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Theorem subfacp1lem6 33779
Description: Lemma for subfacp1 33780. By induction, we cut up the set of all derangements on 𝑁 + 1 according to the 𝑁 possible values of (𝑓‘1) (since (𝑓‘1) ≠ 1), and for each set for fixed 𝑀 = (𝑓‘1), the subset of derangements with (𝑓𝑀) = 1 has size 𝑆(𝑁 − 1) (by subfacp1lem3 33776), while the subset with (𝑓𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 33778). Adding it all up yields the desired equation 𝑁(𝑆(𝑁) + 𝑆(𝑁 − 1)) for the number of derangements on 𝑁 + 1. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
derang.d 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
subfac.n 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
subfacp1lem.a 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}
Assertion
Ref Expression
subfacp1lem6 (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
Distinct variable groups:   𝑓,𝑛,𝑥,𝑦,𝐴   𝑓,𝑁,𝑛,𝑥,𝑦   𝐷,𝑛   𝑆,𝑛,𝑥,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑓)   𝑆(𝑓)

Proof of Theorem subfacp1lem6
Dummy variables 𝑔 𝑚 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano2nn 12165 . . . . 5 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
21nnnn0d 12473 . . . 4 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ0)
3 derang.d . . . . 5 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
4 subfac.n . . . . 5 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
53, 4subfacval 33767 . . . 4 ((𝑁 + 1) ∈ ℕ0 → (𝑆‘(𝑁 + 1)) = (𝐷‘(1...(𝑁 + 1))))
62, 5syl 17 . . 3 (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝐷‘(1...(𝑁 + 1))))
7 fzfid 13878 . . . . 5 (𝑁 ∈ ℕ → (1...(𝑁 + 1)) ∈ Fin)
83derangval 33761 . . . . 5 ((1...(𝑁 + 1)) ∈ Fin → (𝐷‘(1...(𝑁 + 1))) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}))
97, 8syl 17 . . . 4 (𝑁 ∈ ℕ → (𝐷‘(1...(𝑁 + 1))) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}))
10 subfacp1lem.a . . . . 5 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}
1110fveq2i 6845 . . . 4 (♯‘𝐴) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)})
129, 11eqtr4di 2794 . . 3 (𝑁 ∈ ℕ → (𝐷‘(1...(𝑁 + 1))) = (♯‘𝐴))
13 nnuz 12806 . . . . . . . . . . 11 ℕ = (ℤ‘1)
141, 13eleqtrdi 2848 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ‘1))
15 eluzfz1 13448 . . . . . . . . . 10 ((𝑁 + 1) ∈ (ℤ‘1) → 1 ∈ (1...(𝑁 + 1)))
1614, 15syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ → 1 ∈ (1...(𝑁 + 1)))
17 f1of 6784 . . . . . . . . . 10 (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
1817adantr 481 . . . . . . . . 9 ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦) → 𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
19 ffvelcdm 7032 . . . . . . . . . 10 ((𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ 1 ∈ (1...(𝑁 + 1))) → (𝑓‘1) ∈ (1...(𝑁 + 1)))
2019expcom 414 . . . . . . . . 9 (1 ∈ (1...(𝑁 + 1)) → (𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) → (𝑓‘1) ∈ (1...(𝑁 + 1))))
2116, 18, 20syl2im 40 . . . . . . . 8 (𝑁 ∈ ℕ → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦) → (𝑓‘1) ∈ (1...(𝑁 + 1))))
2221ss2abdv 4020 . . . . . . 7 (𝑁 ∈ ℕ → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ (𝑓‘1) ∈ (1...(𝑁 + 1))})
23 fveq1 6841 . . . . . . . . 9 (𝑔 = 𝑓 → (𝑔‘1) = (𝑓‘1))
2423eleq1d 2822 . . . . . . . 8 (𝑔 = 𝑓 → ((𝑔‘1) ∈ (1...(𝑁 + 1)) ↔ (𝑓‘1) ∈ (1...(𝑁 + 1))))
2524cbvabv 2809 . . . . . . 7 {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} = {𝑓 ∣ (𝑓‘1) ∈ (1...(𝑁 + 1))}
2622, 10, 253sstr4g 3989 . . . . . 6 (𝑁 ∈ ℕ → 𝐴 ⊆ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})
27 ssabral 4019 . . . . . 6 (𝐴 ⊆ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} ↔ ∀𝑔𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1)))
2826, 27sylib 217 . . . . 5 (𝑁 ∈ ℕ → ∀𝑔𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1)))
29 rabid2 3436 . . . . 5 (𝐴 = {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} ↔ ∀𝑔𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1)))
3028, 29sylibr 233 . . . 4 (𝑁 ∈ ℕ → 𝐴 = {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})
3130fveq2d 6846 . . 3 (𝑁 ∈ ℕ → (♯‘𝐴) = (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}))
326, 12, 313eqtrd 2780 . 2 (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}))
33 elfz1end 13471 . . . 4 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
341, 33sylib 217 . . 3 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
