Mathbox for Brendan Leahy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem6 Structured version   Visualization version   GIF version

Theorem poimirlem6 32585
 Description: Lemma for poimir 32612 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 3328 . . . . . . . . 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 2706 . . . . . . . 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 7089 . . . . . . 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 7090 . . . . . 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 6113 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
11 f1oeq1 6040 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3319 . . . . 5 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
139, 12sylib 207 . . . 4 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
14 f1of 6050 . . . 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 12241 . . . . . . . . 9 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
1816, 17syl 17 . . . . . . . 8 (𝜑 → (2nd𝑇) ∈ ℕ)
1918nnzd 11357 . . . . . . 7 (𝜑 → (2nd𝑇) ∈ ℤ)
20 peano2zm 11297 . . . . . . 7 ((2nd𝑇) ∈ ℤ → ((2nd𝑇) − 1) ∈ ℤ)
2119, 20syl 17 . . . . . 6 (𝜑 → ((2nd𝑇) − 1) ∈ ℤ)
22 poimir.0 . . . . . . 7 (𝜑𝑁 ∈ ℕ)
2322nnzd 11357 . . . . . 6 (𝜑𝑁 ∈ ℤ)
2421zred 11358 . . . . . . 7 (𝜑 → ((2nd𝑇) − 1) ∈ ℝ)
2518nnred 10912 . . . . . . 7 (𝜑 → (2nd𝑇) ∈ ℝ)
2622nnred 10912 . . . . . . 7 (𝜑𝑁 ∈ ℝ)
2725lem1d 10836 . . . . . . 7 (𝜑 → ((2nd𝑇) − 1) ≤ (2nd𝑇))
28 nnm1nn0 11211 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
2922, 28syl 17 . . . . . . . . 9 (𝜑 → (𝑁 − 1) ∈ ℕ0)
3029nn0red 11229 . . . . . . . 8 (𝜑 → (𝑁 − 1) ∈ ℝ)
31 elfzle2 12216 . . . . . . . . 9 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≤ (𝑁 − 1))
3216, 31syl 17 . . . . . . . 8 (𝜑 → (2nd𝑇) ≤ (𝑁 − 1))
3326lem1d 10836 . . . . . . . 8 (𝜑 → (𝑁 − 1) ≤ 𝑁)
3425, 30, 26, 32, 33letrd 10073 . . . . . . 7 (𝜑 → (2nd𝑇) ≤ 𝑁)
3524, 25, 26, 27, 34letrd 10073 . . . . . 6 (𝜑 → ((2nd𝑇) − 1) ≤ 𝑁)
36 eluz2 11569 . . . . . 6 (𝑁 ∈ (ℤ‘((2nd𝑇) − 1)) ↔ (((2nd𝑇) − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((2nd𝑇) − 1) ≤ 𝑁))
3721, 23, 35, 36syl3anbrc 1239 . . . . 5 (𝜑𝑁 ∈ (ℤ‘((2nd𝑇) − 1)))
38 fzss2 12252 . . . . 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 3569 . . 3 (𝜑𝑀 ∈ (1...𝑁))
4215, 41ffvelrnd 6268 . 2 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁))
43 xp1st 7089 . . . . . . . . . . . . 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 7766 . . . . . . . . . . . 12 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
4644, 45syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
4746adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
48 1ex 9914 . . . . . . . . . . . . . . 15 1 ∈ V
49 fnconstg 6006 . . . . . . . . . . . . . . 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 9913 . . . . . . . . . . . . . . 15 0 ∈ V
52 fnconstg 6006 . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
5655simprbi 479 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
5713, 56syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun (2nd ‘(1st𝑇)))
58 imain 5888 . . . . . . . . . . . . . . 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 12241 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (1...((2nd𝑇) − 1)) → 𝑀 ∈ ℕ)
6140, 60syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℕ)
6261nnred 10912 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℝ)
6362ltm1d 10835 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 1) < 𝑀)
64 fzdisj 12239 . . . . . . . . . . . . . . . . 17 ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
6665imaeq2d 5385 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
67 ima0 5400 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
6866, 67syl6eq 2660 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
6959, 68eqtr3d 2646 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = ∅)
70 fnun 5911 . . . . . . . . . . . . 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 694 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
7261nncnd 10913 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℂ)
73 npcan1 10334 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
75 nnuz 11599 . . . . . . . . . . . . . . . . . . . 20 ℕ = (ℤ‘1)
7661, 75syl6eleq 2698 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘1))
7774, 76eqeltrd 2688 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘1))
78 nnm1nn0 11211 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
7961, 78syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑀 − 1) ∈ ℕ0)
8079nn0zd 11356 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 − 1) ∈ ℤ)
81 uzid 11578 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
8280, 81syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
83 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ‘(𝑀 − 1)))
86 uzss 11584 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘(𝑀 − 1)) → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
8785, 86syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ𝑀) ⊆ (ℤ‘(𝑀 − 1)))
8861nnzd 11357 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℤ)
89 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ (1...((2nd𝑇) − 1)) → 𝑀 ≤ ((2nd𝑇) − 1))
9040, 89syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ≤ ((2nd𝑇) − 1))
9162, 24, 26, 90, 35letrd 10073 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀𝑁)
