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Theorem poimirlem7 35057
Description: Lemma for poimir 35083, similar to poimirlem6 35056, 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 3626 . . . . . . . . 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 2911 . . . . . . . 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 7707 . . . . . . 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 7708 . . . . . 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 6662 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
11 f1oeq1 6583 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3618 . . . . 5 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
139, 12sylib 221 . . . 4 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
14 f1of 6594 . . . 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 12935 . . . . . . . . 9 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
1816, 17syl 17 . . . . . . . 8 (𝜑 → (2nd𝑇) ∈ ℕ)
1918peano2nnd 11646 . . . . . . 7 (𝜑 → ((2nd𝑇) + 1) ∈ ℕ)
2019peano2nnd 11646 . . . . . 6 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℕ)
21 nnuz 12273 . . . . . 6 ℕ = (ℤ‘1)
2220, 21eleqtrdi 2903 . . . . 5 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ (ℤ‘1))
23 fzss1 12945 . . . . 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 3919 . . 3 (𝜑𝑀 ∈ (1...𝑁))
2715, 26ffvelrnd 6833 . 2 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁))
28 xp1st 7707 . . . . . . . . . . . . 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 8416 . . . . . . . . . . . 12 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
3129, 30syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
3231adantr 484 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
33 1ex 10630 . . . . . . . . . . . . . . 15 1 ∈ V
34 fnconstg 6545 . . . . . . . . . . . . . . 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 10628 . . . . . . . . . . . . . . 15 0 ∈ V
37 fnconstg 6545 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 6600 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
4140simprbi 500 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
4213, 41syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun (2nd ‘(1st𝑇)))
43 imain 6413 . . . . . . . . . . . . . . 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))))
45 elfzelz 12906 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → 𝑀 ∈ ℤ)
4625, 45syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℤ)
4746zred 12079 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℝ)
4847ltm1d 11565 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 1) < 𝑀)
49 fzdisj 12933 . . . . . . . . . . . . . . . . 17 ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
5048, 49syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
5150imaeq2d 5900 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
52 ima0 5916 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
5351, 52eqtrdi 2852 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
5444, 53eqtr3d 2838 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = ∅)
55 fnun 6438 . . . . . . . . . . . . 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))))
5639, 54, 55sylancr 590 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
5746zcnd 12080 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℂ)
58 npcan1 11058 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
5957, 58syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
60 1red 10635 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 1 ∈ ℝ)
6120nnred 11644 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℝ)
6219nnred 11644 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
6319nnge1d 11677 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ≤ ((2nd𝑇) + 1))
6462ltp1d 11563 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1))
6560, 62, 61, 63, 64lelttrd 10791 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 1 < (((2nd𝑇) + 1) + 1))
66 elfzle1 12909 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ 𝑀)
6725, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((2nd𝑇) + 1) + 1) ≤ 𝑀)
6860, 61, 47, 65, 67ltletrd 10793 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 1 < 𝑀)
6960, 47, 68ltled 10781 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ≤ 𝑀)
70 elnnz1 12000 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤ 𝑀))
7146, 69, 70sylanbrc 586 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℕ)
7271, 21eleqtrdi 2903 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘1))
7359, 72eqeltrd 2893 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘1))
74 peano2zm 12017 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
7546, 74syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 − 1) ∈ ℤ)
76 uzid 12250 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
77 peano2uz 12293 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
7875, 76, 773syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
7959, 78eqeltrrd 2894 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ‘(𝑀 − 1)))
80 uzss 12257 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘(𝑀 − 1)) → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
8179, 80syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
82 elfzuz3 12903 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → 𝑁 ∈ (ℤ𝑀))
8325, 82syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ𝑀))
8481, 83sseldd 3919 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘(𝑀 − 1)))
85 fzsplit2 12931 . . . . . . . . . . . . . . . . . 18 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
8673, 84, 85syl2anc 587 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
8759oveq1d 7154 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
8887uneq2d 4093 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
8986, 88eqtrd 2836 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
9089imaeq2d 5900 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
