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Theorem poimirlem6 34025
Description: Lemma for poimir 34052 establishing, for a face of a simplex defined by a walk along the edges of an 𝑁-cube, the single dimension in which successive vertices before the opposite vertex differ. (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)))
poimirlem6.3 (𝜑𝑀 ∈ (1...((2nd𝑇) − 1)))
Assertion
Ref Expression
poimirlem6 (𝜑 → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘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 poimirlem6
StepHypRef Expression
1 poimirlem9.1 . . . . . . . 8 (𝜑𝑇𝑆)
2 elrabi 3566 . . . . . . . . 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 2876 . . . . . . . 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 7477 . . . . . . 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 7478 . . . . . 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 6459 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
11 f1oeq1 6380 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3557 . . . . 5 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
139, 12sylib 210 . . . 4 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
14 f1of 6391 . . . 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 12687 . . . . . . . . 9 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
1816, 17syl 17 . . . . . . . 8 (𝜑 → (2nd𝑇) ∈ ℕ)
1918nnzd 11833 . . . . . . 7 (𝜑 → (2nd𝑇) ∈ ℤ)
20 peano2zm 11772 . . . . . . 7 ((2nd𝑇) ∈ ℤ → ((2nd𝑇) − 1) ∈ ℤ)
2119, 20syl 17 . . . . . 6 (𝜑 → ((2nd𝑇) − 1) ∈ ℤ)
22 poimir.0 . . . . . . 7 (𝜑𝑁 ∈ ℕ)
2322nnzd 11833 . . . . . 6 (𝜑𝑁 ∈ ℤ)
2421zred 11834 . . . . . . 7 (𝜑 → ((2nd𝑇) − 1) ∈ ℝ)
2518nnred 11391 . . . . . . 7 (𝜑 → (2nd𝑇) ∈ ℝ)
2622nnred 11391 . . . . . . 7 (𝜑𝑁 ∈ ℝ)
2725lem1d 11311 . . . . . . 7 (𝜑 → ((2nd𝑇) − 1) ≤ (2nd𝑇))
28 nnm1nn0 11685 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
2922, 28syl 17 . . . . . . . . 9 (𝜑 → (𝑁 − 1) ∈ ℕ0)
3029nn0red 11703 . . . . . . . 8 (𝜑 → (𝑁 − 1) ∈ ℝ)
31 elfzle2 12662 . . . . . . . . 9 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≤ (𝑁 − 1))
3216, 31syl 17 . . . . . . . 8 (𝜑 → (2nd𝑇) ≤ (𝑁 − 1))
3326lem1d 11311 . . . . . . . 8 (𝜑 → (𝑁 − 1) ≤ 𝑁)
3425, 30, 26, 32, 33letrd 10533 . . . . . . 7 (𝜑 → (2nd𝑇) ≤ 𝑁)
3524, 25, 26, 27, 34letrd 10533 . . . . . 6 (𝜑 → ((2nd𝑇) − 1) ≤ 𝑁)
36 eluz2 11998 . . . . . 6 (𝑁 ∈ (ℤ‘((2nd𝑇) − 1)) ↔ (((2nd𝑇) − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((2nd𝑇) − 1) ≤ 𝑁))
3721, 23, 35, 36syl3anbrc 1400 . . . . 5 (𝜑𝑁 ∈ (ℤ‘((2nd𝑇) − 1)))
38 fzss2 12698 . . . . 5 (𝑁 ∈ (ℤ‘((2nd𝑇) − 1)) → (1...((2nd𝑇) − 1)) ⊆ (1...𝑁))
3937, 38syl 17 . . . 4 (𝜑 → (1...((2nd𝑇) − 1)) ⊆ (1...𝑁))
40 poimirlem6.3 . . . 4 (𝜑𝑀 ∈ (1...((2nd𝑇) − 1)))
4139, 40sseldd 3821 . . 3 (𝜑𝑀 ∈ (1...𝑁))
4215, 41ffvelrnd 6624 . 2 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁))
43 xp1st 7477 . . . . . . . . . . . . 13 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
447, 43syl 17 . . . . . . . . . . . 12 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
45 elmapfn 8163 . . . . . . . . . . . 12 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
4644, 45syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
4746adantr 474 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
48 1ex 10372 . . . . . . . . . . . . . . 15 1 ∈ V
49 fnconstg 6343 . . . . . . . . . . . . . . 15 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))))
5048, 49ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1)))
51 c0ex 10370 . . . . . . . . . . . . . . 15 0 ∈ V
52 fnconstg 6343 . . . . . . . . . . . . . . 15 (0 ∈ V → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
5351, 52ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))
5450, 53pm3.2i 464 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
55 dff1o3 6397 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
5655simprbi 492 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
5713, 56syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun (2nd ‘(1st𝑇)))
58 imain 6219 . . . . . . . . . . . . . . 15 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
5957, 58syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
60 elfznn 12687 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (1...((2nd𝑇) − 1)) → 𝑀 ∈ ℕ)
6140, 60syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℕ)
6261nnred 11391 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℝ)
6362ltm1d 11310 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 1) < 𝑀)
64 fzdisj 12685 . . . . . . . . . . . . . . . . 17 ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
6665imaeq2d 5720 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
67 ima0 5735 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
6866, 67syl6eq 2829 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
6959, 68eqtr3d 2815 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = ∅)
70 fnun 6243 . . . . . . . . . . . . 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))))
7154, 69, 70sylancr 581 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
7261nncnd 11392 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℂ)
73 npcan1 10800 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
75 nnuz 12029 . . . . . . . . . . . . . . . . . . . 20 ℕ = (ℤ‘1)
7661, 75syl6eleq 2868 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘1))
7774, 76eqeltrd 2858 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘1))
78 nnm1nn0 11685 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