92 eluz2 11569 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
9388, 23, 91, 92syl3anbrc 1239 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ𝑀))
9487, 93sseldd 3569 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘(𝑀 − 1)))
95 fzsplit2 12237 . . . . . . . . . . . . . . . . . 18 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
9677, 94, 95syl2anc 691 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
9774oveq1d 6564 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
9897uneq2d 3729 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
9996, 98eqtrd 2644 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
10099imaeq2d 5385 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
101 imaundi 5464 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
102100, 101syl6eq 2660 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))))
103 f1ofo 6057 . . . . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . . . 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 2646 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁))) = (1...𝑁))
108107fneq2d 5896 . . . . . . . . . . . 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 221 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
110109adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
111 ovex 6577 . . . . . . . . . . 11 (1...𝑁) ∈ V
112111a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → (1...𝑁) ∈ V)
113 inidm 3784 . . . . . . . . . 10 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
114 eqidd 2611 . . . . . . . . . 10 (((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
115 imaundi 5464 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
116 fzpred 12259 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
11793, 116syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
118117imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ((2nd ‘(1st𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))))
119 f1ofn 6051 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
12013, 119syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
121 fnsnfv 6168 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
122120, 41, 121syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {((2nd ‘(1st𝑇))‘𝑀)} = ((2nd ‘(1st𝑇)) “ {𝑀}))
123122uneq1d 3728 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ {𝑀}) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
124115, 118, 1233eqtr4a 2670 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) = ({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
125124xpeq1d 5062 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}))
126 xpundir 5095 . . . . . . . . . . . . . . . 16 (({((2nd ‘(1st𝑇))‘𝑀)} ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
127125, 126syl6eq 2660 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘𝑀)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
128127uneq2d 3729 . . . . . . . . . . . . . 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}))))
129 un12 3733 . . . . . . . . . . . . . 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})))
130128, 129syl6eq 2660 . . . . . . . . . . . . 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}))))
131130fveq1d 6105 . . . . . . . . . . . 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))
132131ad2antrr 758 . . . . . . . . . . 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))
133 fnconstg 6006 . . . . . . . . . . . . . . . . 17 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
13451, 133ax-mp 5 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))
13550, 134pm3.2i 470 . . . . . . . . . . . . . . 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)))
136 imain 5888 . . . . . . . . . . . . . . . . 17 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
13757, 136syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
13879nn0red 11229 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 − 1) ∈ ℝ)
139 peano2re 10088 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
14062, 139syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 + 1) ∈ ℝ)
14162ltp1d 10833 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 < (𝑀 + 1))
142138, 62, 140, 63, 141lttrd 10077 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 − 1) < (𝑀 + 1))
143 fzdisj 12239 . . . . . . . . . . . . . . . . . . 19 ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
144142, 143syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
145144imaeq2d 5385 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
146145, 67syl6eq 2660 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅)
147137, 146eqtr3d 2646 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
148 fnun 5911 . . . . . . . . . . . . . . 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))))
149135, 147, 148sylancr 694 . . . . . . . . . . . . . 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))))
150 imaundi 5464 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
151 imadif 5887 . . . . . . . . . . . . . . . . . 18 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
15257, 151syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})))
153 fzsplit 12238 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
15441, 153syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
155154difeq1d 3689 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}))
