mccllem - Mathbox for Glauco Siliprandi

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Theorem mccllem 46435

Description: * Induction step for mccl 46436. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
mccllem.a	⊢ (𝜑 → 𝐴 ∈ Fin)
mccllem.c	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
mccllem.d	⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
mccllem.b	⊢ (𝜑 → 𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})))
mccllem.6	⊢ (𝜑 → ∀𝑏 ∈ (ℕ₀ ↑_m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ)

**Assertion**
Ref	Expression
mccllem	⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ)

Distinct variable groups: 𝐴,𝑘 𝐵,𝑏,𝑘 𝐶,𝑏,𝑘 𝐷,𝑘 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑏) 𝐴(𝑏) 𝐷(𝑏)

**Proof of Theorem mccllem**
Step	Hyp	Ref	Expression
1		nfv 1947	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑘(!‘(𝐵‘𝐷))
3		mccllem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
4		mccllem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
5		ssfi 9171	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ Fin)
6	3, 4, 5	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin)
7		mccllem.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
8		eldifn 4082	. . . . . 6 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ¬ 𝐷 ∈ 𝐶)
9	7, 8	syl 18	. . . . 5 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶)
10		mccllem.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})))
11		elmapi 8852	. . . . . . . . . 10 ⊢ (𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
12	10, 11	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
13	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
14		elun1 4131	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷}))
15	14	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷}))
16	13, 15	ffvelcdmd 7082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℕ₀)
17	16	faccld 14352	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ)
18	17	nncnd 12277	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ)
19		2fveq3 6887	. . . . 5 ⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷)))
20		snidg 4624	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷})
21	7, 20	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ {𝐷})
22		elun2 4132	. . . . . . . . 9 ⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷}))
23	21, 22	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷}))
24	12, 23	ffvelcdmd 7082	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℕ₀)
25	24	faccld 14352	. . . . . 6 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ)
26	25	nncnd 12277	. . . . 5 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ)
27	1, 2, 6, 7, 9, 18, 19, 26	fprodsplitsn 16082	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))
28	27	oveq2d 7433	. . 3 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))))
29	7	eldifad 3914	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
30		snssi 4749	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
31	29, 30	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
32	4, 31	unssd 4141	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴)
33		ssfi 9171	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin)
34	3, 32, 33	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin)
35	12	ffvelcdmda 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈ ℕ₀)
36	34, 35	fsumnn0cl 15826	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℕ₀)
37	36	faccld 14352	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ)
38	37	nncnd 12277	. . . . . 6 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ)
39	1, 6, 18	fprodclf 16085	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ)
40	39, 26	mulcld 11257	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ)
41	17	nnne0d 12314	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0)
42	6, 18, 41	fprodn0 16072	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0)
43	25	nnne0d 12314	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0)
44	39, 26, 42, 43	mulne0d 11894	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0)
45	38, 40, 44	divcld 12019	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ)
46	45	mullidd 11255	. . . 4 ⊢ (𝜑 → (1 · ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))))
47	46	eqcomd 2768	. . 3 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) = (1 · ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))))
48	6, 16	fsumnn0cl 15826	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℕ₀)
49	48	faccld 14352	. . . . . . . 8 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
50	49	nncnd 12277	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ)
51		nnne0 12298	. . . . . . . 8 ⊢ ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ≠ 0)
52	49, 51	syl 18	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ≠ 0)
53	50, 52	dividd 12017	. . . . . 6 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = 1)
54	53	eqcomd 2768	. . . . 5 ⊢ (𝜑 → 1 = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
55	39, 26	mulcomd 11258	. . . . . . 7 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
56	55	oveq2d 7433	. . . . . 6 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))))
57	38, 26, 39, 43, 42	divdiv1d 12050	. . . . . . 7 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))))
58	57	eqcomd 2768	. . . . . 6 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) = (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
59	56, 58	eqtrd 2797	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) = (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
60	54, 59	oveq12d 7435	. . . 4 ⊢ (𝜑 → (1 · ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))) = (((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))))
61	38, 26, 43	divcld 12019	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ)
62	50, 50, 61, 39, 52, 42	divmul13d 12061	. . . 4 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) = ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))))