7961, 78syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑀 − 1) ∈ ℕ0)
8079nn0zd 11832 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 − 1) ∈ ℤ)
81 uzid 12007 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
8280, 81syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
83 peano2uz 12047 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
8574, 84eqeltrrd 2859 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ‘(𝑀 − 1)))
86 uzss 12013 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘(𝑀 − 1)) → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
8785, 86syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
8861nnzd 11833 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℤ)
89 elfzle2 12662 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ (1...((2nd𝑇) − 1)) → 𝑀 ≤ ((2nd𝑇) − 1))
9040, 89syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ≤ ((2nd𝑇) − 1))
9162, 24, 26, 90, 35letrd 10533 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀𝑁)
92 eluz2 11998 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
9388, 23, 91, 92syl3anbrc 1400 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ𝑀))
9487, 93sseldd 3821 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘(𝑀 − 1)))
95 fzsplit2 12683 . . . . . . . . . . . . . . . . . 18 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
9677, 94, 95syl2anc 579 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
9774oveq1d 6937 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
9897uneq2d 3989 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
9996, 98eqtrd 2813 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
10099imaeq2d 5720 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
101 imaundi 5799 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
102100, 101syl6eq 2829 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
103 f1ofo 6398 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
10413, 103syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
105 foima 6371 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
106104, 105syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
107102, 106eqtr3d 2815 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = (1...𝑁))
108107fneq2d 6227 . . . . . . . . . . . 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)))
10971, 108mpbid 224 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
110109adantr 474 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
111 ovexd 6956 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1...𝑁) ∈ V)
112 inidm 4042 . . . . . . . . . 10 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
113 eqidd 2778 . . . . . . . . . 10 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
114 imaundi 5799 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
115 fzpred 12706 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
11693, 115syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
117116imaeq2d 5720 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))))
118 f1ofn 6392 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
11913, 118syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
120 fnsnfv 6518 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
121119, 41, 120syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
122121uneq1d 3988 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
123114, 117, 1223eqtr4a 2839 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
124123xpeq1d 5384 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}))
125 xpundir 5418 . . . . . . . . . . . . . . . 16 (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
126124, 125syl6eq 2829 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
127126uneq2d 3989 . . . . . . . . . . . . . 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}))))
128 un12 3993 . . . . . . . . . . . . . 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})))
129127, 128syl6eq 2829 . . . . . . . . . . . . 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}))))
130129fveq1d 6448 . . . . . . . . . . . 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))
131130ad2antrr 716 . . . . . . . . . . 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))
132 fnconstg 6343 . . . . . . . . . . . . . . . . 17 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
13351, 132ax-mp 5 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))
13450, 133pm3.2i 464 . . . . . . . . . . . . . . 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)))
135 imain 6219 . . . . . . . . . . . . . . . . 17 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
13657, 135syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
13779nn0red 11703 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 − 1) ∈ ℝ)
138 peano2re 10549 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
13962, 138syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 + 1) ∈ ℝ)
14062ltp1d 11308 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 < (𝑀 + 1))
141137, 62, 139, 63, 140lttrd 10537 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 − 1) < (𝑀 + 1))
142 fzdisj 12685 . . . . . . . . . . . . . . . . . . 19 ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
143141, 142syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
144143imaeq2d 5720 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
145144, 67syl6eq 2829 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅)
146136, 145eqtr3d 2815 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
