subfacp1lem6 - Mathbox for Mario Carneiro

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Theorem subfacp1lem6 32041

Description: Lemma for subfacp1 32042. 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 (𝑓‘𝑀) = 1 has size 𝑆(𝑁 − 1) (by subfacp1lem3 32038), while the subset with (𝑓‘𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 32040). 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:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 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))(𝑓‘𝑦) ≠ 𝑦)}

**Assertion**
Ref	Expression
subfacp1lem6	⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (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 ℎ 𝑚 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2nn 11498	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnnn0d 11803	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
3		derang.d	. . . . 5 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
4		subfac.n	. . . . 5 ⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))
5	3, 4	subfacval 32029	. . . 4 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑆‘(𝑁 + 1)) = (𝐷‘(1...(𝑁 + 1))))
6	2, 5	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝐷‘(1...(𝑁 + 1))))
7		fzfid 13191	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 + 1)) ∈ Fin)
8	3	derangval 32023	. . . . 5 ⊢ ((1...(𝑁 + 1)) ∈ Fin → (𝐷‘(1...(𝑁 + 1))) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}))
9	7, 8	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐷‘(1...(𝑁 + 1))) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}))
10		subfacp1lem.a	. . . . 5 ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}
11	10	fveq2i 6541	. . . 4 ⊢ (♯‘𝐴) = (♯‘{𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)})
12	9, 11	syl6eqr 2849	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐷‘(1...(𝑁 + 1))) = (♯‘𝐴))
13		nnuz 12130	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
14	1, 13	syl6eleq 2893	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ_≥‘1))
15		eluzfz1 12764	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑁 + 1)))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...(𝑁 + 1)))
17		f1of 6483	. . . . . . . . . 10 ⊢ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
18	17	adantr 481	. . . . . . . . 9 ⊢ ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) → 𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
19		ffvelrn 6714	. . . . . . . . . 10 ⊢ ((𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ 1 ∈ (1...(𝑁 + 1))) → (𝑓‘1) ∈ (1...(𝑁 + 1)))
20	19	expcom 414	. . . . . . . . 9 ⊢ (1 ∈ (1...(𝑁 + 1)) → (𝑓:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) → (𝑓‘1) ∈ (1...(𝑁 + 1))))
21	16, 18, 20	syl2im 40	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) → (𝑓‘1) ∈ (1...(𝑁 + 1))))
22	21	ss2abdv 3965	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ (𝑓‘1) ∈ (1...(𝑁 + 1))})
23		fveq1 6537	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝑔‘1) = (𝑓‘1))
24	23	eleq1d 2867	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → ((𝑔‘1) ∈ (1...(𝑁 + 1)) ↔ (𝑓‘1) ∈ (1...(𝑁 + 1))))
25	24	cbvabv 2927	. . . . . . 7 ⊢ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} = {𝑓 ∣ (𝑓‘1) ∈ (1...(𝑁 + 1))}
26	22, 10, 25	3sstr4g 3933	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐴 ⊆ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})
27		ssabral 3963	. . . . . 6 ⊢ (𝐴 ⊆ {𝑔 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} ↔ ∀𝑔 ∈ 𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1)))
28	26, 27	sylib 219	. . . . 5 ⊢ (𝑁 ∈ ℕ → ∀𝑔 ∈ 𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1)))
29		rabid2 3340	. . . . 5 ⊢ (𝐴 = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))} ↔ ∀𝑔 ∈ 𝐴 (𝑔‘1) ∈ (1...(𝑁 + 1)))
30	28, 29	sylibr 235	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝐴 = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})
31	30	fveq2d 6542	. . 3 ⊢ (𝑁 ∈ ℕ → (♯‘𝐴) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}))
32	6, 12, 31	3eqtrd 2835	. 2 ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}))
33		elfz1end 12787	. . . 4 ⊢ ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
34	1, 33	sylib 219	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
35		eleq1 2870	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ∈ (1...(𝑁 + 1)) ↔ 1 ∈ (1...(𝑁 + 1))))
36		oveq2 7024	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (1...𝑥) = (1...1))
37		1z 11861	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
38		fzsn 12799	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → (1...1) = {1})
39	37, 38	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (1...1) = {1}
40	36, 39	syl6eq 2847	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (1...𝑥) = {1})
41	40	eleq2d 2868	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ {1}))
42		fvex 6551	. . . . . . . . . . . 12 ⊢ (𝑔‘1) ∈ V
43	42	elsn 4487	. . . . . . . . . . 11 ⊢ ((𝑔‘1) ∈ {1} ↔ (𝑔‘1) = 1)
44	41, 43	syl6bb 288	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) = 1))
45	44	rabbidv 3425	. . . . . . . . 9 ⊢ (𝑥 = 1 → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1})