35 eleq1 2825 . . . . . . 7 (𝑥 = 1 → (𝑥 ∈ (1...(𝑁 + 1)) ↔ 1 ∈ (1...(𝑁 + 1))))
36 oveq2 7365 . . . . . . . . . . . . 13 (𝑥 = 1 → (1...𝑥) = (1...1))
37 1z 12533 . . . . . . . . . . . . . 14 1 ∈ ℤ
38 fzsn 13483 . . . . . . . . . . . . . 14 (1 ∈ ℤ → (1...1) = {1})
3937, 38ax-mp 5 . . . . . . . . . . . . 13 (1...1) = {1}
4036, 39eqtrdi 2792 . . . . . . . . . . . 12 (𝑥 = 1 → (1...𝑥) = {1})
4140eleq2d 2823 . . . . . . . . . . 11 (𝑥 = 1 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ {1}))
42 fvex 6855 . . . . . . . . . . . 12 (𝑔‘1) ∈ V
4342elsn 4601 . . . . . . . . . . 11 ((𝑔‘1) ∈ {1} ↔ (𝑔‘1) = 1)
4441, 43bitrdi 286 . . . . . . . . . 10 (𝑥 = 1 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) = 1))
4544rabbidv 3415 . . . . . . . . 9 (𝑥 = 1 → {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔𝐴 ∣ (𝑔‘1) = 1})
4645fveq2d 6846 . . . . . . . 8 (𝑥 = 1 → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔𝐴 ∣ (𝑔‘1) = 1}))
47 oveq1 7364 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 − 1) = (1 − 1))
48 1m1e0 12225 . . . . . . . . . 10 (1 − 1) = 0
4947, 48eqtrdi 2792 . . . . . . . . 9 (𝑥 = 1 → (𝑥 − 1) = 0)
5049oveq1d 7372 . . . . . . . 8 (𝑥 = 1 → ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (0 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
5146, 50eqeq12d 2752 . . . . . . 7 (𝑥 = 1 → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
5235, 51imbi12d 344 . . . . . 6 (𝑥 = 1 → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) ↔ (1 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
5352imbi2d 340 . . . . 5 (𝑥 = 1 → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → (1 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))))
54 eleq1 2825 . . . . . . 7 (𝑥 = 𝑚 → (𝑥 ∈ (1...(𝑁 + 1)) ↔ 𝑚 ∈ (1...(𝑁 + 1))))
55 oveq2 7365 . . . . . . . . . . 11 (𝑥 = 𝑚 → (1...𝑥) = (1...𝑚))
5655eleq2d 2823 . . . . . . . . . 10 (𝑥 = 𝑚 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...𝑚)))
5756rabbidv 3415 . . . . . . . . 9 (𝑥 = 𝑚 → {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)})
5857fveq2d 6846 . . . . . . . 8 (𝑥 = 𝑚 → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}))
59 oveq1 7364 . . . . . . . . 9 (𝑥 = 𝑚 → (𝑥 − 1) = (𝑚 − 1))
6059oveq1d 7372 . . . . . . . 8 (𝑥 = 𝑚 → ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
6158, 60eqeq12d 2752 . . . . . . 7 (𝑥 = 𝑚 → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
6254, 61imbi12d 344 . . . . . 6 (𝑥 = 𝑚 → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) ↔ (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
6362imbi2d 340 . . . . 5 (𝑥 = 𝑚 → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))))
64 eleq1 2825 . . . . . . 7 (𝑥 = (𝑚 + 1) → (𝑥 ∈ (1...(𝑁 + 1)) ↔ (𝑚 + 1) ∈ (1...(𝑁 + 1))))
65 oveq2 7365 . . . . . . . . . . 11 (𝑥 = (𝑚 + 1) → (1...𝑥) = (1...(𝑚 + 1)))
6665eleq2d 2823 . . . . . . . . . 10 (𝑥 = (𝑚 + 1) → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...(𝑚 + 1))))
6766rabbidv 3415 . . . . . . . . 9 (𝑥 = (𝑚 + 1) → {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))})
6867fveq2d 6846 . . . . . . . 8 (𝑥 = (𝑚 + 1) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}))
69 oveq1 7364 . . . . . . . . 9 (𝑥 = (𝑚 + 1) → (𝑥 − 1) = ((𝑚 + 1) − 1))
7069oveq1d 7372 . . . . . . . 8 (𝑥 = (𝑚 + 1) → ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
7168, 70eqeq12d 2752 . . . . . . 7 (𝑥 = (𝑚 + 1) → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
7264, 71imbi12d 344 . . . . . 6 (𝑥 = (𝑚 + 1) → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) ↔ ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
7372imbi2d 340 . . . . 5 (𝑥 = (𝑚 + 1) → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))))
74 eleq1 2825 . . . . . . 7 (𝑥 = (𝑁 + 1) → (𝑥 ∈ (1...(𝑁 + 1)) ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1))))
75 oveq2 7365 . . . . . . . . . . 11 (𝑥 = (𝑁 + 1) → (1...𝑥) = (1...(𝑁 + 1)))
7675eleq2d 2823 . . . . . . . . . 10 (𝑥 = (𝑁 + 1) → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...(𝑁 + 1))))
7776rabbidv 3415 . . . . . . . . 9 (𝑥 = (𝑁 + 1) → {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})
7877fveq2d 6846 . . . . . . . 8 (𝑥 = (𝑁 + 1) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}))
79 oveq1 7364 . . . . . . . . 9 (𝑥 = (𝑁 + 1) → (𝑥 − 1) = ((𝑁 + 1) − 1))
8079oveq1d 7372 . . . . . . . 8 (𝑥 = (𝑁 + 1) → ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