91 imaundi 5979 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
9290, 91eqtrdi 2852 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
93 f1ofo 6601 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
9413, 93syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
95 foima 6574 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9694, 95syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9792, 96eqtr3d 2838 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = (1...𝑁))
9897fneq2d 6421 . . . . . . . . . . . 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)))
9956, 98mpbid 235 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
10099adantr 484 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
101 ovexd 7174 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1...𝑁) ∈ V)
102 inidm 4148 . . . . . . . . . 10 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
103 eqidd 2802 . . . . . . . . . 10 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
104 imaundi 5979 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
105 fzpred 12954 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
10683, 105syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
107106imaeq2d 5900 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))))
108 f1ofn 6595 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
10913, 108syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
110 fnsnfv 6722 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
111109, 26, 110syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
112111uneq1d 4092 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
113104, 107, 1123eqtr4a 2862 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
114113xpeq1d 5552 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}))
115 xpundir 5589 . . . . . . . . . . . . . . . 16 (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
116114, 115eqtrdi 2852 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
117116uneq2d 4093 . . . . . . . . . . . . . 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}))))
118 un12 4097 . . . . . . . . . . . . . 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})))
119117, 118eqtrdi 2852 . . . . . . . . . . . . 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}))))
120119fveq1d 6651 . . . . . . . . . . . 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))
121120ad2antrr 725 . . . . . . . . . . 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))
122 fnconstg 6545 . . . . . . . . . . . . . . . . 17 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
12336, 122ax-mp 5 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))
12435, 123pm3.2i 474 . . . . . . . . . . . . . . 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)))
125 imain 6413 . . . . . . . . . . . . . . . . 17 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
12642, 125syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
12775zred 12079 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 − 1) ∈ ℝ)
128 peano2re 10806 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
12947, 128syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 + 1) ∈ ℝ)
13047ltp1d 11563 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 < (𝑀 + 1))
131127, 47, 129, 48, 130lttrd 10794 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 − 1) < (𝑀 + 1))
132 fzdisj 12933 . . . . . . . . . . . . . . . . . . 19 ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
133131, 132syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
134133imaeq2d 5900 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
135134, 52eqtrdi 2852 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅)
136126, 135eqtr3d 2838 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
137 fnun 6438 . . . . . . . . . . . . . . 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))))
138124, 136, 137sylancr 590 . . . . . . . . . . . . . 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))))
139 imaundi 5979 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
140 imadif 6412 . . . . . . . . . . . . . . . . . 18 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
14142, 140syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
142 fzsplit 12932 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
14326, 142syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
144143difeq1d 4052 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}))
145 difundir 4210 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀}))
146 fzsplit2 12931 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
14773, 79, 146syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
14859oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀))
149 fzsn 12948 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
15046, 149syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑀...𝑀) = {𝑀})
151148, 150eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀})
152151uneq2d 4093 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀}))
153147, 152eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀}))
154153difeq1d 4052 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}))
155 difun2 4390 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀})
156127, 47ltnled 10780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1)))
15748, 156mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1))
158 elfzle2 12910 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1))
159157, 158nsyl 142 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1)))
160 difsn 4694 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
161159, 160syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
162155, 161syl5eq 2848 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1)))
163154, 162eqtrd 2836 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1)))
16447, 129ltnled 10780 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
