mccllem - Mathbox for Glauco Siliprandi

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Theorem mccllem 41885

Description: * Induction step for mccl 41886. (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 1915	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑘(!‘(𝐵‘𝐷))
3		mccllem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
4		mccllem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
5		ssfi 8740	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ Fin)
6	3, 4, 5	syl2anc 586	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin)
7		mccllem.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
8		eldifn 4106	. . . . . 6 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ¬ 𝐷 ∈ 𝐶)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶)
10		mccllem.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})))
11		elmapi 8430	. . . . . . . . . 10 ⊢ (𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
13	12	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
14		elun1 4154	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷}))
15	14	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷}))
16	13, 15	ffvelrnd 6854	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℕ₀)
17	16	faccld 13647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ)
18	17	nncnd 11656	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ)
19		2fveq3 6677	. . . . 5 ⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷)))
20		snidg 4601	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷})
21	7, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ {𝐷})
22		elun2 4155	. . . . . . . . 9 ⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷}))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷}))
24	12, 23	ffvelrnd 6854	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℕ₀)
25	24	faccld 13647	. . . . . 6 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ)
26	25	nncnd 11656	. . . . 5 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ)
27	1, 2, 6, 7, 9, 18, 19, 26	fprodsplitsn 15345	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))
28	27	oveq2d 7174	. . 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 3950	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
30		snssi 4743	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
32	4, 31	unssd 4164	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴)
33		ssfi 8740	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin)
34	3, 32, 33	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin)
35	12	ffvelrnda 6853	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈ ℕ₀)
36	34, 35	fsumnn0cl 15095	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℕ₀)
37	36	faccld 13647	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ)
38	37	nncnd 11656	. . . . . 6 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ)
39	1, 6, 18	fprodclf 15348	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ)
40	39, 26	mulcld 10663	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ)
41	17	nnne0d 11690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0)
42	6, 18, 41	fprodn0 15335	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0)
43	25	nnne0d 11690	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0)
44	39, 26, 42, 43	mulne0d 11294	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0)
45	38, 40, 44	divcld 11418	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ)
46	45	mulid2d 10661	. . . 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 2829	. . 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 15095	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℕ₀)
49	48	faccld 13647	. . . . . . . 8 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
50	49	nncnd 11656	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ)
51		nnne0 11674	. . . . . . . 8 ⊢ ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ≠ 0)
52	49, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ≠ 0)
53	50, 52	dividd 11416	. . . . . 6 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = 1)
54	53	eqcomd 2829	. . . . 5 ⊢ (𝜑 → 1 = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
55	39, 26	mulcomd 10664	. . . . . . 7 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
56	55	oveq2d 7174	. . . . . 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 11449	. . . . . . 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 2829	. . . . . 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 2858	. . . . 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 7176	. . . 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 11418	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ)
62	50, 50, 61, 39, 52, 42	divmul13d 11460	. . . 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 2858	. . 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 2862	. 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 11449	. . . . 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 3909	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘)
67	16	nn0cnd 11960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ)
68		csbeq1a 3899	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
69		csbfv 6717	. . . . . . . . . . . . 13 ⊢ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)
70	69	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷))
71	24	nn0cnd 11960	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ)
72	70, 71	eqeltrd 2915	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ)
73	1, 66, 6, 29, 9, 67, 68, 72	fsumsplitsn 15102	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
74	73	oveq1d 7173	. . . . . . . . 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 11960	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ)
76	75, 72	pncan2d 11001	. . . . . . . . 9 ⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
77	74, 76, 70	3eqtrrd 2863	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
78	77	fveq2d 6676	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
79	78	oveq1d 7173	. . . . . 6 ⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
80	79	oveq2d 7174	. . . . 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 11996	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
82	36	nn0zd 12088	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ)
83	48	nn0zd 12088	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ)
84	81, 82, 83	3jca 1124	. . . . . . . . 9 ⊢ (𝜑 → (0 ∈ ℤ ∧ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ))
85	48	nn0ge0d 11961	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
86	24	nn0ge0d 11961	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐷))
87	70	eqcomd 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
88	86, 87	breqtrd 5094	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
89	48	nn0red 11959	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ)
90	24	nn0red 11959	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ)
91	70, 90	eqeltrd 2915	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ)
92	89, 91	addge01d 11230	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ↔ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))))
93	88, 92	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
94	73	eqcomd 2829	. . . . . . . . . 10 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
95	93, 94	breqtrd 5094	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
96	84, 85, 95	jca32 518	. . . . . . . 8 ⊢ (𝜑 → ((0 ∈ ℤ ∧ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ) ∧ (0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))))
97		elfz2 12902	. . . . . . . 8 ⊢ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ↔ ((0 ∈ ℤ ∧ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ) ∧ (0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))))
98	96, 97	sylibr 236	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)))
99		bcval2 13668	. . . . . . 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)))))
100	98, 99	syl 17	. . . . . 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)))))
101	100	eqcomd 2829	. . . . 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)))
102	65, 80, 101	3eqtrd 2862	. . . 4 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
103		bccl2 13686	. . . . 5 ⊢ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
104	98, 103	syl 17	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
105	102, 104	eqeltrd 2915	. . 3 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) ∈ ℕ)
106		mccllem.6	. . . 4 ⊢ (𝜑 → ∀𝑏 ∈ (ℕ₀ ↑_m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ)
107		ssun1 4150	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷})
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷}))
109		elmapssres 8433	. . . . 5 ⊢ ((𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
110	10, 108, 109	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
111		fveq1 6671	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
112	111	adantr 483	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
113		fvres 6691	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
114	113	adantl 484	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
115	112, 114	eqtrd 2858	. . . . . . . . 9 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘))
116	115	sumeq2dv 15062	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
117	116	fveq2d 6676	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
118	115	fveq2d 6676	. . . . . . . 8 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘)))
119	118	prodeq2dv 15279	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))
120	117, 119	oveq12d 7176	. . . . . 6 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
121	120	eleq1d 2899	. . . . 5 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ ↔ ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ))
122	121	rspccva 3624	. . . 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)
123	106, 110, 122	syl2anc 586	. . 3 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)
124	105, 123	nnmulcld 11693	. 2 ⊢ (𝜑 → ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) ∈ ℕ)
125	64, 124	eqeltrd 2915	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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ⦋csb 3885 ∖ cdif 3935 ∪ cun 3936 ⊆ wss 3938 {csn 4569 class class class wbr 5068 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑_m cmap 8408 Fincfn 8511 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 ℕ₀cn0 11900 ℤcz 11984 ...cfz 12895 !cfa 13636 Ccbc 13665 Σcsu 15044 ∏cprod 15261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-prod 15262

This theorem is referenced by: mccl 41886

W3C validator