8178, 80eqeq12d 2752 . . . . . . 7 (𝑥 = (𝑁 + 1) → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
8274, 81imbi12d 344 . . . . . 6 (𝑥 = (𝑁 + 1) → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) ↔ ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
8382imbi2d 340 . . . . 5 (𝑥 = (𝑁 + 1) → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))))
84 hash0 14267 . . . . . . 7 (♯‘∅) = 0
85 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑦 = 1 → (𝑓𝑦) = (𝑓‘1))
86 id 22 . . . . . . . . . . . . . . . 16 (𝑦 = 1 → 𝑦 = 1)
8785, 86neeq12d 3005 . . . . . . . . . . . . . . 15 (𝑦 = 1 → ((𝑓𝑦) ≠ 𝑦 ↔ (𝑓‘1) ≠ 1))
8887rspcv 3577 . . . . . . . . . . . . . 14 (1 ∈ (1...(𝑁 + 1)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦 → (𝑓‘1) ≠ 1))
8916, 88syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦 → (𝑓‘1) ≠ 1))
9089adantld 491 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦) → (𝑓‘1) ≠ 1))
9190ss2abdv 4020 . . . . . . . . . . 11 (𝑁 ∈ ℕ → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ (𝑓‘1) ≠ 1})
92 df-ne 2944 . . . . . . . . . . . . 13 ((𝑔‘1) ≠ 1 ↔ ¬ (𝑔‘1) = 1)
9323neeq1d 3003 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → ((𝑔‘1) ≠ 1 ↔ (𝑓‘1) ≠ 1))
9492, 93bitr3id 284 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (¬ (𝑔‘1) = 1 ↔ (𝑓‘1) ≠ 1))
9594cbvabv 2809 . . . . . . . . . . 11 {𝑔 ∣ ¬ (𝑔‘1) = 1} = {𝑓 ∣ (𝑓‘1) ≠ 1}
9691, 10, 953sstr4g 3989 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝐴 ⊆ {𝑔 ∣ ¬ (𝑔‘1) = 1})
97 ssabral 4019 . . . . . . . . . 10 (𝐴 ⊆ {𝑔 ∣ ¬ (𝑔‘1) = 1} ↔ ∀𝑔𝐴 ¬ (𝑔‘1) = 1)
9896, 97sylib 217 . . . . . . . . 9 (𝑁 ∈ ℕ → ∀𝑔𝐴 ¬ (𝑔‘1) = 1)
99 rabeq0 4344 . . . . . . . . 9 ({𝑔𝐴 ∣ (𝑔‘1) = 1} = ∅ ↔ ∀𝑔𝐴 ¬ (𝑔‘1) = 1)
10098, 99sylibr 233 . . . . . . . 8 (𝑁 ∈ ℕ → {𝑔𝐴 ∣ (𝑔‘1) = 1} = ∅)
101100fveq2d 6846 . . . . . . 7 (𝑁 ∈ ℕ → (♯‘{𝑔𝐴 ∣ (𝑔‘1) = 1}) = (♯‘∅))
102 nnnn0 12420 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
1033, 4subfacf 33769 . . . . . . . . . . . 12 𝑆:ℕ0⟶ℕ0
104103ffvelcdmi 7034 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑆𝑁) ∈ ℕ0)
105102, 104syl 17 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑆𝑁) ∈ ℕ0)
106 nnm1nn0 12454 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
107103ffvelcdmi 7034 . . . . . . . . . . 11 ((𝑁 − 1) ∈ ℕ0 → (𝑆‘(𝑁 − 1)) ∈ ℕ0)
108106, 107syl 17 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑆‘(𝑁 − 1)) ∈ ℕ0)
109105, 108nn0addcld 12477 . . . . . . . . 9 (𝑁 ∈ ℕ → ((𝑆𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℕ0)
110109nn0cnd 12475 . . . . . . . 8 (𝑁 ∈ ℕ → ((𝑆𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℂ)
111110mul02d 11353 . . . . . . 7 (𝑁 ∈ ℕ → (0 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = 0)
11284, 101, 1113eqtr4a 2802 . . . . . 6 (𝑁 ∈ ℕ → (♯‘{𝑔𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
113112a1d 25 . . . . 5 (𝑁 ∈ ℕ → (1 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
114 simplr 767 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℕ)
115114, 13eleqtrdi 2848 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (ℤ‘1))
116 peano2fzr 13454 . . . . . . . . . . 11 ((𝑚 ∈ (ℤ‘1) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (1...(𝑁 + 1)))
117115, 116sylancom 588 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (1...(𝑁 + 1)))
118117ex 413 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → 𝑚 ∈ (1...(𝑁 + 1))))
119118imim1d 82 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
120 oveq1 7364 . . . . . . . . . . 11 ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = (((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
121 elfzp1 13491 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (ℤ‘1) → ((𝑔‘1) ∈ (1...(𝑚 + 1)) ↔ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))))
122115, 121syl 17 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑔‘1) ∈ (1...(𝑚 + 1)) ↔ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))))
123122rabbidv 3415 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))} = {𝑔𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))})
124 unrab 4265 . . . . . . . . . . . . . . 15 ({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = {𝑔𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))}
125123, 124eqtr4di 2794 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))} = ({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))
126125fveq2d 6846 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (♯‘({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