147 fnun 6243 . . . . . . . . . . . . . . 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))))
148134, 146, 147sylancr 581 . . . . . . . . . . . . . 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))))
149 imaundi 5799 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
150 imadif 6218 . . . . . . . . . . . . . . . . . 18 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
15157, 150syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
152 fzsplit 12684 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
15341, 152syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
154153difeq1d 3949 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}))
155 difundir 4106 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀}))
156 fzsplit2 12683 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
15777, 85, 156syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
15874oveq1d 6937 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀))
159 fzsn 12700 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
16088, 159syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑀...𝑀) = {𝑀})
161158, 160eqtrd 2813 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀})
162161uneq2d 3989 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀}))
163157, 162eqtrd 2813 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀}))
164163difeq1d 3949 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}))
165 difun2 4271 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀})
166137, 62ltnled 10523 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1)))
16763, 166mpbid 224 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1))
168 elfzle2 12662 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1))
169167, 168nsyl 138 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1)))
170 difsn 4560 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
171169, 170syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
172165, 171syl5eq 2825 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1)))
173164, 172eqtrd 2813 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1)))
17462, 139ltnled 10523 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
175140, 174mpbid 224 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
176 elfzle1 12661 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀)
177175, 176nsyl 138 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁))
178 difsn 4560 . . . . . . . . . . . . . . . . . . . . . 22 𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
179177, 178syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
180173, 179uneq12d 3990 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
181155, 180syl5eq 2825 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
182154, 181eqtrd 2813 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
183182imaeq2d 5720 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))))
184121eqcomd 2783 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑀}) = {((2nd ‘(1st𝑇))‘𝑀)})
185106, 184difeq12d 3951 . . . . . . . . . . . . . . . . 17 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
186151, 183, 1853eqtr3d 2821 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
187149, 186syl5eqr 2827 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
188187fneq2d 6227 . . . . . . . . . . . . . 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)})))
189148, 188mpbid 224 . . . . . . . . . . . . 13 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
190 eldifsn 4549 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)))
191190biimpri 220 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
192191ancoms 452 . . . . . . . . . . . . 13 ((𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
193 disjdif 4263 . . . . . . . . . . . . . 14 ({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅
194 fnconstg 6343 . . . . . . . . . . . . . . . 16 (0 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
19551, 194ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
196 fvun2 6530 . . . . . . . . . . . . . . 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))
197195, 196mp3an1 1521 . . . . . . . . . . . . . 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))
198193, 197mpanr1 693 . . . . . . . . . . . . 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))
199189, 192, 198syl2an 589 . . . . . . . . . . . 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))
200199anassrs 461 . . . . . . . . . . 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))
201131, 200eqtrd 2813 . . . . . . . . . 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))
20247, 110, 111, 111, 112, 113, 201ofval 7183 . . . . . . . . 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)))
203 fnconstg 6343 . . . . . . . . . . . . . . 15 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)))
20448, 203ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀))
205204, 133pm3.2i 464 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
206 imain 6219 . . . . . . . . . . . . . . 15 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
20757, 206syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
208 fzdisj 12685 . . . . . . . . . . . . . . . . 17 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
209140, 208syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
210209imaeq2d 5720 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
211210, 67syl6eq 2829 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