46	45	fveq2d 6542	. . . . . . . 8 ⊢ (𝑥 = 1 → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}))
47		oveq1 7023	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 − 1) = (1 − 1))
48		1m1e0 11557	. . . . . . . . . 10 ⊢ (1 − 1) = 0
49	47, 48	syl6eq 2847	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑥 − 1) = 0)
50	49	oveq1d 7031	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
51	46, 50	eqeq12d 2810	. . . . . . 7 ⊢ (𝑥 = 1 → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
52	35, 51	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 1 → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ (1 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
53	52	imbi2d 342	. . . . 5 ⊢ (𝑥 = 1 → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → (1 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))))
54		eleq1 2870	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑥 ∈ (1...(𝑁 + 1)) ↔ 𝑚 ∈ (1...(𝑁 + 1))))
55		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑚 → (1...𝑥) = (1...𝑚))
56	55	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...𝑚)))
57	56	rabbidv 3425	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)})
58	57	fveq2d 6542	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}))
59		oveq1 7023	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → (𝑥 − 1) = (𝑚 − 1))
60	59	oveq1d 7031	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
61	58, 60	eqeq12d 2810	. . . . . . 7 ⊢ (𝑥 = 𝑚 → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
62	54, 61	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝑚 → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
63	62	imbi2d 342	. . . . 5 ⊢ (𝑥 = 𝑚 → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))))
64		eleq1 2870	. . . . . . 7 ⊢ (𝑥 = (𝑚 + 1) → (𝑥 ∈ (1...(𝑁 + 1)) ↔ (𝑚 + 1) ∈ (1...(𝑁 + 1))))
65		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑚 + 1) → (1...𝑥) = (1...(𝑚 + 1)))
66	65	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑥 = (𝑚 + 1) → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...(𝑚 + 1))))
67	66	rabbidv 3425	. . . . . . . . 9 ⊢ (𝑥 = (𝑚 + 1) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))})
68	67	fveq2d 6542	. . . . . . . 8 ⊢ (𝑥 = (𝑚 + 1) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}))
69		oveq1 7023	. . . . . . . . 9 ⊢ (𝑥 = (𝑚 + 1) → (𝑥 − 1) = ((𝑚 + 1) − 1))
70	69	oveq1d 7031	. . . . . . . 8 ⊢ (𝑥 = (𝑚 + 1) → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
71	68, 70	eqeq12d 2810	. . . . . . 7 ⊢ (𝑥 = (𝑚 + 1) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
72	64, 71	imbi12d 346	. . . . . 6 ⊢ (𝑥 = (𝑚 + 1) → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
73	72	imbi2d 342	. . . . 5 ⊢ (𝑥 = (𝑚 + 1) → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))))
74		eleq1 2870	. . . . . . 7 ⊢ (𝑥 = (𝑁 + 1) → (𝑥 ∈ (1...(𝑁 + 1)) ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1))))
75		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑁 + 1) → (1...𝑥) = (1...(𝑁 + 1)))
76	75	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑥 = (𝑁 + 1) → ((𝑔‘1) ∈ (1...𝑥) ↔ (𝑔‘1) ∈ (1...(𝑁 + 1))))
77	76	rabbidv 3425	. . . . . . . . 9 ⊢ (𝑥 = (𝑁 + 1) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)} = {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))})
78	77	fveq2d 6542	. . . . . . . 8 ⊢ (𝑥 = (𝑁 + 1) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}))
79		oveq1 7023	. . . . . . . . 9 ⊢ (𝑥 = (𝑁 + 1) → (𝑥 − 1) = ((𝑁 + 1) − 1))
80	79	oveq1d 7031	. . . . . . . 8 ⊢ (𝑥 = (𝑁 + 1) → ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
81	78, 80	eqeq12d 2810	. . . . . . 7 ⊢ (𝑥 = (𝑁 + 1) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
82	74, 81	imbi12d 346	. . . . . 6 ⊢ (𝑥 = (𝑁 + 1) → ((𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) ↔ ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
83	82	imbi2d 342	. . . . 5 ⊢ (𝑥 = (𝑁 + 1) → ((𝑁 ∈ ℕ → (𝑥 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑥)}) = ((𝑥 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) ↔ (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))))
84		hash0 13578	. . . . . . 7 ⊢ (♯‘∅) = 0
85		fveq2 6538	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 1 → (𝑓‘𝑦) = (𝑓‘1))
86		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 1 → 𝑦 = 1)
87	85, 86	neeq12d 3045	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 1 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑓‘1) ≠ 1))
88	87	rspcv 3555	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (1...(𝑁 + 1)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 → (𝑓‘1) ≠ 1))
89	16, 88	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 → (𝑓‘1) ≠ 1))
90	89	adantld 491	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) → (𝑓‘1) ≠ 1))
91	90	ss2abdv 3965	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ (𝑓‘1) ≠ 1})
92		df-ne 2985	. . . . . . . . . . . . 13 ⊢ ((𝑔‘1) ≠ 1 ↔ ¬ (𝑔‘1) = 1)
93	23	neeq1d 3043	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → ((𝑔‘1) ≠ 1 ↔ (𝑓‘1) ≠ 1))
94	92, 93	syl5bbr 286	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (¬ (𝑔‘1) = 1 ↔ (𝑓‘1) ≠ 1))
95	94	cbvabv 2927	. . . . . . . . . . 11 ⊢ {𝑔 ∣ ¬ (𝑔‘1) = 1} = {𝑓 ∣ (𝑓‘1) ≠ 1}
