Theorem fprodle 11581

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 11494	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
2		prodeq1 11494	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
3	1, 2	breq12d 3995	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ ∅ 𝐵 ≤ ∏𝑘 ∈ ∅ 𝐶))
4		prodeq1 11494	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
5		prodeq1 11494	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝑦 𝐶)
6	4, 5	breq12d 3995	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶))
7		prodeq1 11494	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
8		prodeq1 11494	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
9	7, 8	breq12d 3995	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
10		prodeq1 11494	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
11		prodeq1 11494	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝐴 𝐶)
12	10, 11	breq12d 3995	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶))
13		prod0 11526	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
14		prod0 11526	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
15	13, 14	eqtr4i 2189	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝐵 = ∏𝑘 ∈ ∅ 𝐶
16		1re 7898	. . . . 5 ⊢ 1 ∈ ℝ
17	13, 16	eqeltri 2239	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 ∈ ℝ
18	17	eqlei 7992	. . 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 1516	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑦 ∈ Fin
22	20, 21	nfan 1553	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ Fin)
23		nfv 1516	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))
24	22, 23	nfan 1553	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
25		simplr 520	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
26		simplll 523	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
27		simplrl 525	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑦 ⊆ 𝐴)
28		simpr 109	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝑦)
29	27, 28	sseldd 3143	. . . . . . . 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 409	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℝ)
32	24, 25, 31	fprodreclf 11555	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℝ)
33	32	adantr 274	. . . . 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 409	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐶 ∈ ℝ)
36	24, 25, 35	fprodreclf 11555	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐶 ∈ ℝ)
37	36	adantr 274	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∏𝑘 ∈ 𝑦 𝐶 ∈ ℝ)
38		simpll 519	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑)
39		simprr 522	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
40	39	eldifad 3127	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
41	30	ex 114	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ ℝ))
42	20, 41	ralrimi 2537	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℝ)
43		nfv 1516	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝐵 ∈ ℝ
44		nfcsb1v 3078	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
45	44	nfel1 2319	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ
46		csbeq1a 3054	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
47	46	eleq1d 2235	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ))
48	43, 45, 47	cbvral 2688	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℝ ↔ ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)
49	42, 48	sylib 121	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)
50		rsp 2513	. . . . . . . 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 274	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)
54	34	ex 114	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℝ))
55	20, 54	ralrimi 2537	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℝ)
56		nfv 1516	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝐶 ∈ ℝ
57		nfcsb1v 3078	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
58	57	nfel1 2319	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ
59		csbeq1a 3054	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
60	59	eleq1d 2235	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ))
61	56, 58, 60	cbvral 2688	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ ℝ ↔ ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ)
62	55, 61	sylib 121	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ)
63		rsp 2513	. . . . . . . 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 274	. . . . 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 409	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 0 ≤ 𝐵)
69	24, 25, 31, 68	fprodge0 11578	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 0 ≤ ∏𝑘 ∈ 𝑦 𝐵)
70	69	adantr 274	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → 0 ≤ ∏𝑘 ∈ 𝑦 𝐵)
71	67	ex 114	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 0 ≤ 𝐵))
72	20, 71	ralrimi 2537	. . . . . . . 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 2308	. . . . . . . . 9 ⊢ Ⅎ𝑘0
75		nfcv 2308	. . . . . . . . 9 ⊢ Ⅎ𝑘 ≤
76	74, 75, 44	nfbr 4028	. . . . . . . 8 ⊢ Ⅎ𝑘0 ≤ ⦋𝑧 / 𝑘⦌𝐵
77	46	breq2d 3994	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑧 / 𝑘⦌𝐵))
78	76, 77	rspc 2824	. . . . . . 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 274	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → 0 ≤ ⦋𝑧 / 𝑘⦌𝐵)
81		simpr 109	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶)
82	40	adantr 274	. . . . . 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 114	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ≤ 𝐶))
85	20, 84	ralrimi 2537	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ≤ 𝐶)
86	85	ad3antrrr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∀𝑘 ∈ 𝐴 𝐵 ≤ 𝐶)
87	44, 75, 57	nfbr 4028	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶
88	46, 59	breq12d 3995	. . . . . . 7 ⊢ (𝑘 = 𝑧 → (𝐵 ≤ 𝐶 ↔ ⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶))
89	87, 88	rspc 2824	. . . . . 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 8837	. . . 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 3128	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
93	30	recnd 7927	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
94	26, 29, 93	syl2anc 409	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
95	52	recnd 7927	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
96	24, 44, 25, 39, 92, 94, 46, 95	fprodsplitsn 11574	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
97	35	recnd 7927	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐶 ∈ ℂ)
98	65	recnd 7927	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
99	24, 57, 25, 39, 92, 97, 59, 98	fprodsplitsn 11574	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (∏𝑘 ∈ 𝑦 𝐶 · ⦋𝑧 / 𝑘⦌𝐶))
100	96, 99	breq12d 3995	. . . . 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 274	. . . 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 166	. . 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 114	. 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 6858	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  ∀wral 2444  ⦋csb 3045   ∖ cdif 3113   ∪ cun 3114   ⊆ wss 3116  ∅c0 3409  {csn 3576   class class class wbr 3982  (class class class)co 5842  Fincfn 6706  ℂcc 7751  ℝcr 7752  0cc0 7753  1c1 7754   · cmul 7758   ≤ cle 7934  ∏cprod 11491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-proddc 11492

This theorem is referenced by: (None)