Theorem fprodle 11632

Description: If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
fprodle.kph	⊢ Ⅎ𝑘𝜑
fprodle.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodle.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
fprodle.0l3b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
fprodle.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)
fprodle.blec	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)

Assertion

Ref	Expression
fprodle	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶)

Distinct variable group:   𝐴,𝑘

Allowed substitution hints:   𝜑(𝑘)   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem fprodle

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodeq1 11545	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
2		prodeq1 11545	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
3	1, 2	breq12d 4013	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ ∅ 𝐵 ≤ ∏𝑘 ∈ ∅ 𝐶))
4		prodeq1 11545	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
5		prodeq1 11545	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝑦 𝐶)
6	4, 5	breq12d 4013	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶))
7		prodeq1 11545	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
8		prodeq1 11545	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
9	7, 8	breq12d 4013	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
10		prodeq1 11545	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
11		prodeq1 11545	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝐴 𝐶)
12	10, 11	breq12d 4013	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶))
13		prod0 11577	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
14		prod0 11577	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
15	13, 14	eqtr4i 2201	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝐵 = ∏𝑘 ∈ ∅ 𝐶
16		1re 7947	. . . . 5 ⊢ 1 ∈ ℝ
17	13, 16	eqeltri 2250	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 ∈ ℝ
18	17	eqlei 8041	. . 3 ⊢ (∏𝑘 ∈ ∅ 𝐵 = ∏𝑘 ∈ ∅ 𝐶 → ∏𝑘 ∈ ∅ 𝐵 ≤ ∏𝑘 ∈ ∅ 𝐶)
19	15, 18	mp1i 10	. 2 ⊢ (𝜑 → ∏𝑘 ∈ ∅ 𝐵 ≤ ∏𝑘 ∈ ∅ 𝐶)
20		fprodle.kph	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
21		nfv 1528	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑦 ∈ Fin
22	20, 21	nfan 1565	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ Fin)
23		nfv 1528	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))
24	22, 23	nfan 1565	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
25		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
26		simplll 533	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
27		simplrl 535	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑦 ⊆ 𝐴)
28		simpr 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝑦)
29	27, 28	sseldd 3156	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
30		fprodle.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
31	26, 29, 30	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℝ)
32	24, 25, 31	fprodreclf 11606	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℝ)
33	32	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℝ)
34		fprodle.c	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)
35	26, 29, 34	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐶 ∈ ℝ)
36	24, 25, 35	fprodreclf 11606	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐶 ∈ ℝ)
37	36	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∏𝑘 ∈ 𝑦 𝐶 ∈ ℝ)
38		simpll 527	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑)
39		simprr 531	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
40	39	eldifad 3140	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
41	30	ex 115	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ ℝ))
42	20, 41	ralrimi 2548	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℝ)
43		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝐵 ∈ ℝ
44		nfcsb1v 3090	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
45	44	nfel1 2330	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ
46		csbeq1a 3066	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
47	46	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ))
48	43, 45, 47	cbvral 2699	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℝ ↔ ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)
49	42, 48	sylib 122	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)
50		rsp 2524	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ → (𝑧 ∈ 𝐴 → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ))
51	49, 50	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ))
52	38, 40, 51	sylc 62	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)
53	52	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)
54	34	ex 115	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℝ))
55	20, 54	ralrimi 2548	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℝ)