212207, 211eqtr3d 2815 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
213 fnun 6243 . . . . . . . . . . . . 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))))
214205, 212, 213sylancr 581 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
215153imaeq2d 5720 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
216 imaundi 5799 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
217215, 216syl6eq 2829 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
218217, 106eqtr3d 2815 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
219218fneq2d 6227 . . . . . . . . . . . 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)))
220214, 219mpbid 224 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
221220adantr 474 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
222 imaundi 5799 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀}))
223163imaeq2d 5720 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})))
224121uneq2d 3989 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀})))
225222, 223, 2243eqtr4a 2839 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}))
226225xpeq1d 5384 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}))
227 xpundir 5418 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1}))
228226, 227syl6eq 2829 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1})))
229228uneq1d 3988 . . . . . . . . . . . . . 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})))
230 un23 3994 . . . . . . . . . . . . . . 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}))
231230equncomi 3981 . . . . . . . . . . . . . 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})))
232229, 231syl6eq 2829 . . . . . . . . . . . . 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}))))
233232fveq1d 6448 . . . . . . . . . . . 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))
234233ad2antrr 716 . . . . . . . . . . 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))
235 fnconstg 6343 . . . . . . . . . . . . . . . 16 (1 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
23648, 235ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
237 fvun2 6530 . . . . . . . . . . . . . . 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))
238236, 237mp3an1 1521 . . . . . . . . . . . . . 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))
239193, 238mpanr1 693 . . . . . . . . . . . . 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))
240189, 192, 239syl2an 589 . . . . . . . . . . . 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))
241240anassrs 461 . . . . . . . . . . 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))
242234, 241eqtrd 2813 . . . . . . . . . 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))
24347, 221, 111, 111, 112, 113, 242ofval 7183 . . . . . . . . 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)))
244202, 243eqtr4d 2816 . . . . . . . 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))
245244an32s 642 . . . . . . 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))
246245anasss 460 . . . . . 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))
247 fveq2 6446 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
248247breq2d 4898 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
249248ifbid 4328 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
250249csbeq1d 3757 . . . . . . . . . . . . . . 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}))))
251 2fveq3 6451 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
252 2fveq3 6451 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
253252imaeq1d 5719 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
254253xpeq1d 5384 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
255252imaeq1d 5719 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
256255xpeq1d 5384 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
257254, 256uneq12d 3990 . . . . . . . . . . . . . . . . 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})))
258251, 257oveq12d 6940 . . . . . . . . . . . . . . . 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}))))
259258csbeq2dv 4216 . . . . . . . . . . . . . . 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}))))
260250, 259eqtrd 2813 . . . . . . . . . . . . . 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}))))
261260mpteq2dv 4980 . . . . . . . . . . . . 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})))))
262261eqeq2d 2787 . . . . . . . . . . . 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}))))))
263262, 3elrab2 3575 . . . . . . . . . . 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}))))))
264263simprbi 492 . . . . . . . . . 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})))))
2651, 264syl 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})))))
266 breq1 4889 . . . . . . . . . . . . 13 (𝑦 = (𝑀 − 1) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < (2nd𝑇)))
267 id 22 . . . . . . . . . . . . 13 (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1))
268266, 267ifbieq1d 4329 . . . . . . . . . . . 12 (𝑦 = (𝑀 − 1) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2nd𝑇), (𝑀 − 1), (𝑦 + 1)))
26925ltm1d 11310 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) − 1) < (2nd𝑇))
27062, 24, 25, 90, 269lelttrd 10534 . . . . . . . . . . . . . 14 (𝜑𝑀 < (2nd𝑇))
271137, 62, 25, 63, 270lttrd 10537 . . . . . . . . . . . . 13 (𝜑 → (𝑀 − 1) < (2nd𝑇))
272271iftrued 4314 . . . . . . . . . . . 12 (𝜑 → if((𝑀 − 1) < (2nd𝑇), (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1))