127 fzfi 13877 . . . . . . . . . . . . . . . . 17 (1...(𝑁 + 1)) ∈ Fin
128 deranglem 33760 . . . . . . . . . . . . . . . . 17 ((1...(𝑁 + 1)) ∈ Fin → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)} ∈ Fin)
129127, 128ax-mp 5 . . . . . . . . . . . . . . . 16 {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)} ∈ Fin
13010, 129eqeltri 2834 . . . . . . . . . . . . . . 15 𝐴 ∈ Fin
131 ssrab2 4037 . . . . . . . . . . . . . . 15 {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ⊆ 𝐴
132 ssfi 9117 . . . . . . . . . . . . . . 15 ((𝐴 ∈ Fin ∧ {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ⊆ 𝐴) → {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin)
133130, 131, 132mp2an 690 . . . . . . . . . . . . . 14 {𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin
134 ssrab2 4037 . . . . . . . . . . . . . . 15 {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ⊆ 𝐴
135 ssfi 9117 . . . . . . . . . . . . . . 15 ((𝐴 ∈ Fin ∧ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ⊆ 𝐴) → {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin)
136130, 134, 135mp2an 690 . . . . . . . . . . . . . 14 {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin
137 inrab 4266 . . . . . . . . . . . . . . 15 ({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = {𝑔𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))}
138 fzp1disj 13500 . . . . . . . . . . . . . . . . . 18 ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
13942elsn 4601 . . . . . . . . . . . . . . . . . . . 20 ((𝑔‘1) ∈ {(𝑚 + 1)} ↔ (𝑔‘1) = (𝑚 + 1))
140 inelcm 4424 . . . . . . . . . . . . . . . . . . . 20 (((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) ∈ {(𝑚 + 1)}) → ((1...𝑚) ∩ {(𝑚 + 1)}) ≠ ∅)
141139, 140sylan2br 595 . . . . . . . . . . . . . . . . . . 19 (((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)) → ((1...𝑚) ∩ {(𝑚 + 1)}) ≠ ∅)
142141necon2bi 2974 . . . . . . . . . . . . . . . . . 18 (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ → ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)))
143138, 142ax-mp 5 . . . . . . . . . . . . . . . . 17 ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))
144143rgenw 3068 . . . . . . . . . . . . . . . 16 𝑔𝐴 ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))
145 rabeq0 4344 . . . . . . . . . . . . . . . 16 ({𝑔𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))} = ∅ ↔ ∀𝑔𝐴 ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)))
146144, 145mpbir 230 . . . . . . . . . . . . . . 15 {𝑔𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))} = ∅
147137, 146eqtri 2764 . . . . . . . . . . . . . 14 ({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ∅
148 hashun 14282 . . . . . . . . . . . . . 14 (({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin ∧ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin ∧ ({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ∅) → (♯‘({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
149133, 136, 147, 148mp3an 1461 . . . . . . . . . . . . 13 (♯‘({𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))
150126, 149eqtrdi 2792 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
151 nncn 12161 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
152151ad2antlr 725 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℂ)
153 ax-1cn 11109 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
154153a1i 11 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 1 ∈ ℂ)
155152, 154, 154addsubd 11533 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑚 + 1) − 1) = ((𝑚 − 1) + 1))
156155oveq1d 7372 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) + 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
157 subcl 11400 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑚 − 1) ∈ ℂ)
158152, 153, 157sylancl 586 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 − 1) ∈ ℂ)
159109ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑆𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℕ0)
160159nn0cnd 12475 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑆𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℂ)
161158, 154, 160adddird 11180 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 − 1) + 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) + (1 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