165130, 164mpbid 235 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
166 elfzle1 12909 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀)
167165, 166nsyl 142 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁))
168 difsn 4694 . . . . . . . . . . . . . . . . . . . . . 22 𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
169167, 168syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
170163, 169uneq12d 4094 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
171145, 170syl5eq 2848 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
172144, 171eqtrd 2836 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
173172imaeq2d 5900 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))))
174111eqcomd 2807 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑀}) = {((2nd ‘(1st𝑇))‘𝑀)})
17596, 174difeq12d 4054 . . . . . . . . . . . . . . . . 17 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
176141, 173, 1753eqtr3d 2844 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
177139, 176syl5eqr 2850 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
178177fneq2d 6421 . . . . . . . . . . . . . 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)})))
179138, 178mpbid 235 . . . . . . . . . . . . 13 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
180 eldifsn 4683 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)))
181180biimpri 231 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
182181ancoms 462 . . . . . . . . . . . . 13 ((𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
183 disjdif 4382 . . . . . . . . . . . . . 14 ({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅
184 fnconstg 6545 . . . . . . . . . . . . . . . 16 (0 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
18536, 184ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
186 fvun2 6734 . . . . . . . . . . . . . . 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))
187185, 186mp3an1 1445 . . . . . . . . . . . . . 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))
188183, 187mpanr1 702 . . . . . . . . . . . . 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))
189179, 182, 188syl2an 598 . . . . . . . . . . . 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))
190189anassrs 471 . . . . . . . . . . 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))
191121, 190eqtrd 2836 . . . . . . . . . 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))
19232, 100, 101, 101, 102, 103, 191ofval 7402 . . . . . . . . 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)))
193 fnconstg 6545 . . . . . . . . . . . . . . 15 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)))
19433, 193ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀))
195194, 123pm3.2i 474 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
196 imain 6413 . . . . . . . . . . . . . . 15 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
19742, 196syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
198 fzdisj 12933 . . . . . . . . . . . . . . . . 17 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
199130, 198syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
200199imaeq2d 5900 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
201200, 52eqtrdi 2852 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
202197, 201eqtr3d 2838 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
203 fnun 6438 . . . . . . . . . . . . 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))))
204195, 202, 203sylancr 590 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
205143imaeq2d 5900 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
206 imaundi 5979 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
207205, 206eqtrdi 2852 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
208207, 96eqtr3d 2838 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
209208fneq2d 6421 . . . . . . . . . . . 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)))
210204, 209mpbid 235 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
211210adantr 484 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
212 imaundi 5979 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀}))
213153imaeq2d 5900 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})))
214111uneq2d 4093 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀})))
215212, 213, 2143eqtr4a 2862 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}))
216215xpeq1d 5552 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}))
217 xpundir 5589 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1}))
218216, 217eqtrdi 2852 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1})))
219218uneq1d 4092 . . . . . . . . . . . . . 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})))
220 un23 4098 . . . . . . . . . . . . . . 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}))
221220equncomi 4085 . . . . . . . . . . . . . 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})))
222219, 221eqtrdi 2852 . . . . . . . . . . . . 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}))))
223222fveq1d 6651 . . . . . . . . . . . 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))
224223ad2antrr 725 . . . . . . . . . . 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))
225 fnconstg 6545 . . . . . . . . . . . . . . . 16 (1 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
22633, 225ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
227 fvun2 6734 . . . . . . . . . . . . . . 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))
228226, 227mp3an1 1445 . . . . . . . . . . . . . 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))
229183, 228mpanr1 702 . . . . . . . . . . . . 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))
230179, 182, 229syl2an 598 . . . . . . . . . . . 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))
231230anassrs 471 . . . . . . . . . . 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))