273268, 272sylan9eqr 2835 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1))
274273csbeq1d 3757 . . . . . . . . . 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 − 1) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
275 oveq2 6930 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
276275imaeq2d 5720 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 − 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))))
277276xpeq1d 5384 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 − 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
278277adantl 475 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
279 oveq1 6929 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
280279, 74sylan9eqr 2835 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
281280oveq1d 6937 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
282281imaeq2d 5720 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = (𝑀 − 1)) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
283282xpeq1d 5384 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))
284278, 283uneq12d 3990 . . . . . . . . . . . . 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})))
285284oveq2d 6938 . . . . . . . . . . . 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}))))
28679, 285csbied 3777 . . . . . . . . . . 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}))))
287286adantr 474 . . . . . . . . . 10 ((𝜑𝑦 = (𝑀 − 1)) → (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}))))
288274, 287eqtrd 2813 . . . . . . . . 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))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
289 1red 10377 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
29062, 26, 289, 91lesub1dd 10991 . . . . . . . . . 10 (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1))
291 elfz2nn0 12749 . . . . . . . . . 10 ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑀 − 1) ≤ (𝑁 − 1)))
29279, 29, 290, 291syl3anbrc 1400 . . . . . . . . 9 (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
293 ovexd 6956 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V)
294265, 288, 292, 293fvmptd 6548 . . . . . . . 8 (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
295294fveq1d 6448 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
296295adantr 474 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
297 breq1 4889 . . . . . . . . . . . . 13 (𝑦 = 𝑀 → (𝑦 < (2nd𝑇) ↔ 𝑀 < (2nd𝑇)))
298 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝑀𝑦 = 𝑀)
299297, 298ifbieq1d 4329 . . . . . . . . . . . 12 (𝑦 = 𝑀 → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if(𝑀 < (2nd𝑇), 𝑀, (𝑦 + 1)))
300270iftrued 4314 . . . . . . . . . . . 12 (𝜑 → if(𝑀 < (2nd𝑇), 𝑀, (𝑦 + 1)) = 𝑀)
301299, 300sylan9eqr 2835 . . . . . . . . . . 11 ((𝜑𝑦 = 𝑀) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
302301csbeq1d 3757 . . . . . . . . . 10 ((𝜑𝑦 = 𝑀) → 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}))))
303 oveq2 6930 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
304303imaeq2d 5720 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑀)))
305304xpeq1d 5384 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}))
306 oveq1 6929 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
307306oveq1d 6937 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
308307imaeq2d 5720 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
309308xpeq1d 5384 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
310305, 309uneq12d 3990 . . . . . . . . . . . . . 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})))
311310oveq2d 6938 . . . . . . . . . . . . 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}))))
312311adantl 475 . . . . . . . . . . . 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}))))
31340, 312csbied 3777 . . . . . . . . . . 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}))))
314313adantr 474 . . . . . . . . . 10 ((𝜑𝑦 = 𝑀) → 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}))))
315302, 314eqtrd 2813 . . . . . . . . 9 ((𝜑𝑦 = 𝑀) → 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}))))
31629nn0zd 11832 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℤ)
31725, 26, 289, 34lesub1dd 10991 . . . . . . . . . . . . 13 (𝜑 → ((2nd𝑇) − 1) ≤ (𝑁 − 1))
318 eluz2 11998 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ (ℤ‘((2nd𝑇) − 1)) ↔ (((2nd𝑇) − 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ ((2nd𝑇) − 1) ≤ (𝑁 − 1)))
31921, 316, 317, 318syl3anbrc 1400 . . . . . . . . . . . 12 (𝜑 → (𝑁 − 1) ∈ (ℤ‘((2nd𝑇) − 1)))
320 fzss2 12698 . . . . . . . . . . . 12 ((𝑁 − 1) ∈ (ℤ‘((2nd𝑇) − 1)) → (1...((2nd𝑇) − 1)) ⊆ (1...(𝑁 − 1)))
321319, 320syl 17 . . . . . . . . . . 11 (𝜑 → (1...((2nd𝑇) − 1)) ⊆ (1...(𝑁 − 1)))
322 fz1ssfz0 12754 . . . . . . . . . . 11 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
323321, 322syl6ss 3832 . . . . . . . . . 10 (𝜑 → (1...((2nd𝑇) − 1)) ⊆ (0...(𝑁 − 1)))
324323, 40sseldd 3821 . . . . . . . . 9 (𝜑𝑀 ∈ (0...(𝑁 − 1)))
325 ovexd 6956 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
326265, 315, 324, 325fvmptd 6548 . . . . . . . 8 (𝜑 → (𝐹𝑀) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
327326fveq1d 6448 . . . . . . 7 (𝜑 → ((𝐹𝑀)‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
328327adantr 474 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹𝑀)‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
329246, 296, 3283eqtr4d 2823 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹𝑀)‘𝑛))
