd 2737 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
374 fzss1 13480 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘1) → (𝑀...𝑁) ⊆ (1...𝑁))
37571, 374syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁))
376 eluzfz1 13448 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
37782, 376syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (𝑀...𝑁))
378 fnfvima 7183 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
379108, 375, 377, 378syl3anc 1371 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
380 fvun2 6933 . . . . . . . . . . . . 13 (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)))
38135, 38, 380mp3an12 1451 . . . . . . . . . . . 12 (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)))
38253, 379, 381syl2anc 584 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)))
38336fvconst2 7153 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
384379, 383syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
385382, 384eqtrd 2776 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
386385adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
38731, 98, 372, 372, 101, 373, 386ofval 7628 . . . . . . . 8 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 0))
38827, 387mpdan 685 . . . . . . 7 (𝜑 → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 0))
389367nn0cnd 12475 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℂ)
390389addid1d 11355 . . . . . . 7 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
391371, 388, 3903eqtrd 2780 . . . . . 6 (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
392357fveq1d 6844 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)))
393 fzss2 13481 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → (1...𝑀) ⊆ (1...𝑁))
39482, 393syl 17 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ⊆ (1...𝑁))
395 elfz1end 13471 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
39670, 395sylib 217 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (1...𝑀))
397 fnfvima 7183 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
398108, 394, 396, 397syl3anc 1371 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
399 fvun1 6932 . . . . . . . . . . . . 13 (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)))
400193, 122, 399mp3an12 1451 . . . . . . . . . . . 12 (((((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)))
401201, 398, 400syl2anc 584 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)))
40233fvconst2 7153 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
403398, 402syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
404401, 403eqtrd 2776 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
405404adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
40631, 209, 372, 372, 101, 373, 405ofval 7628 . . . . . . . 8 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
40727, 406mpdan 685 . . . . . . 7 (𝜑 → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
408392, 407eqtrd 2776 . . . . . 6 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
409370, 391, 4083netr4d 3021 . . . . 5 (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)))
410 fveq2 6842 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)))
411 fveq2 6842 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)))
412410, 411neeq12d 3005 . . . . 5 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀))))
413409, 412syl5ibrcom 246 . . . 4 (𝜑 → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)))
414413adantr 481 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)))
415362, 414impbid 211 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
41627, 415riota5 7343 1 (𝜑 → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wne 2943  {crab 3407  Vcvv 3445  csb 3855  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  crio 7312  (class class class)co 7357  f cof 7615  1st c1st 7919  2nd c2nd 7920  m cmap 8765  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  ..^cfzo 13567
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568
This theorem is referenced by:  poimirlem8  36086
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