mccllem - Mathbox for Glauco Siliprandi

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Theorem mccllem 46172

Description: * Induction step for mccl 46173. (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 1937	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑘(!‘(𝐵‘𝐷))
3		mccllem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
4		mccllem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
5		ssfi 9145	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ Fin)
6	3, 4, 5	syl2anc 595	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin)
7		mccllem.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
8		eldifn 4088	. . . . . 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 8834	. . . . . . . . . 10 ⊢ (𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
12	10, 11	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
13	12	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
14		elun1 4137	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷}))
15	14	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷}))
16	13, 15	ffvelcdmd 7070	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℕ₀)
17	16	faccld 14308	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ)
18	17	nncnd 12237	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ)
19		2fveq3 6876	. . . . 5 ⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷)))
20		snidg 4622	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷})
21	7, 20	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ {𝐷})
22		elun2 4138	. . . . . . . . 9 ⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷}))
23	21, 22	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷}))
24	12, 23	ffvelcdmd 7070	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℕ₀)
25	24	faccld 14308	. . . . . 6 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ)
26	25	nncnd 12237	. . . . 5 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ)
27	1, 2, 6, 7, 9, 18, 19, 26	fprodsplitsn 16031	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))
28	27	oveq2d 7416	. . 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 3919	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
30		snssi 4747	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
31	29, 30	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
32	4, 31	unssd 4147	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴)
33		ssfi 9145	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin)
34	3, 32, 33	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin)
35	12	ffvelcdmda 7069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈ ℕ₀)
36	34, 35	fsumnn0cl 15775	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℕ₀)
37	36	faccld 14308	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ)
38	37	nncnd 12237	. . . . . 6 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ)
39	1, 6, 18	fprodclf 16034	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ)
40	39, 26	mulcld 11217	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ)
41	17	nnne0d 12274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0)
42	6, 18, 41	fprodn0 16021	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0)
43	25	nnne0d 12274	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0)
44	39, 26, 42, 43	mulne0d 11854	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0)
45	38, 40, 44	divcld 11979	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ)
46	45	mullidd 11215	. . . 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 2771	. . 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 15775	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℕ₀)
49	48	faccld 14308	. . . . . . . 8 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
50	49	nncnd 12237	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ)
51		nnne0 12258	. . . . . . . 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 11977	. . . . . 6 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = 1)
54	53	eqcomd 2771	. . . . 5 ⊢ (𝜑 → 1 = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
55	39, 26	mulcomd 11218	. . . . . . 7 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
56	55	oveq2d 7416	. . . . . 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 12010	. . . . . . 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 2771	. . . . . 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 2800	. . . . 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 7418	. . . 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 11979	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ)
62	50, 50, 61, 39, 52, 42	divmul13d 12021	. . . 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 2800	. . 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 2804	. 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 12010	. . . . 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 3879	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘)
67	16	nn0cnd 12555	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ)
68		csbeq1a 3869	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
69		csbfv 6918	. . . . . . . . . . . . 13 ⊢ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)
70	69	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷))
71	24	nn0cnd 12555	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ)
72	70, 71	eqeltrd 2865	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ)
73	1, 66, 6, 29, 9, 67, 68, 72	fsumsplitsn 15783	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
74	73	oveq1d 7415	. . . . . . . . 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 12555	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ)
76	75, 72	pncan2d 11559	. . . . . . . . 9 ⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
77	74, 76, 70	3eqtrrd 2805	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
78	77	fveq2d 6875	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
79	78	oveq1d 7415	. . . . . 6 ⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
80	79	oveq2d 7416	. . . . 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 12591	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
82	36	nn0zd 12604	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ)
83	48	nn0zd 12604	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ)
84	48	nn0ge0d 12556	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
85	24	nn0ge0d 12556	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐷))
86	70	eqcomd 2771	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
87	85, 86	breqtrd 5130	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
88	48	nn0red 12554	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ)
89	24	nn0red 12554	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ)
90	70, 89	eqeltrd 2865	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ)
91	88, 90	addge01d 11790	. . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
94	92, 93	breqtrd 5130	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
95	81, 82, 83, 84, 94	elfzd 13531	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)))
96		bcval2 14329	. . . . . . 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 2771	. . . . 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 2804	. . . 4 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
100		bccl2 14347	. . . . 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 2865	. . 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 4133	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷})
105	104	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷}))
106		elmapssres 8852	. . . . 5 ⊢ ((𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
107	10, 105, 106	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
108		fveq1 6870	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
109	108	adantr 485	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
110		fvres 6890	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
111	110	adantl 486	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
112	109, 111	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘))
113	112	sumeq2dv 15741	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
114	113	fveq2d 6875	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
115	112	fveq2d 6875	. . . . . . . 8 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘)))
116	115	prodeq2dv 15964	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))
117	114, 116	oveq12d 7418	. . . . . 6 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
118	117	eleq1d 2850	. . . . 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 3583	. . . 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 595	. . 3 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)
121	102, 120	nnmulcld 12277	. 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 2865	1 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ⦋csb 3855 ∖ cdif 3904 ∪ cun 3905 ⊆ wss 3907 {csn 4585 class class class wbr 5104 ↾ cres 5653 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ↑_m cmap 8812 Fincfn 8931 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 ≤ cle 11232 − cmin 11429 / cdiv 11859 ℕcn 12221 ℕ₀cn0 12492 ...cfz 13523 !cfa 14297 Ccbc 14326 Σcsu 15725 ∏cprod 15945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-n0 12493 df-z 12580 df-uz 12851 df-rp 13005 df-fz 13524 df-fzo 13671 df-seq 14026 df-exp 14086 df-fac 14298 df-bc 14327 df-hash 14355 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15527 df-sum 15726 df-prod 15946

This theorem is referenced by: mccl 46173

W3C validator