156 difundir 3839 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀}))
157 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
15877, 85, 157syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
15974oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀))
160 fzsn 12254 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
16188, 160syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑀...𝑀) = {𝑀})
162159, 161eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀})
163162uneq2d 3729 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀}))
164158, 163eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀}))
165164difeq1d 3689 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}))
166 difun2 4000 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀})
167138, 62ltnled 10063 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1)))
16863, 167mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1))
169 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1))
170168, 169nsyl 134 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1)))
171 difsn 4269 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
172170, 171syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
173166, 172syl5eq 2656 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1)))
174165, 173eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1)))
17562, 140ltnled 10063 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
176141, 175mpbid 221 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
177 elfzle1 12215 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀)
178176, 177nsyl 134 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁))
179 difsn 4269 . . . . . . . . . . . . . . . . . . . . . 22 𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
180178, 179syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
181174, 180uneq12d 3730 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
182156, 181syl5eq 2656 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
183155, 182eqtrd 2644 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
184183imaeq2d 5385 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))))
185122eqcomd 2616 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑀}) = {((2nd ‘(1st𝑇))‘𝑀)})
186106, 185difeq12d 3691 . . . . . . . . . . . . . . . . 17 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
187152, 184, 1863eqtr3d 2652 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
188150, 187syl5eqr 2658 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
189188fneq2d 5896 . . . . . . . . . . . . . 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)})))
190149, 189mpbid 221 . . . . . . . . . . . . 13 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
191 eldifsn 4260 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)))
192191biimpri 217 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
193192ancoms 468 . . . . . . . . . . . . 13 ((𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)}))
194 disjdif 3992 . . . . . . . . . . . . . 14 ({((2nd ‘(1st𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑀)})) = ∅
195 fnconstg 6006 . . . . . . . . . . . . . . . 16 (0 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
19651, 195ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
197 fvun2 6180 . . . . . . . . . . . . . . 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))
198196, 197mp3an1 1403 . . . . . . . . . . . . . 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))
199194, 198mpanr1 715 . . . . . . . . . . . . 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))
200190, 193, 199syl2an 493 . . . . . . . . . . . 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))
201200anassrs 678 . . . . . . . . . . 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))
202132, 201eqtrd 2644 . . . . . . . . . 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))
20347, 110, 112, 112, 113, 114, 202ofval 6804 . . . . . . . . 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)))
204 fnconstg 6006 . . . . . . . . . . . . . . 15 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)))
20548, 204ax-mp 5 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀))
206205, 134pm3.2i 470 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
207 imain 5888 . . . . . . . . . . . . . . 15 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
20857, 207syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
209 fzdisj 12239 . . . . . . . . . . . . . . . . 17 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
210141, 209syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
211210imaeq2d 5385 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
212211, 67syl6eq 2660 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
213208, 212eqtr3d 2646 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
214 fnun 5911 . . . . . . . . . . . . 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))))
215206, 213, 214sylancr 694 . . . . . . . . . . . 12 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
216154imaeq2d 5385 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
217 imaundi 5464 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
218216, 217syl6eq 2660 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
219218, 106eqtr3d 2646 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