56		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝐶 ∈ ℝ
57		nfcsb1v 3090	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
58	57	nfel1 2330	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ
59		csbeq1a 3066	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
60	59	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ))
61	56, 58, 60	cbvral 2699	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ ℝ ↔ ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ)
62	55, 61	sylib 122	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ)
63		rsp 2524	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ → (𝑧 ∈ 𝐴 → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ))
64	62, 63	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ))
65	38, 40, 64	sylc 62	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ)
66	65	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ)
67		fprodle.0l3b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
68	26, 29, 67	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 0 ≤ 𝐵)
69	24, 25, 31, 68	fprodge0 11629	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 0 ≤ ∏𝑘 ∈ 𝑦 𝐵)
70	69	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → 0 ≤ ∏𝑘 ∈ 𝑦 𝐵)
71	67	ex 115	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 0 ≤ 𝐵))
72	20, 71	ralrimi 2548	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 0 ≤ 𝐵)
73	38, 72	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 0 ≤ 𝐵)
74		nfcv 2319	. . . . . . . . 9 ⊢ Ⅎ𝑘0
75		nfcv 2319	. . . . . . . . 9 ⊢ Ⅎ𝑘 ≤
76	74, 75, 44	nfbr 4046	. . . . . . . 8 ⊢ Ⅎ𝑘0 ≤ ⦋𝑧 / 𝑘⦌𝐵
77	46	breq2d 4012	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑧 / 𝑘⦌𝐵))
78	76, 77	rspc 2835	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 0 ≤ 𝐵 → 0 ≤ ⦋𝑧 / 𝑘⦌𝐵))
79	40, 73, 78	sylc 62	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 0 ≤ ⦋𝑧 / 𝑘⦌𝐵)
80	79	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → 0 ≤ ⦋𝑧 / 𝑘⦌𝐵)
81		simpr 110	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶)
82	40	adantr 276	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → 𝑧 ∈ 𝐴)
83		fprodle.blec	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)
84	83	ex 115	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ≤ 𝐶))
85	20, 84	ralrimi 2548	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ≤ 𝐶)
86	85	ad3antrrr 492	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∀𝑘 ∈ 𝐴 𝐵 ≤ 𝐶)
87	44, 75, 57	nfbr 4046	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶
88	46, 59	breq12d 4013	. . . . . . 7 ⊢ (𝑘 = 𝑧 → (𝐵 ≤ 𝐶 ↔ ⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶))
89	87, 88	rspc 2835	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ≤ 𝐶 → ⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶))
90	82, 86, 89	sylc 62	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶)
91	33, 37, 53, 66, 70, 80, 81, 90	lemul12ad 8888	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵) ≤ (∏𝑘 ∈ 𝑦 𝐶 · ⦋𝑧 / 𝑘⦌𝐶))
92	39	eldifbd 3141	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
93	30	recnd 7976	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
94	26, 29, 93	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
95	52	recnd 7976	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
96	24, 44, 25, 39, 92, 94, 46, 95	fprodsplitsn 11625	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
97	35	recnd 7976	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐶 ∈ ℂ)
98	65	recnd 7976	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
99	24, 57, 25, 39, 92, 97, 59, 98	fprodsplitsn 11625	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (∏𝑘 ∈ 𝑦 𝐶 · ⦋𝑧 / 𝑘⦌𝐶))
100	96, 99	breq12d 4013	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 ↔ (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵) ≤ (∏𝑘 ∈ 𝑦 𝐶 · ⦋𝑧 / 𝑘⦌𝐶)))
101	100	adantr 276	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 ↔ (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵) ≤ (∏𝑘 ∈ 𝑦 𝐶 · ⦋𝑧 / 𝑘⦌𝐶)))
102	91, 101	mpbird 167	. . 3 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
103	102	ex 115	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶 → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
104		fprodle.a	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
105	3, 6, 9, 12, 19, 103, 104	findcard2sd 6886	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  Ⅎwnf 1460   ∈ wcel 2148  ∀wral 2455  ⦋csb 3057   ∖ cdif 3126   ∪ cun 3127   ⊆ wss 3129  ∅c0 3422  {csn 3591   class class class wbr 4000  (class class class)co 5869  Fincfn 6734  ℂcc 7800  ℝcr 7801  0cc0 7802  1c1 7803   · cmul 7807   ≤ cle 7983  ∏cprod 11542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-ico 9881  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543

This theorem is referenced by: (None)