96	91, 10, 95	3sstr4g 3933	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝐴 ⊆ {𝑔 ∣ ¬ (𝑔‘1) = 1})
97		ssabral 3963	. . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝑔 ∣ ¬ (𝑔‘1) = 1} ↔ ∀𝑔 ∈ 𝐴 ¬ (𝑔‘1) = 1)
98	96, 97	sylib 219	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ∀𝑔 ∈ 𝐴 ¬ (𝑔‘1) = 1)
99		rabeq0 4258	. . . . . . . . 9 ⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1} = ∅ ↔ ∀𝑔 ∈ 𝐴 ¬ (𝑔‘1) = 1)
100	98, 99	sylibr 235	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1} = ∅)
101	100	fveq2d 6542	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) = (♯‘∅))
102		nnnn0 11752	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
103	3, 4	subfacf 32031	. . . . . . . . . . . 12 ⊢ 𝑆:ℕ₀⟶ℕ₀
104	103	ffvelrni 6715	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘𝑁) ∈ ℕ₀)
105	102, 104	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑆‘𝑁) ∈ ℕ₀)
106		nnm1nn0 11786	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
107	103	ffvelrni 6715	. . . . . . . . . . 11 ⊢ ((𝑁 − 1) ∈ ℕ₀ → (𝑆‘(𝑁 − 1)) ∈ ℕ₀)
108	106, 107	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 − 1)) ∈ ℕ₀)
109	105, 108	nn0addcld 11807	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℕ₀)
110	109	nn0cnd 11805	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℂ)
111	110	mul02d 10685	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = 0)
112	84, 101, 111	3eqtr4a 2857	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
113	112	a1d 25	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = 1}) = (0 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
114		simplr 765	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℕ)
115	114, 13	syl6eleq 2893	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (ℤ_≥‘1))
116		peano2fzr 12770	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (ℤ_≥‘1) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (1...(𝑁 + 1)))
117	115, 116	sylancom 588	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ (1...(𝑁 + 1)))
118	117	ex 413	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → 𝑚 ∈ (1...(𝑁 + 1))))
119	118	imim1d 82	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
120		oveq1 7023	. . . . . . . . . . 11 ⊢ ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
121		elfzp1 12807	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ_≥‘1) → ((𝑔‘1) ∈ (1...(𝑚 + 1)) ↔ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))))
122	115, 121	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑔‘1) ∈ (1...(𝑚 + 1)) ↔ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))))
123	122	rabbidv 3425	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))} = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))})
124		unrab 4194	. . . . . . . . . . . . . . 15 ⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∨ (𝑔‘1) = (𝑚 + 1))}
125	123, 124	syl6eqr 2849	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))} = ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))
126	125	fveq2d 6542	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (♯‘({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
127		fzfi 13190	. . . . . . . . . . . . . . . . 17 ⊢ (1...(𝑁 + 1)) ∈ Fin
128		deranglem 32022	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑁 + 1)) ∈ Fin → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
129	127, 128	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin
130	10, 129	eqeltri 2879	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ∈ Fin
131		ssrab2 3977	. . . . . . . . . . . . . . 15 ⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ⊆ 𝐴
132		ssfi 8584	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin)
133	130, 131, 132	mp2an 688	. . . . . . . . . . . . . 14 ⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin
134		ssrab2 3977	. . . . . . . . . . . . . . 15 ⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ⊆ 𝐴
135		ssfi 8584	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin)
136	130, 134, 135	mp2an 688	. . . . . . . . . . . . . 14 ⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin
137		inrab 4195	. . . . . . . . . . . . . . 15 ⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))}
138		fzp1disj 12816	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
139	42	elsn 4487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘1) ∈ {(𝑚 + 1)} ↔ (𝑔‘1) = (𝑚 + 1))
140		inelcm 4328	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) ∈ {(𝑚 + 1)}) → ((1...𝑚) ∩ {(𝑚 + 1)}) ≠ ∅)
141	139, 140	sylan2br 594	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)) → ((1...𝑚) ∩ {(𝑚 + 1)}) ≠ ∅)
142	141	necon2bi 3014	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ → ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)))
143	138, 142	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))
144	143	rgenw 3117	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑔 ∈ 𝐴 ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))
145		rabeq0 4258	. . . . . . . . . . . . . . . 16 ⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))} = ∅ ↔ ∀𝑔 ∈ 𝐴 ¬ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1)))