232224, 231eqtrd 2836 . . . . . . . . . 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))
23332, 211, 101, 101, 102, 103, 232ofval 7402 . . . . . . . . 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)))
234192, 233eqtr4d 2839 . . . . . . . 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))
235234an32s 651 . . . . . . 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))
236235anasss 470 . . . . . 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))
237 fveq2 6649 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
238237breq2d 5045 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
239238ifbid 4450 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
240239csbeq1d 3835 . . . . . . . . . . . . . . 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}))))
241 2fveq3 6654 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
242 2fveq3 6654 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
243242imaeq1d 5899 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
244243xpeq1d 5552 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
245242imaeq1d 5899 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
246245xpeq1d 5552 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
247244, 246uneq12d 4094 . . . . . . . . . . . . . . . . 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})))
248241, 247oveq12d 7157 . . . . . . . . . . . . . . . 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}))))
249248csbeq2dv 3838 . . . . . . . . . . . . . . 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}))))
250240, 249eqtrd 2836 . . . . . . . . . . . . . 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}))))
251250mpteq2dv 5129 . . . . . . . . . . . . 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})))))
252251eqeq2d 2812 . . . . . . . . . . . 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}))))))
253252, 3elrab2 3634 . . . . . . . . . . 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}))))))
254253simprbi 500 . . . . . . . . . 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})))))
2551, 254syl 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})))))
256 breq1 5036 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 2) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 2) < (2nd𝑇)))
257256adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 2)) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 2) < (2nd𝑇)))
258 oveq1 7146 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 2) → (𝑦 + 1) = ((𝑀 − 2) + 1))
259 sub1m1 11881 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℂ → ((𝑀 − 1) − 1) = (𝑀 − 2))
26057, 259syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑀 − 1) − 1) = (𝑀 − 2))
261260oveq1d 7154 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑀 − 1) − 1) + 1) = ((𝑀 − 2) + 1))
26275zcnd 12080 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 − 1) ∈ ℂ)
263 npcan1 11058 . . . . . . . . . . . . . . . 16 ((𝑀 − 1) ∈ ℂ → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1))
264262, 263syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1))
265261, 264eqtr3d 2838 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 − 2) + 1) = (𝑀 − 1))
266258, 265sylan9eqr 2858 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 2)) → (𝑦 + 1) = (𝑀 − 1))
267257, 266ifbieq2d 4453 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 2) < (2nd𝑇), 𝑦, (𝑀 − 1)))
26818nncnd 11645 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℂ)
269 add1p1 11880 . . . . . . . . . . . . . . . . 17 ((2nd𝑇) ∈ ℂ → (((2nd𝑇) + 1) + 1) = ((2nd𝑇) + 2))
270268, 269syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) + 1) = ((2nd𝑇) + 2))
271270, 67eqbrtrrd 5057 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) + 2) ≤ 𝑀)
27218nnred 11644 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℝ)
273 2re 11703 . . . . . . . . . . . . . . . . . 18 2 ∈ ℝ
274 leaddsub 11109 . . . . . . . . . . . . . . . . . 18 (((2nd𝑇) ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 2) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 2)))
275273, 274mp3an2 1446 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 2) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 2)))
276272, 47, 275syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 2) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 2)))
27760, 47posdifd 11220 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1 < 𝑀 ↔ 0 < (𝑀 − 1)))
27868, 277mpbid 235 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 0 < (𝑀 − 1))
279 elnnz 11983 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 − 1) ∈ ℕ ↔ ((𝑀 − 1) ∈ ℤ ∧ 0 < (𝑀 − 1)))
28075, 278, 279sylanbrc 586 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 − 1) ∈ ℕ)
281 nnm1nn0 11930 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 − 1) ∈ ℕ → ((𝑀 − 1) − 1) ∈ ℕ0)
282280, 281syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑀 − 1) − 1) ∈ ℕ0)
283260, 282eqeltrrd 2894 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 − 2) ∈ ℕ0)
284283nn0red 11948 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 2) ∈ ℝ)
285272, 284lenltd 10779 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) ≤ (𝑀 − 2) ↔ ¬ (𝑀 − 2) < (2nd𝑇)))
286276, 285bitrd 282 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 2) ≤ 𝑀 ↔ ¬ (𝑀 − 2) < (2nd𝑇)))
287271, 286mpbid 235 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑀 − 2) < (2nd𝑇))
288287iffalsed 4439 . . . . . . . . . . . . 13 (𝜑 → if((𝑀 − 2) < (2nd𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1))
289288adantr 484 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 2)) → if((𝑀 − 2) < (2nd𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1))
290267, 289eqtrd 2836 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1))
291290csbeq1d 3835 . . . . . . . . . 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}))))
