mccllem - Mathbox for Glauco Siliprandi

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Theorem mccllem 40467

Description: * Induction step for mccl 40468. (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 2009	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑘(!‘(𝐵‘𝐷))
3		mccllem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
4		mccllem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
5		ssfi 8387	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ Fin)
6	3, 4, 5	syl2anc 579	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin)
7		mccllem.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
8		eldifn 3895	. . . . . 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 8082	. . . . . . . . . 10 ⊢ (𝐵 ∈ (ℕ₀ ↑_𝑚 (𝐶 ∪ {𝐷})) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
13	12	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
14		elun1 3942	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷}))
15	14	adantl 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷}))
16	13, 15	ffvelrnd 6550	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℕ₀)
17	16	faccld 13275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ)
18	17	nncnd 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ)
19		2fveq3 6380	. . . . 5 ⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷)))
20		snidg 4364	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷})
21	7, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ {𝐷})
22		elun2 3943	. . . . . . . . 9 ⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷}))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷}))
24	12, 23	ffvelrnd 6550	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℕ₀)
25	24	faccld 13275	. . . . . 6 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ)
26	25	nncnd 11292	. . . . 5 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ)
27	1, 2, 6, 7, 9, 18, 19, 26	fprodsplitsn 15002	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))
28	27	oveq2d 6858	. . 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 3744	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
30		snssi 4493	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
32	4, 31	unssd 3951	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴)
33		ssfi 8387	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin)
34	3, 32, 33	syl2anc 579	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin)
35	12	ffvelrnda 6549	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈ ℕ₀)
36	34, 35	fsumnn0cl 14752	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℕ₀)
37	36	faccld 13275	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ)
38	37	nncnd 11292	. . . . . 6 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ)
39	1, 6, 18	fprodclf 15005	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ)
40	39, 26	mulcld 10314	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ)
41	17	nnne0d 11322	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0)
42	6, 18, 41	fprodn0 14992	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0)
43	25	nnne0d 11322	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0)
44	39, 26, 42, 43	mulne0d 10933	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0)
45	38, 40, 44	divcld 11055	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ)
46	45	mulid2d 10312	. . . 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 14752	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℕ₀)
49	48	faccld 13275	. . . . . . . 8 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
50	49	nncnd 11292	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ)
51		nnne0 11310	. . . . . . . 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 11053	. . . . . 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 10315	. . . . . . 7 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
56	55	oveq2d 6858	. . . . . 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 11086	. . . . . . 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 2799	. . . . 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 6860	. . . 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 11055	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ)
62	50, 50, 61, 39, 52, 42	divmul13d 11097	. . . 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 2799	. . 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 2803	. 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 11086	. . . . 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 3707	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘)
67	16	nn0cnd 11600	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ)
68		csbeq1a 3700	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
69		csbfv 6421	. . . . . . . . . . . . 13 ⊢ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)
70	69	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷))
71	24	nn0cnd 11600	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ)
72	70, 71	eqeltrd 2844	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ)
73	1, 66, 6, 29, 9, 67, 68, 72	fsumsplitsn 14759	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
74	73	oveq1d 6857	. . . . . . . . 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 11600	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ)
76	75, 72	pncan2d 10648	. . . . . . . . 9 ⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
77	74, 76, 70	3eqtrrd 2804	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
78	77	fveq2d 6379	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
79	78	oveq1d 6857	. . . . . 6 ⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
80	79	oveq2d 6858	. . . . 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 11636	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
82	36	nn0zd 11727	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ)
83	48	nn0zd 11727	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ)
84	81, 82, 83	3jca 1158	. . . . . . . . 9 ⊢ (𝜑 → (0 ∈ ℤ ∧ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ))
85	48	nn0ge0d 11601	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
86	24	nn0ge0d 11601	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐷))
87	70	eqcomd 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
88	86, 87	breqtrd 4835	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
89	48	nn0red 11599	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ)
90	24	nn0red 11599	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ)
91	70, 90	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ)
92	89, 91	addge01d 10869	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ↔ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))))
93	88, 92	mpbid 223	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
94	73	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
95	93, 94	breqtrd 4835	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
96	84, 85, 95	jca32 511	. . . . . . . 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 12540	. . . . . . . 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 225	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)))
99		bcval2 13296	. . . . . . 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 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)))
102	65, 80, 101	3eqtrd 2803	. . . 4 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
103		bccl2 13314	. . . . 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 2844	. . 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 3938	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷})
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷}))
109		elmapssres 8085	. . . . 5 ⊢ ((𝐵 ∈ (ℕ₀ ↑_𝑚 (𝐶 ∪ {𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_𝑚 𝐶))
110	10, 108, 109	syl2anc 579	. . . 4 ⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_𝑚 𝐶))
111		fveq1 6374	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
112	111	adantr 472	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
113		fvres 6394	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
114	113	adantl 473	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
115	112, 114	eqtrd 2799	. . . . . . . . 9 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘))
116	115	sumeq2dv 14718	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
117	116	fveq2d 6379	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
118	115	fveq2d 6379	. . . . . . . 8 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘)))
119	118	prodeq2dv 14936	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))
120	117, 119	oveq12d 6860	. . . . . 6 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
121	120	eleq1d 2829	. . . . 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 3460	. . . 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 579	. . 3 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)
124	105, 123	nnmulcld 11325	. 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 2844	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 384 ∧ w3a 1107 = wceq 1652 ∈ wcel 2155 ≠ wne 2937 ∀wral 3055 ⦋csb 3691 ∖ cdif 3729 ∪ cun 3730 ⊆ wss 3732 {csn 4334 class class class wbr 4809 ↾ cres 5279 ⟶wf 6064 ‘cfv 6068 (class class class)co 6842 ↑_𝑚 cmap 8060 Fincfn 8160 ℂcc 10187 ℝcr 10188 0cc0 10189 1c1 10190 + caddc 10192 · cmul 10194 ≤ cle 10329 − cmin 10520 / cdiv 10938 ℕcn 11274 ℕ₀cn0 11538 ℤcz 11624 ...cfz 12533 !cfa 13264 Ccbc 13293 Σcsu 14701 ∏cprod 14918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4930 ax-sep 4941 ax-nul 4949 ax-pow 5001 ax-pr 5062 ax-un 7147 ax-inf2 8753 ax-cnex 10245 ax-resscn 10246 ax-1cn 10247 ax-icn 10248 ax-addcl 10249 ax-addrcl 10250 ax-mulcl 10251 ax-mulrcl 10252 ax-mulcom 10253 ax-addass 10254 ax-mulass 10255 ax-distr 10256 ax-i2m1 10257 ax-1ne0 10258 ax-1rid 10259 ax-rnegex 10260 ax-rrecex 10261 ax-cnre 10262 ax-pre-lttri 10263 ax-pre-lttrn 10264 ax-pre-ltadd 10265 ax-pre-mulgt0 10266 ax-pre-sup 10267