330329expr 450 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹𝑀)‘𝑛)))
331330necon1d 2990 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) → 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
332 elmapi 8162 . . . . . . . . . . 11 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
33344, 332syl 17 . . . . . . . . . 10 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
334333, 42ffvelrnd 6624 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾))
335 elfzonn0 12832 . . . . . . . . 9 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
336334, 335syl 17 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
337336nn0red 11703 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℝ)
338337ltp1d 11308 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) < (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
339337, 338ltned 10512 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
340294fveq1d 6448 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)))
341 ovexd 6956 . . . . . . . . 9 (𝜑 → (1...𝑁) ∈ V)
342 eqidd 2778 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
343 fzss1 12697 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘1) → (𝑀...𝑁) ⊆ (1...𝑁))
34476, 343syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁))
345 eluzfz1 12665 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
34693, 345syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (𝑀...𝑁))
347 fnfvima 6769 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
348119, 344, 346, 347syl3anc 1439 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
349 fvun2 6530 . . . . . . . . . . . . 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)))
35050, 53, 349mp3an12 1524 . . . . . . . . . . . 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)))
35169, 348, 350syl2anc 579 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)))
35251fvconst2 6741 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
353348, 352syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
354351, 353eqtrd 2813 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
355354adantr 474 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
35646, 109, 341, 341, 112, 342, 355ofval 7183 . . . . . . . 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))
35742, 356mpdan 677 . . . . . . 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))
358336nn0cnd 11704 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℂ)
359358addid1d 10576 . . . . . . 7 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
360340, 357, 3593eqtrd 2817 . . . . . 6 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
361326fveq1d 6448 . . . . . . 7 (𝜑 → ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)))
362 fzss2 12698 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → (1...𝑀) ⊆ (1...𝑁))
36393, 362syl 17 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ⊆ (1...𝑁))
364 elfz1end 12688 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
36561, 364sylib 210 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (1...𝑀))
366 fnfvima 6769 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
367119, 363, 365, 366syl3anc 1439 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
368 fvun1 6529 . . . . . . . . . . . . 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)))
369204, 133, 368mp3an12 1524 . . . . . . . . . . . 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)))
370212, 367, 369syl2anc 579 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)))
37148fvconst2 6741 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
372367, 371syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
373370, 372eqtrd 2813 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
374373adantr 474 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
37546, 220, 341, 341, 112, 342, 374ofval 7183 . . . . . . . 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))
37642, 375mpdan 677 . . . . . . 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))
377361, 376eqtrd 2813 . . . . . 6 (𝜑 → ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
378339, 360, 3773netr4d 3045 . . . . 5 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)))
379 fveq2 6446 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)))
380 fveq2 6446 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)))
381379, 380neeq12d 3029 . . . . 5 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀))))
382378, 381syl5ibrcom 239 . . . 4 (𝜑 → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)))
383382adantr 474 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)))
384331, 383impbid 204 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
38542, 384riota5 6909 1 (𝜑 → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1601  wcel 2106  {cab 2762  wne 2968  {crab 3093  Vcvv 3397  csb 3750  cdif 3788  cun 3789  cin 3790  wss 3791  c0 4140  ifcif 4306  {csn 4397   class class class wbr 4886  cmpt 4965   × cxp 5353  ccnv 5354  cima 5358  Fun wfun 6129   Fn wfn 6130  wf 6131  ontowfo 6133  1-1-ontowf1o 6134  cfv 6135  crio 6882  (class class class)co 6922  𝑓 cof 7172  1st c1st 7443  2nd c2nd 7444  𝑚 cmap 8140  cc 10270  cr 10271  0cc0 10272  1c1 10273   + caddc 10275   < clt 10411  cle 10412  cmin 10606  cn 11374  0cn0 11642  cz 11728  cuz 11992  ...cfz 12643  ..^cfzo 12784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785
This theorem is referenced by:  poimirlem8  34027
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