162160mulid2d 11173 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (1 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))
163 exmidne 2953 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔‘(𝑚 + 1)) = 1 ∨ (𝑔‘(𝑚 + 1)) ≠ 1)
164 orcom 868 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔‘(𝑚 + 1)) = 1 ∨ (𝑔‘(𝑚 + 1)) ≠ 1) ↔ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1))
165163, 164mpbi 229 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1)
166165biantru 530 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔‘1) = (𝑚 + 1) ↔ ((𝑔‘1) = (𝑚 + 1) ∧ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1)))
167 andi 1006 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔‘1) = (𝑚 + 1) ∧ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1)) ↔ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)))
168166, 167bitri 274 . . . . . . . . . . . . . . . . . . . 20 ((𝑔‘1) = (𝑚 + 1) ↔ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)))
169168rabbii 3413 . . . . . . . . . . . . . . . . . . 19 {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} = {𝑔𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))}
170 unrab 4265 . . . . . . . . . . . . . . . . . . 19 ({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = {𝑔𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))}
171169, 170eqtr4i 2767 . . . . . . . . . . . . . . . . . 18 {𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} = ({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})
172171fveq2i 6845 . . . . . . . . . . . . . . . . 17 (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = (♯‘({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}))
173 ssrab2 4037 . . . . . . . . . . . . . . . . . . 19 {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ⊆ 𝐴
174 ssfi 9117 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ Fin ∧ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ⊆ 𝐴) → {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin)
175130, 173, 174mp2an 690 . . . . . . . . . . . . . . . . . 18 {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin
176 ssrab2 4037 . . . . . . . . . . . . . . . . . . 19 {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ⊆ 𝐴
177 ssfi 9117 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ Fin ∧ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ⊆ 𝐴) → {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin)
178130, 176, 177mp2an 690 . . . . . . . . . . . . . . . . . 18 {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin
179 inrab 4266 . . . . . . . . . . . . . . . . . . 19 ({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = {𝑔𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))}
180 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1) → (𝑔‘(𝑚 + 1)) = 1)
181180necon3ai 2968 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔‘(𝑚 + 1)) ≠ 1 → ¬ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))
182181adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) → ¬ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))
183 imnan 400 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) → ¬ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)) ↔ ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)))
184182, 183mpbi 229 . . . . . . . . . . . . . . . . . . . . 21 ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))
185184rgenw 3068 . . . . . . . . . . . . . . . . . . . 20 𝑔𝐴 ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))
186 rabeq0 4344 . . . . . . . . . . . . . . . . . . . 20 ({𝑔𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} = ∅ ↔ ∀𝑔𝐴 ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)))
187185, 186mpbir 230 . . . . . . . . . . . . . . . . . . 19 {𝑔𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} = ∅
188179, 187eqtri 2764 . . . . . . . . . . . . . . . . . 18 ({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = ∅
189 hashun 14282 . . . . . . . . . . . . . . . . . 18 (({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin ∧ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin ∧ ({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = ∅) → (♯‘({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})))
190175, 178, 188, 189mp3an 1461 . . . . . . . . . . . . . . . . 17 (♯‘({𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}))