220219fneq2d 5896 . . . . . . . . . . . 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)))
221215, 220mpbid 221 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
222221adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
223 imaundi 5464 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀}))
224164imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})))
225122uneq2d 3729 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑀})))
226223, 224, 2253eqtr4a 2670 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}))
227226xpeq1d 5062 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}))
228 xpundir 5095 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑀)}) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1}))
229227, 228syl6eq 2660 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑀)} × {1})))
230229uneq1d 3728 . . . . . . . . . . . . . 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})))
231 un23 3734 . . . . . . . . . . . . . . 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}))
232231equncomi 3721 . . . . . . . . . . . . . 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})))
233230, 232syl6eq 2660 . . . . . . . . . . . . 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}))))
234233fveq1d 6105 . . . . . . . . . . . 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))
235234ad2antrr 758 . . . . . . . . . . 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))
236 fnconstg 6006 . . . . . . . . . . . . . . . 16 (1 ∈ V → ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)})
23748, 236ax-mp 5 . . . . . . . . . . . . . . 15 ({((2nd ‘(1st𝑇))‘𝑀)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑀)}
238 fvun2 6180 . . . . . . . . . . . . . . 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))
239237, 238mp3an1 1403 . . . . . . . . . . . . . 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))
240194, 239mpanr1 715 . . . . . . . . . . . . 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))
241190, 193, 240syl2an 493 . . . . . . . . . . . 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))
242241anassrs 678 . . . . . . . . . . 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))
243235, 242eqtrd 2644 . . . . . . . . . 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))
24447, 222, 112, 112, 113, 114, 243ofval 6804 . . . . . . . . 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)))
245203, 244eqtr4d 2647 . . . . . . . 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))
246245an32s 842 . . . . . . 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))
247246anasss 677 . . . . . 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))
248 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
249248breq2d 4595 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
250249ifbid 4058 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
251250csbeq1d 3506 . . . . . . . . . . . . . . 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}))))
252 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
253252fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
254252fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
255254imaeq1d 5384 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
256255xpeq1d 5062 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
257254imaeq1d 5384 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
258257xpeq1d 5062 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
259256, 258uneq12d 3730 . . . . . . . . . . . . . . . . 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})))
260253, 259oveq12d 6567 . . . . . . . . . . . . . . . 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}))))
261260csbeq2dv 3944 . . . . . . . . . . . . . . 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}))))
262251, 261eqtrd 2644 . . . . . . . . . . . . . 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}))))
263262mpteq2dv 4673 . . . . . . . . . . . . 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})))))
264263eqeq2d 2620 . . . . . . . . . . . 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}))))))
265264, 3elrab2 3333 . . . . . . . . . . 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}))))))
266265simprbi 479 . . . . . . . . . 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})))))
2671, 266syl 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})))))
268 breq1 4586 . . . . . . . . . . . . 13 (𝑦 = (𝑀 − 1) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < (2nd𝑇)))
269 id 22 . . . . . . . . . . . . 13 (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1))
270268, 269ifbieq1d 4059 . . . . . . . . . . . 12 (𝑦 = (𝑀 − 1) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2nd𝑇), (𝑀 − 1), (𝑦 + 1)))
27125ltm1d 10835 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) − 1) < (2nd𝑇))
27262, 24, 25, 90, 271lelttrd 10074 . . . . . . . . . . . . . 14 (𝜑𝑀 < (2nd𝑇))
273138, 62, 25, 63, 272lttrd 10077 . . . . . . . . . . . . 13 (𝜑 → (𝑀 − 1) < (2nd𝑇))
274273iftrued 4044 . . . . . . . . . . . 12 (𝜑 → if((𝑀 − 1) < (2nd𝑇), (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1))
275270, 274sylan9eqr 2666 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1))
276275csbeq1d 3506 . . . . . . . . . 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}))))
277 oveq2 6557 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