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rmo 3063 df-rab 3064 df-v 3352 df-sbc 3597 df-csb 3692 df-dif 3735 df-un 3737 df-in 3739 df-ss 3746 df-pss 3748 df-nul 4080 df-if 4244 df-pw 4317 df-sn 4335 df-pr 4337 df-tp 4339 df-op 4341 df-uni 4595 df-int 4634 df-iun 4678 df-br 4810 df-opab 4872 df-mpt 4889 df-tr 4912 df-id 5185 df-eprel 5190 df-po 5198 df-so 5199 df-fr 5236 df-se 5237 df-we 5238 df-xp 5283 df-rel 5284 df-cnv 5285 df-co 5286 df-dm 5287 df-rn 5288 df-res 5289 df-ima 5290 df-pred 5865 df-ord 5911 df-on 5912 df-lim 5913 df-suc 5914 df-iota 6031 df-fun 6070 df-fn 6071 df-f 6072 df-f1 6073 df-fo 6074 df-f1o 6075 df-fv 6076 df-isom 6077 df-riota 6803 df-ov 6845 df-oprab 6846 df-mpt2 6847 df-om 7264 df-1st 7366 df-2nd 7367 df-wrecs 7610 df-recs 7672 df-rdg 7710 df-1o 7764 df-oadd 7768 df-er 7947 df-map 8062 df-en 8161 df-dom 8162 df-sdom 8163 df-fin 8164 df-sup 8555 df-oi 8622 df-card 9016 df-pnf 10330 df-mnf 10331 df-xr 10332 df-ltxr 10333 df-le 10334 df-sub 10522 df-neg 10523 df-div 10939 df-nn 11275 df-2 11335 df-3 11336 df-n0 11539 df-z 11625 df-uz 11887 df-rp 12029 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-cj 14124 df-re 14125 df-im 14126 df-sqrt 14260 df-abs 14261 df-clim 14504 df-sum 14702 df-prod 14919

This theorem is referenced by: mccl 40468

W3C validator