mccllem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mccllem		Structured version Visualization version GIF version

Theorem mccllem 43393

Description: * Induction step for mccl 43394. (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 1916	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑘(!‘(𝐵‘𝐷))
3		mccllem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
4		mccllem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
5		ssfi 9016	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ Fin)
6	3, 4, 5	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin)
7		mccllem.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))
8		eldifn 4072	. . . . . 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 8686	. . . . . . . . . 10 ⊢ (𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
13	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ₀)
14		elun1 4120	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷}))
15	14	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷}))
16	13, 15	ffvelcdmd 7001	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℕ₀)
17	16	faccld 14077	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ)
18	17	nncnd 12068	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ)
19		2fveq3 6816	. . . . 5 ⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷)))
20		snidg 4604	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷})
21	7, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ {𝐷})
22		elun2 4121	. . . . . . . . 9 ⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷}))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷}))
24	12, 23	ffvelcdmd 7001	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℕ₀)
25	24	faccld 14077	. . . . . 6 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ)
26	25	nncnd 12068	. . . . 5 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ)
27	1, 2, 6, 7, 9, 18, 19, 26	fprodsplitsn 15775	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))
28	27	oveq2d 7332	. . 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 3908	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
30		snssi 4752	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐷} ⊆ 𝐴)
32	4, 31	unssd 4130	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴)
33		ssfi 9016	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin)
34	3, 32, 33	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin)
35	12	ffvelcdmda 7000	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈ ℕ₀)
36	34, 35	fsumnn0cl 15524	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℕ₀)
37	36	faccld 14077	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ)
38	37	nncnd 12068	. . . . . 6 ⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ)
39	1, 6, 18	fprodclf 15778	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ)
40	39, 26	mulcld 11074	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ)
41	17	nnne0d 12102	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0)
42	6, 18, 41	fprodn0 15765	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0)
43	25	nnne0d 12102	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0)
44	39, 26, 42, 43	mulne0d 11706	. . . . . 6 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0)
45	38, 40, 44	divcld 11830	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ)
46	45	mulid2d 11072	. . . 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 2742	. . 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 15524	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℕ₀)
49	48	faccld 14077	. . . . . . . 8 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
50	49	nncnd 12068	. . . . . . 7 ⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ)
51		nnne0 12086	. . . . . . . 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 11828	. . . . . 6 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = 1)
54	53	eqcomd 2742	. . . . 5 ⊢ (𝜑 → 1 = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
55	39, 26	mulcomd 11075	. . . . . . 7 ⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
56	55	oveq2d 7332	. . . . . 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 11861	. . . . . . 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 2742	. . . . . 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 2776	. . . . 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 7334	. . . 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 11830	. . . . 5 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ)
62	50, 50, 61, 39, 52, 42	divmul13d 11872	. . . 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 2776	. . 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 2780	. 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 11861	. . . . 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 3866	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘)
67	16	nn0cnd 12374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ)
68		csbeq1a 3855	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
69		csbfv 6858	. . . . . . . . . . . . 13 ⊢ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)
70	69	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷))
71	24	nn0cnd 12374	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ)
72	70, 71	eqeltrd 2837	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ)
73	1, 66, 6, 29, 9, 67, 68, 72	fsumsplitsn 15532	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
74	73	oveq1d 7331	. . . . . . . . 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 12374	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ)
76	75, 72	pncan2d 11413	. . . . . . . . 9 ⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
77	74, 76, 70	3eqtrrd 2781	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
78	77	fveq2d 6815	. . . . . . 7 ⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
79	78	oveq1d 7331	. . . . . 6 ⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))
80	79	oveq2d 7332	. . . . 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 12410	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
82	36	nn0zd 12503	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ)
83	48	nn0zd 12503	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ)
84	48	nn0ge0d 12375	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
85	24	nn0ge0d 12375	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐷))
86	70	eqcomd 2742	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
87	85, 86	breqtrd 5112	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))
88	48	nn0red 12373	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ)
89	24	nn0red 12373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ)
90	70, 89	eqeltrd 2837	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ)
91	88, 90	addge01d 11642	. . . . . . . . . 10 ⊢ (𝜑 → (0 ≤ ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ↔ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))))
92	87, 91	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))
93	73	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
94	92, 93	breqtrd 5112	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))
95	81, 82, 83, 84, 94	elfzd 13326	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)))
96		bcval2 14098	. . . . . . 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 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)))))
98	97	eqcomd 2742	. . . . 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 2780	. . . 4 ⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
100		bccl2 14116	. . . . 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 17	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ)
102	99, 101	eqeltrd 2837	. . 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 4116	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷})
105	104	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷}))
106		elmapssres 8704	. . . . 5 ⊢ ((𝐵 ∈ (ℕ₀ ↑_m (𝐶 ∪ {𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
107	10, 105, 106	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ₀ ↑_m 𝐶))
108		fveq1 6810	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
109	108	adantr 481	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘))
110		fvres 6830	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
111	110	adantl 482	. . . . . . . . . 10 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘))
112	109, 111	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘))
113	112	sumeq2dv 15491	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))
114	113	fveq2d 6815	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))
115	112	fveq2d 6815	. . . . . . . 8 ⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘)))
116	115	prodeq2dv 15709	. . . . . . 7 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))
117	114, 116	oveq12d 7334	. . . . . 6 ⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))
118	117	eleq1d 2821	. . . . 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 3568	. . . 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 584	. . 3 ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)
121	102, 120	nnmulcld 12105	. 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 2837	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 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∀wral 3061 ⦋csb 3841 ∖ cdif 3893 ∪ cun 3894 ⊆ wss 3896 {csn 4570 class class class wbr 5086 ↾ cres 5609 ⟶wf 6461 ‘cfv 6465 (class class class)co 7316 ↑_m cmap 8664 Fincfn 8782 ℂcc 10948 ℝcr 10949 0cc0 10950 1c1 10951 + caddc 10953 · cmul 10955 ≤ cle 11089 − cmin 11284 / cdiv 11711 ℕcn 12052 ℕ₀cn0 12312 ...cfz 13318 !cfa 14066 Ccbc 14095 Σcsu 15473 ∏cprod 15691

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-inf2 9476 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-pre-sup 11028

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-om 7759 df-1st 7877 df-2nd 7878 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-er 8547 df-map 8666 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-sup 9277 df-oi 9345 df-card 9774 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-nn 12053 df-2 12115 df-3 12116 df-n0 12313 df-z 12399 df-uz 12662 df-rp 12810 df-fz 13319 df-fzo 13462 df-seq 13801 df-exp 13862 df-fac 14067 df-bc 14096 df-hash 14124 df-cj 14886 df-re 14887 df-im 14888 df-sqrt 15022 df-abs 15023 df-clim 15273 df-sum 15474 df-prod 15692

This theorem is referenced by: mccl 43394

W3C validator