292 oveq2 7147 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
293292imaeq2d 5900 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 − 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))))
294293xpeq1d 5552 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 − 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
295294adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
296 oveq1 7146 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
297296, 59sylan9eqr 2858 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
298297oveq1d 7154 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
299298imaeq2d 5900 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = (𝑀 − 1)) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
300299xpeq1d 5552 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))
301295, 300uneq12d 4094 . . . . . . . . . . . . 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})))
302301oveq2d 7155 . . . . . . . . . . . 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}))))
30375, 302csbied 3867 . . . . . . . . . . 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}))))
304303adantr 484 . . . . . . . . . 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}))))
305291, 304eqtrd 2836 . . . . . . . . 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}))))
306 poimir.0 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ)
307 nnm1nn0 11930 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
308306, 307syl 17 . . . . . . . . . 10 (𝜑 → (𝑁 − 1) ∈ ℕ0)
309306nnred 11644 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℝ)
31047lem1d 11566 . . . . . . . . . . . . 13 (𝜑 → (𝑀 − 1) ≤ 𝑀)
311 elfzle2 12910 . . . . . . . . . . . . . 14 (𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → 𝑀𝑁)
31225, 311syl 17 . . . . . . . . . . . . 13 (𝜑𝑀𝑁)
313127, 47, 309, 310, 312letrd 10790 . . . . . . . . . . . 12 (𝜑 → (𝑀 − 1) ≤ 𝑁)
314127, 309, 60, 313lesub1dd 11249 . . . . . . . . . . 11 (𝜑 → ((𝑀 − 1) − 1) ≤ (𝑁 − 1))
315260, 314eqbrtrrd 5057 . . . . . . . . . 10 (𝜑 → (𝑀 − 2) ≤ (𝑁 − 1))
316 elfz2nn0 12997 . . . . . . . . . 10 ((𝑀 − 2) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 2) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑀 − 2) ≤ (𝑁 − 1)))
317283, 308, 315, 316syl3anbrc 1340 . . . . . . . . 9 (𝜑 → (𝑀 − 2) ∈ (0...(𝑁 − 1)))
318 ovexd 7174 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V)
319255, 305, 317, 318fvmptd 6756 . . . . . . . 8 (𝜑 → (𝐹‘(𝑀 − 2)) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
320319fveq1d 6651 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
321320adantr 484 . . . . . 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))
322 breq1 5036 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 1) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < (2nd𝑇)))
323322adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < (2nd𝑇)))
324 oveq1 7146 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1))
325324, 59sylan9eqr 2858 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀)
326323, 325ifbieq2d 4453 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2nd𝑇), 𝑦, 𝑀))
32762lep1d 11564 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) + 1) ≤ (((2nd𝑇) + 1) + 1))
32862, 61, 47, 327, 67letrd 10790 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) + 1) ≤ 𝑀)
329 1re 10634 . . . . . . . . . . . . . . . . . 18 1 ∈ ℝ
330 leaddsub 11109 . . . . . . . . . . . . . . . . . 18 (((2nd𝑇) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 1) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 1)))
331329, 330mp3an2 1446 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd𝑇) + 1) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 1)))
332272, 47, 331syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) ≤ 𝑀 ↔ (2nd𝑇) ≤ (𝑀 − 1)))
333272, 127lenltd 10779 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < (2nd𝑇)))
334332, 333bitrd 282 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 1) ≤ 𝑀 ↔ ¬ (𝑀 − 1) < (2nd𝑇)))
335328, 334mpbid 235 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑀 − 1) < (2nd𝑇))
336335iffalsed 4439 . . . . . . . . . . . . 13 (𝜑 → if((𝑀 − 1) < (2nd𝑇), 𝑦, 𝑀) = 𝑀)
337336adantr 484 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < (2nd𝑇), 𝑦, 𝑀) = 𝑀)
338326, 337eqtrd 2836 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
339338csbeq1d 3835 . . . . . . . . . 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}))))
340 oveq2 7147 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
341340imaeq2d 5900 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑀)))
342341xpeq1d 5552 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}))
343 oveq1 7146 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
344343oveq1d 7154 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
345344imaeq2d 5900 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
346345xpeq1d 5552 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
347342, 346uneq12d 4094 . . . . . . . . . . . . . 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})))
348347oveq2d 7155 . . . . . . . . . . . . 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}))))
349348adantl 485 . . . . . . . . . . . 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}))))
35025, 349csbied 3867 . . . . . . . . . . 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}))))
351350adantr 484 . . . . . . . . . 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}))))
352339, 351eqtrd 2836 . . . . . . . . 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}))))
353280nnnn0d 11947 . . . . . . . . . 10 (𝜑 → (𝑀 − 1) ∈ ℕ0)
35447, 309, 60, 312lesub1dd 11249 . . . . . . . . . 10 (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1))
355 elfz2nn0 12997 . . . . . . . . . 10 ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑀 − 1) ≤ (𝑁 − 1)))
356353, 308, 354, 355syl3anbrc 1340 . . . . . . . . 9 (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