146	144, 145	mpbir 232	. . . . . . . . . . . . . . 15 ⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) ∈ (1...𝑚) ∧ (𝑔‘1) = (𝑚 + 1))} = ∅
147	137, 146	eqtri 2819	. . . . . . . . . . . . . 14 ⊢ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ∅
148		hashun 13591	. . . . . . . . . . . . . 14 ⊢ (({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} ∈ Fin ∧ ({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∩ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ∅) → (♯‘({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
149	133, 136, 147, 148	mp3an 1453	. . . . . . . . . . . . 13 ⊢ (♯‘({𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)} ∪ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))
150	126, 149	syl6eq 2847	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
151		nncn 11494	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
152	151	ad2antlr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℂ)
153		ax-1cn 10441	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
154	153	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 1 ∈ ℂ)
155	152, 154, 154	addsubd 10866	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑚 + 1) − 1) = ((𝑚 − 1) + 1))
156	155	oveq1d 7031	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) + 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
157		subcl 10732	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑚 − 1) ∈ ℂ)
158	152, 153, 157	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 − 1) ∈ ℂ)
159	109	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℕ₀)
160	159	nn0cnd 11805	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))) ∈ ℂ)
161	158, 154, 160	adddird 10512	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 − 1) + 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (1 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
162	160	mulid2d 10505	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (1 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))
163		exmidne 2994	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔‘(𝑚 + 1)) = 1 ∨ (𝑔‘(𝑚 + 1)) ≠ 1)
164		orcom 865	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑔‘(𝑚 + 1)) = 1 ∨ (𝑔‘(𝑚 + 1)) ≠ 1) ↔ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1))
165	163, 164	mpbi 231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1)
166	165	biantru 530	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔‘1) = (𝑚 + 1) ↔ ((𝑔‘1) = (𝑚 + 1) ∧ ((𝑔‘(𝑚 + 1)) ≠ 1 ∨ (𝑔‘(𝑚 + 1)) = 1)))
167		andi 1002	. . . . . . . . . . . . . . . . . . . . 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)))
168	166, 167	bitri 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘1) = (𝑚 + 1) ↔ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)))
169	168	rabbii 3419	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} = {𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))}
170		unrab 4194	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = {𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∨ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))}
171	169, 170	eqtr4i 2822	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)} = ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})
172	171	fveq2i 6541	. . . . . . . . . . . . . . . . 17 ⊢ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = (♯‘({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}))
173		ssrab2 3977	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ⊆ 𝐴
174		ssfi 8584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin)
175	130, 173, 174	mp2an 688	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin
176		ssrab2 3977	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ⊆ 𝐴
177		ssfi 8584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ⊆ 𝐴) → {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin)
178	130, 176, 177	mp2an 688	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin
179		inrab 4195	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = {𝑔 ∈ 𝐴 ∣ (((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)
181	180	necon3ai 3009	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔‘(𝑚 + 1)) ≠ 1 → ¬ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))
182	181	adantl 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)))
184	182, 183	mpbi 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))
185	184	rgenw 3117	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑔 ∈ 𝐴 ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))
186		rabeq0 4258	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} = ∅ ↔ ∀𝑔 ∈ 𝐴 ¬ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)))
187	185, 186	mpbir 232	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑔 ∈ 𝐴 ∣ (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ∧ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1))} = ∅
188	179, 187	eqtri 2819	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = ∅
189		hashun 13591	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∈ Fin ∧ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} ∈ Fin ∧ ({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∩ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = ∅) → (♯‘({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})))