191172, 190eqtri 2764 . . . . . . . . . . . . . . . 16 (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ((♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}))
192 simpll 765 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑁 ∈ ℕ)
193 nnne0 12187 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → 𝑚 ≠ 0)
194 0p1e1 12275 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 + 1) = 1
195194eqeq2i 2749 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 + 1) = (0 + 1) ↔ (𝑚 + 1) = 1)
196 0cn 11147 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ∈ ℂ
197 addcan2 11340 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑚 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0))
198196, 153, 197mp3an23 1453 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℂ → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0))
199151, 198syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0))
200195, 199bitr3id 284 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → ((𝑚 + 1) = 1 ↔ 𝑚 = 0))
201200necon3bbid 2981 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → (¬ (𝑚 + 1) = 1 ↔ 𝑚 ≠ 0))
202193, 201mpbird 256 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → ¬ (𝑚 + 1) = 1)
203202ad2antlr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ¬ (𝑚 + 1) = 1)
20414adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (𝑁 + 1) ∈ (ℤ‘1))
205 elfzp12 13520 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 + 1) ∈ (ℤ‘1) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) ↔ ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))))
206204, 205syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) ↔ ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))))
207206biimpa 477 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1))))
208207ord 862 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (¬ (𝑚 + 1) = 1 → (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1))))
209203, 208mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))
210 df-2 12216 . . . . . . . . . . . . . . . . . . . 20 2 = (1 + 1)
211210oveq1i 7367 . . . . . . . . . . . . . . . . . . 19 (2...(𝑁 + 1)) = ((1 + 1)...(𝑁 + 1))
212209, 211eleqtrrdi 2849 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 + 1) ∈ (2...(𝑁 + 1)))
213 ovex 7390 . . . . . . . . . . . . . . . . . 18 (𝑚 + 1) ∈ V
214 eqid 2736 . . . . . . . . . . . . . . . . . 18 ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) = ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})
215 fveq1 6841 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = → (𝑔‘1) = (‘1))
216215eqeq1d 2738 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = → ((𝑔‘1) = (𝑚 + 1) ↔ (‘1) = (𝑚 + 1)))
217 fveq1 6841 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = → (𝑔‘(𝑚 + 1)) = (‘(𝑚 + 1)))
218217neeq1d 3003 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = → ((𝑔‘(𝑚 + 1)) ≠ 1 ↔ (‘(𝑚 + 1)) ≠ 1))
219216, 218anbi12d 631 . . . . . . . . . . . . . . . . . . 19 (𝑔 = → (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ↔ ((‘1) = (𝑚 + 1) ∧ (‘(𝑚 + 1)) ≠ 1)))
220219cbvrabv 3417 . . . . . . . . . . . . . . . . . 18 {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} = {𝐴 ∣ ((‘1) = (𝑚 + 1) ∧ (‘(𝑚 + 1)) ≠ 1)}
221 eqid 2736 . . . . . . . . . . . . . . . . . 18 (( I ↾ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})) ∪ {⟨1, (𝑚 + 1)⟩, ⟨(𝑚 + 1), 1⟩}) = (( I ↾ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})) ∪ {⟨1, (𝑚 + 1)⟩, ⟨(𝑚 + 1), 1⟩})
222 f1oeq1 6772 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ↔ 𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))))
223 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑦 → (𝑔𝑧) = (𝑔𝑦))
224 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑦𝑧 = 𝑦)
225223, 224neeq12d 3005 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑦 → ((𝑔𝑧) ≠ 𝑧 ↔ (𝑔𝑦) ≠ 𝑦))
226225cbvralvw 3225 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑧 ∈ (2...(𝑁 + 1))(𝑔𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑔𝑦) ≠ 𝑦)
227 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = 𝑓 → (𝑔𝑦) = (𝑓𝑦))
228227neeq1d 3003 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = 𝑓 → ((𝑔𝑦) ≠ 𝑦 ↔ (𝑓𝑦) ≠ 𝑦))
229228ralbidv 3174 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (∀𝑦 ∈ (2...(𝑁 + 1))(𝑔𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦))
230226, 229bitrid 282 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (∀𝑧 ∈ (2...(𝑁 + 1))(𝑔𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦))