357 ovexd 7174 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
358255, 352, 356, 357fvmptd 6756 . . . . . . . 8 (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
359358fveq1d 6651 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
360359adantr 484 . . . . . 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))
361236, 321, 3603eqtr4d 2846 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛))
362361expr 460 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛)))
363362necon1d 3012 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) → 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
364 elmapi 8415 . . . . . . . . . . 11 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
36529, 364syl 17 . . . . . . . . . 10 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
366365, 27ffvelrnd 6833 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾))
367 elfzonn0 13081 . . . . . . . . 9 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
368366, 367syl 17 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
369368nn0red 11948 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℝ)
370369ltp1d 11563 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) < (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
371369, 370ltned 10769 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
372319fveq1d 6651 . . . . . . 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)))
373 ovexd 7174 . . . . . . . . 9 (𝜑 → (1...𝑁) ∈ V)
374 eqidd 2802 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
375 fzss1 12945 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘1) → (𝑀...𝑁) ⊆ (1...𝑁))
37672, 375syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁))
377 eluzfz1 12913 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
37883, 377syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (𝑀...𝑁))
379 fnfvima 6977 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
380109, 376, 378, 379syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
381 fvun2 6734 . . . . . . . . . . . . 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)))
38235, 38, 381mp3an12 1448 . . . . . . . . . . . 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)))
38354, 380, 382syl2anc 587 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)))
38436fvconst2 6947 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
385380, 384syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
386383, 385eqtrd 2836 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
387386adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
38831, 99, 373, 373, 102, 374, 387ofval 7402 . . . . . . . 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))
38927, 388mpdan 686 . . . . . . 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))
390368nn0cnd 11949 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℂ)
391390addid1d 10833 . . . . . . 7 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
392372, 389, 3913eqtrd 2840 . . . . . 6 (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
393358fveq1d 6651 . . . . . . 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)))
394 fzss2 12946 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → (1...𝑀) ⊆ (1...𝑁))
39583, 394syl 17 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ⊆ (1...𝑁))
396 elfz1end 12936 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
39771, 396sylib 221 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (1...𝑀))
398 fnfvima 6977 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
399109, 395, 397, 398syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
400 fvun1 6733 . . . . . . . . . . . . 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)))
401194, 123, 400mp3an12 1448 . . . . . . . . . . . 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)))
402202, 399, 401syl2anc 587 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)))
40333fvconst2 6947 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
404399, 403syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
405402, 404eqtrd 2836 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
406405adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
40731, 210, 373, 373, 102, 374, 406ofval 7402 . . . . . . . 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))
40827, 407mpdan 686 . . . . . . 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))
409393, 408eqtrd 2836 . . . . . 6 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
410371, 392, 4093netr4d 3067 . . . . 5 (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)))
411 fveq2 6649 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 2))‘((2nd ‘(1st𝑇))‘𝑀)))
412 fveq2 6649 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)))
413411, 412neeq12d 3051 . . . . 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))))
414410, 413syl5ibrcom 250 . . . 4 (𝜑 → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)))
415414adantr 484 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)))
416363, 415impbid 215 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
41727, 416riota5 7126 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 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wne 2990  {crab 3113  Vcvv 3444  csb 3831  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  ifcif 4428  {csn 4528   class class class wbr 5033  cmpt 5113   × cxp 5521  ccnv 5522  cima 5526  Fun wfun 6322   Fn wfn 6323  wf 6324  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  crio 7096  (class class class)co 7139  f cof 7391  1st c1st 7673  2nd c2nd 7674  m cmap 8393  cc 10528  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   < clt 10668  cle 10669  cmin 10863  cn 11629  2c2 11684  0cn0 11889  cz 11973  cuz 12235  ...cfz 12889  ..^cfzo 13032
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033
This theorem is referenced by:  poimirlem8  35058
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