190	175, 178, 188, 189	mp3an 1453	. . . . . . . . . . . . . . . . 17 ⊢ (♯‘({𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} ∪ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}))
191	172, 190	eqtri 2819	. . . . . . . . . . . . . . . 16 ⊢ (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ((♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}))
192		simpll 763	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → 𝑁 ∈ ℕ)
193		nnne0 11519	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
194		0p1e1 11607	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 + 1) = 1
195	194	eqeq2i 2807	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 + 1) = (0 + 1) ↔ (𝑚 + 1) = 1)
196		0cn 10479	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ∈ ℂ
197		addcan2 10672	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0))
198	196, 153, 197	mp3an23 1445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0))
199	151, 198	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) = (0 + 1) ↔ 𝑚 = 0))
200	195, 199	syl5bbr 286	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) = 1 ↔ 𝑚 = 0))
201	200	necon3bbid 3021	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ → (¬ (𝑚 + 1) = 1 ↔ 𝑚 ≠ 0))
202	193, 201	mpbird 258	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ → ¬ (𝑚 + 1) = 1)
203	202	ad2antlr 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ¬ (𝑚 + 1) = 1)
204	14	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (𝑁 + 1) ∈ (ℤ_≥‘1))
205		elfzp12 12836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘1) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) ↔ ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))))
206	204, 205	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) ↔ ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))))
207	206	biimpa 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((𝑚 + 1) = 1 ∨ (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1))))
208	207	ord 859	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (¬ (𝑚 + 1) = 1 → (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1))))
209	203, 208	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 + 1) ∈ ((1 + 1)...(𝑁 + 1)))
210		df-2 11548	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 = (1 + 1)
211	210	oveq1i 7026	. . . . . . . . . . . . . . . . . . 19 ⊢ (2...(𝑁 + 1)) = ((1 + 1)...(𝑁 + 1))
212	209, 211	syl6eleqr 2894	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (𝑚 + 1) ∈ (2...(𝑁 + 1)))
213		ovex 7048	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 + 1) ∈ V
214		eqid 2795	. . . . . . . . . . . . . . . . . 18 ⊢ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) = ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})
215		fveq1 6537	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = ℎ → (𝑔‘1) = (ℎ‘1))
216	215	eqeq1d 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = ℎ → ((𝑔‘1) = (𝑚 + 1) ↔ (ℎ‘1) = (𝑚 + 1)))
217		fveq1 6537	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = ℎ → (𝑔‘(𝑚 + 1)) = (ℎ‘(𝑚 + 1)))
218	217	neeq1d 3043	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = ℎ → ((𝑔‘(𝑚 + 1)) ≠ 1 ↔ (ℎ‘(𝑚 + 1)) ≠ 1))
219	216, 218	anbi12d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = ℎ → (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1) ↔ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) ≠ 1)))
220	219	cbvrabv 3434	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)} = {ℎ ∈ 𝐴 ∣ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) ≠ 1)}
221		eqid 2795	. . . . . . . . . . . . . . . . . 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 6472	. . . . . . . . . . . . . . . . . . . 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 6538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑦 → (𝑔‘𝑧) = (𝑔‘𝑦))
224		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
225	223, 224	neeq12d 3045	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((𝑔‘𝑧) ≠ 𝑧 ↔ (𝑔‘𝑦) ≠ 𝑦))
226	225	cbvralv 3403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑧 ∈ (2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑔‘𝑦) ≠ 𝑦)
227		fveq1 6537	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑦) = (𝑓‘𝑦))
228	227	neeq1d 3043	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑦) ≠ 𝑦 ↔ (𝑓‘𝑦) ≠ 𝑦))
229	228	ralbidv 3164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑓 → (∀𝑦 ∈ (2...(𝑁 + 1))(𝑔‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦))
230	226, 229	syl5bb 284	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑓 → (∀𝑧 ∈ (2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦))
231	222, 230	anbi12d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑓 → ((𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑧 ∈ (2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧) ↔ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)))
232	231	cbvabv 2927	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∣ (𝑔:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑧 ∈ (2...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧)} = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}