278277imaeq2d 5385 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 − 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))))
279278xpeq1d 5062 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 − 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
280279adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}))
281 oveq1 6556 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
282281, 74sylan9eqr 2666 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
283282oveq1d 6564 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
284283imaeq2d 5385 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = (𝑀 − 1)) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
285284xpeq1d 5062 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))
286280, 285uneq12d 3730 . . . . . . . . . . . . 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})))
287286oveq2d 6565 . . . . . . . . . . . 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}))))
28879, 287csbied 3526 . . . . . . . . . . 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}))))
289288adantr 480 . . . . . . . . . 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}))))
290276, 289eqtrd 2644 . . . . . . . . 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}))))
291 1red 9934 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
29262, 26, 291, 91lesub1dd 10522 . . . . . . . . . 10 (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1))
293 elfz2nn0 12300 . . . . . . . . . 10 ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑀 − 1) ≤ (𝑁 − 1)))
29479, 29, 292, 293syl3anbrc 1239 . . . . . . . . 9 (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
295 ovex 6577 . . . . . . . . . 10 ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V
296295a1i 11 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V)
297267, 290, 294, 296fvmptd 6197 . . . . . . . 8 (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))))
298297fveq1d 6105 . . . . . . 7 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
299298adantr 480 . . . . . 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))
300 breq1 4586 . . . . . . . . . . . . 13 (𝑦 = 𝑀 → (𝑦 < (2nd𝑇) ↔ 𝑀 < (2nd𝑇)))
301 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝑀𝑦 = 𝑀)
302300, 301ifbieq1d 4059 . . . . . . . . . . . 12 (𝑦 = 𝑀 → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if(𝑀 < (2nd𝑇), 𝑀, (𝑦 + 1)))
303272iftrued 4044 . . . . . . . . . . . 12 (𝜑 → if(𝑀 < (2nd𝑇), 𝑀, (𝑦 + 1)) = 𝑀)
304302, 303sylan9eqr 2666 . . . . . . . . . . 11 ((𝜑𝑦 = 𝑀) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
305304csbeq1d 3506 . . . . . . . . . 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}))))
306 oveq2 6557 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
307306imaeq2d 5385 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑀)))
308307xpeq1d 5062 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}))
309 oveq1 6556 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
310309oveq1d 6564 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
311310imaeq2d 5385 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
312311xpeq1d 5062 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
313308, 312uneq12d 3730 . . . . . . . . . . . . . 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})))
314313oveq2d 6565 . . . . . . . . . . . . 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}))))
315314adantl 481 . . . . . . . . . . . 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}))))
31640, 315csbied 3526 . . . . . . . . . . 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}))))
317316adantr 480 . . . . . . . . . 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}))))
318305, 317eqtrd 2644 . . . . . . . . 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}))))
31929nn0zd 11356 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℤ)
32025, 26, 291, 34lesub1dd 10522 . . . . . . . . . . . . 13 (𝜑 → ((2nd𝑇) − 1) ≤ (𝑁 − 1))
321 eluz2 11569 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ (ℤ‘((2nd𝑇) − 1)) ↔ (((2nd𝑇) − 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ ((2nd𝑇) − 1) ≤ (𝑁 − 1)))
32221, 319, 320, 321syl3anbrc 1239 . . . . . . . . . . . 12 (𝜑 → (𝑁 − 1) ∈ (ℤ‘((2nd𝑇) − 1)))
323 fzss2 12252 . . . . . . . . . . . 12 ((𝑁 − 1) ∈ (ℤ‘((2nd𝑇) − 1)) → (1...((2nd𝑇) − 1)) ⊆ (1...(𝑁 − 1)))
324322, 323syl 17 . . . . . . . . . . 11 (𝜑 → (1...((2nd𝑇) − 1)) ⊆ (1...(𝑁 − 1)))
325 1eluzge0 11608 . . . . . . . . . . . 12 1 ∈ (ℤ‘0)
326 fzss1 12251 . . . . . . . . . . . 12 (1 ∈ (ℤ‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
327325, 326ax-mp 5 . . . . . . . . . . 11 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
328324, 327syl6ss 3580 . . . . . . . . . 10 (𝜑 → (1...((2nd𝑇) − 1)) ⊆ (0...(𝑁 − 1)))
329328, 40sseldd 3569 . . . . . . . . 9 (𝜑𝑀 ∈ (0...(𝑁 − 1)))
330 ovex 6577 . . . . . . . . . 10 ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V
331330a1i 11 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
332267, 318, 329, 331fvmptd 6197 . . . . . . . 8 (𝜑 → (𝐹𝑀) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
333332fveq1d 6105 . . . . . . 7 (𝜑 → ((𝐹𝑀)‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
334333adantr 480 . . . . . 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))
335247, 299, 3343eqtr4d 2654 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹𝑀)‘𝑛))
336335expr 641 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹𝑀)‘𝑛)))