231222, 230anbi12d 631 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑧 ∈ (2...(𝑁 + 1))(𝑔𝑧) ≠ 𝑧) ↔ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)))
232231cbvabv 2809 . . . . . . . . . . . . . . . . . 18 {𝑔 ∣ (𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑧 ∈ (2...(𝑁 + 1))(𝑔𝑧) ≠ 𝑧)} = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}
2333, 4, 10, 192, 212, 213, 214, 220, 221, 232subfacp1lem5 33778 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) = (𝑆𝑁))
234217eqeq1d 2738 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = → ((𝑔‘(𝑚 + 1)) = 1 ↔ (‘(𝑚 + 1)) = 1))
235216, 234anbi12d 631 . . . . . . . . . . . . . . . . . . 19 (𝑔 = → (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1) ↔ ((‘1) = (𝑚 + 1) ∧ (‘(𝑚 + 1)) = 1)))
236235cbvrabv 3417 . . . . . . . . . . . . . . . . . 18 {𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} = {𝐴 ∣ ((‘1) = (𝑚 + 1) ∧ (‘(𝑚 + 1)) = 1)}
237 f1oeq1 6772 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ↔ 𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})))
238225cbvralvw 3225 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔𝑦) ≠ 𝑦)
239228ralbidv 3174 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓𝑦) ≠ 𝑦))
240238, 239bitrid 282 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓𝑦) ≠ 𝑦))
241237, 240anbi12d 631 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔𝑧) ≠ 𝑧) ↔ (𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓𝑦) ≠ 𝑦)))
242241cbvabv 2809 . . . . . . . . . . . . . . . . . 18 {𝑔 ∣ (𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔𝑧) ≠ 𝑧)} = {𝑓 ∣ (𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓𝑦) ≠ 𝑦)}
2433, 4, 10, 192, 212, 213, 214, 236, 242subfacp1lem3 33776 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = (𝑆‘(𝑁 − 1)))
244233, 243oveq12d 7375 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))
245191, 244eqtrid 2788 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))
246162, 245eqtr4d 2779 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (1 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))
247246oveq2d 7373 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) + (1 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) = (((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
248156, 161, 2473eqtrd 2780 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
249150, 248eqeq12d 2752 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) ↔ ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = (((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))))
250120, 249syl5ibr 245 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
251250ex 413 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → ((♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
252251a2d 29 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
253119, 252syld 47 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
254253expcom 414 . . . . . 6 (𝑚 ∈ ℕ → (𝑁 ∈ ℕ → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))))
255254a2d 29 . . . . 5 (𝑚 ∈ ℕ → ((𝑁 ∈ ℕ → (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))) → (𝑁 ∈ ℕ → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))))
25653, 63, 73, 83, 113, 255nnind 12171 . . . 4 ((𝑁 + 1) ∈ ℕ → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))))
2571, 256mpcom 38 . . 3 (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1))))))
25834, 257mpd 15 . 2 (𝑁 ∈ ℕ → (♯‘{𝑔𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
259 nncn 12161 . . . 4 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
260 pncan 11407 . . . 4 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
261259, 153, 260sylancl 586 . . 3 (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
262261oveq1d 7372 . 2 (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))) = (𝑁 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
26332, 258, 2623eqtrd 2780 1 (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  {crab 3407  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  {cpr 4588  cop 4592  cmpt 5188   I cid 5530  cres 5635  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  cmin 11385  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  chash 14230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231
This theorem is referenced by:  subfacp1  33780
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