233	3, 4, 10, 192, 212, 213, 214, 220, 221, 232	subfacp1lem5 32040	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) = (𝑆‘𝑁))
234	217	eqeq1d 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = ℎ → ((𝑔‘(𝑚 + 1)) = 1 ↔ (ℎ‘(𝑚 + 1)) = 1))
235	216, 234	anbi12d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = ℎ → (((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1) ↔ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) = 1)))
236	235	cbvrabv 3434	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)} = {ℎ ∈ 𝐴 ∣ ((ℎ‘1) = (𝑚 + 1) ∧ (ℎ‘(𝑚 + 1)) = 1)}
237		f1oeq1 6472	. . . . . . . . . . . . . . . . . . . 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)})))
238	225	cbvralv 3403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑦) ≠ 𝑦)
239	228	ralbidv 3164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑓 → (∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦))
240	238, 239	syl5bb 284	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑓 → (∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦))
241	237, 240	anbi12d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑓 → ((𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧) ↔ (𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦)))
242	241	cbvabv 2927	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∣ (𝑔:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑧 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑔‘𝑧) ≠ 𝑧)} = {𝑓 ∣ (𝑓:((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})–1-1-onto→((2...(𝑁 + 1)) ∖ {(𝑚 + 1)}) ∧ ∀𝑦 ∈ ((2...(𝑁 + 1)) ∖ {(𝑚 + 1)})(𝑓‘𝑦) ≠ 𝑦)}
243	3, 4, 10, 192, 212, 213, 214, 236, 242	subfacp1lem3 32038	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)}) = (𝑆‘(𝑁 − 1)))
244	233, 243	oveq12d 7034	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) ≠ 1)}) + (♯‘{𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = (𝑚 + 1) ∧ (𝑔‘(𝑚 + 1)) = 1)})) = ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))
245	191, 244	syl5eq 2843	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}) = ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))
246	162, 245	eqtr4d 2834	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (1 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))
247	246	oveq2d 7032	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (1 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
248	156, 161, 247	3eqtrd 2835	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})))
249	150, 248	eqeq12d 2810	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) ↔ ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)})) = (((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) + (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) = (𝑚 + 1)}))))
250	120, 249	syl5ibr 247	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 + 1) ∈ (1...(𝑁 + 1))) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
251	250	ex 413	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → ((♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
252	251	a2d 29	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
253	119, 252	syld 47	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
254	253	expcom 414	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝑁 ∈ ℕ → ((𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))))
255	254	a2d 29	. . . . 5 ⊢ (𝑚 ∈ ℕ → ((𝑁 ∈ ℕ → (𝑚 ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...𝑚)}) = ((𝑚 − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))) → (𝑁 ∈ ℕ → ((𝑚 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑚 + 1))}) = (((𝑚 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))))
256	53, 63, 73, 83, 113, 255	nnind 11504	. . . 4 ⊢ ((𝑁 + 1) ∈ ℕ → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))))
257	1, 256	mpcom 38	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (1...(𝑁 + 1)) → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))))
258	34, 257	mpd 15	. 2 ⊢ (𝑁 ∈ ℕ → (♯‘{𝑔 ∈ 𝐴 ∣ (𝑔‘1) ∈ (1...(𝑁 + 1))}) = (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
259		nncn 11494	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
260		pncan 10739	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
261	259, 153, 260	sylancl 586	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
262	261	oveq1d 7031	. 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
263	32, 258, 262	3eqtrd 2835	1 ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 = wceq 1522 ∈ wcel 2081 {cab 2775 ≠ wne 2984 ∀wral 3105 {crab 3109 ∖ cdif 3856 ∪ cun 3857 ∩ cin 3858 ⊆ wss 3859 ∅c0 4211 {csn 4472 {cpr 4474 ⟨cop 4478 ↦ cmpt 5041 I cid 5347 ↾ cres 5445 ⟶wf 6221 –1-1-onto→wf1o 6224 ‘cfv 6225 (class class class)co 7016 Fincfn 8357 ℂcc 10381 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 − cmin 10717 ℕcn 11486 2c2 11540 ℕ₀cn0 11745 ℤcz 11829 ℤ_≥cuz 12093 ...cfz 12742 ♯chash 13540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-dju 9176 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-xnn0 11816 df-z 11830 df-uz 12094 df-fz 12743 df-hash 13541

This theorem is referenced by: subfacp1 32042

W3C validator