337336necon1d 2804 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) → 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
338 elmapi 7765 . . . . . . . . . . 11 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
33944, 338syl 17 . . . . . . . . . 10 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
340339, 42ffvelrnd 6268 . . . . . . . . 9 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾))
341 elfzonn0 12380 . . . . . . . . 9 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
342340, 341syl 17 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℕ0)
343342nn0red 11229 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℝ)
344343ltp1d 10833 . . . . . . 7 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) < (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
345343, 344ltned 10052 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
346297fveq1d 6105 . . . . . . 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)))
347111a1i 11 . . . . . . . . 9 (𝜑 → (1...𝑁) ∈ V)
348 eqidd 2611 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
349 fzss1 12251 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘1) → (𝑀...𝑁) ⊆ (1...𝑁))
35076, 349syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁))
351 eluzfz1 12219 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
35293, 351syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (𝑀...𝑁))
353 fnfvima 6400 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
354120, 350, 352, 353syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)))
355 fvun2 6180 . . . . . . . . . . . . 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)))
35650, 53, 355mp3an12 1406 . . . . . . . . . . . 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)))
35769, 354, 356syl2anc 691 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)))
35851fvconst2 6374 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
359354, 358syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
360357, 359eqtrd 2644 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
361360adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 0)
36246, 109, 347, 347, 113, 348, 361ofval 6804 . . . . . . . 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))
36342, 362mpdan 699 . . . . . . 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))
364342nn0cnd 11230 . . . . . . . 8 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) ∈ ℂ)
365364addid1d 10115 . . . . . . 7 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
366346, 363, 3653eqtrd 2648 . . . . . 6 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)))
367332fveq1d 6105 . . . . . . 7 (𝜑 → ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑀)))
368 fzss2 12252 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → (1...𝑀) ⊆ (1...𝑁))
36993, 368syl 17 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ⊆ (1...𝑁))
370 elfz1end 12242 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
37161, 370sylib 207 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (1...𝑀))
372 fnfvima 6400 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
373120, 369, 371, 372syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → ((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))
374 fvun1 6179 . . . . . . . . . . . . 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)))
375205, 134, 374mp3an12 1406 . . . . . . . . . . . 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)))
376213, 373, 375syl2anc 691 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)))
37748fvconst2 6374 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇))‘𝑀) ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
378373, 377syl 17 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
379376, 378eqtrd 2644 . . . . . . . . . 10 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
380379adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑀)) = 1)
38146, 221, 347, 347, 113, 348, 380ofval 6804 . . . . . . . 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))
38242, 381mpdan 699 . . . . . . 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))
383367, 382eqtrd 2644 . . . . . 6 (𝜑 → ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑀)) + 1))
384345, 366, 3833netr4d 2859 . . . . 5 (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)))
385 fveq2 6103 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)))
386 fveq2 6103 . . . . . 6 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀)))
387385, 386neeq12d 2843 . . . . 5 (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘((2nd ‘(1st𝑇))‘𝑀)) ≠ ((𝐹𝑀)‘((2nd ‘(1st𝑇))‘𝑀))))
388384, 387syl5ibrcom 236 . . . 4 (𝜑 → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)))
389388adantr 480 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 = ((2nd ‘(1st𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)))
390337, 389impbid 201 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ 𝑛 = ((2nd ‘(1st𝑇))‘𝑀)))
39142, 390riota5 6536 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 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  {crab 2900  Vcvv 3173  ⦋csb 3499   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   ∘𝑓 cof 6793  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335 This theorem is referenced by:  poimirlem8  32587
 Copyright terms: Public domain W3C validator