63	60, 62	eqtrd 2797	. . 3 ⊢ (𝜑 → (1 · ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))) = ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))))
64	28, 47, 63	3eqtrd 2801	. 2 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) = ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))))
65	38, 26, 50, 43, 52	divdiv1d 12050	. . . . 5 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))))
66		nfcsb1v 3874	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘)
67	16	nn0cnd 12595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ)
68		csbeq1a 3864	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
69		csbfv 6929	. . . . . . . . . . . . 13 ⊢ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)
70	69	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷))
71	24	nn0cnd 12595	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ)
72	70, 71	eqeltrd 2862	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ)
73	1, 66, 6, 29, 9, 67, 68, 72	fsumsplitsn 15834	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
74	73	oveq1d 7432	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
75	48	nn0cnd 12595	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ)
76	75, 72	pncan2d 11599	. . . . . . . . 9 ⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
77	74, 76, 70	3eqtrrd 2802	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
78	77	fveq2d 6886	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
79	78	oveq1d 7432	. . . . . 6 ⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
80	79	oveq2d 7433	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))))
81		0zd 12631	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
82	36	nn0zd 12644	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ)
83	48	nn0zd 12644	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ)
84	48	nn0ge0d 12596	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
85	24	nn0ge0d 12596	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐷))
86	70	eqcomd 2768	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
87	85, 86	breqtrd 5135	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
88	48	nn0red 12594	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ)
89	24	nn0red 12594	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ)
90	70, 89	eqeltrd 2862	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ)
91	88, 90	addge01d 11830	. . . . . . . . . 10 ⊢ (𝜑 → (0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ↔ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))))
92	87, 91	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
93	73	eqcomd 2768	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
94	92, 93	breqtrd 5135	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
95	81, 82, 83, 84, 94	elfzd 13573	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)))
96		bcval2 14373	. . . . . . 7 ⊢ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))))
97	95, 96	syl 18	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))))
98	97	eqcomd 2768	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
99	65, 80, 98	3eqtrd 2801	. . . 4 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
100		bccl2 14391	. . . . 5 ⊢ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
101	95, 100	syl 18	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
102	99, 101	eqeltrd 2862	. . 3 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) ∈ ℕ)
103		mccllem.6	. . . 4 ⊢ (𝜑 → ∀𝑏 ∈ (ℕ₀ ↑_m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ)
104		ssun1 4127	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷})
105	104	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷}))
106		elmapssres 8877	. . . . 5 ⊢ ((𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
107	10, 105, 106	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
108		fveq1 6881	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
109	108	adantr 486	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
110		fvres 6901	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
111	110	adantl 487	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
112	109, 111	eqtrd 2797	. . . . . . . . 9 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘))
113	112	sumeq2dv 15793	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
114	113	fveq2d 6886	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
115	112	fveq2d 6886	. . . . . . . 8 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘)))
116	115	prodeq2dv 16015	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))
117	114, 116	oveq12d 7435	. . . . . 6 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
118	117	eleq1d 2847	. . . . 5 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ ↔ ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ))
119	118	rspccva 3578	. . . 4 ⊢ ((∀𝑏 ∈ (ℕ₀ ↑_m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ ∧ (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶)) → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)
120	103, 107, 119	syl2anc 596	. . 3 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)
121	102, 120	nnmulcld 12317	. 2 ⊢ (𝜑 → ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) ∈ ℕ)
122	64, 121	eqeltrd 2862	1 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ⦋csb 3850 ∖ cdif 3899 ∪ cun 3900 ⊆ wss 3902 {csn 4587 class class class wbr 5107 ↾ cres 5661 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 ↑_m cmap 8830 Fincfn 8956 ℂcc 11126 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 · cmul 11133 ≤ cle 11272 − cmin 11469 / cdiv 11899 ℕcn 12261 ℕ₀cn0 12532 ...cfz 13565 !cfa 14341 Ccbc 14370 Σcsu 15777 ∏cprod 15996

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-z 12620 df-uz 12892 df-rp 13047 df-fz 13566 df-fzo 13714 df-seq 14070 df-exp 14130 df-fac 14342 df-bc 14371 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-sum 15778 df-prod 15997

This theorem is used by: mccl 46436

W3C validator