Theorem fprodle 11893

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 11806	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
2		prodeq1 11806	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
3	1, 2	breq12d 4056	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ ∅ 𝐵 ≤ ∏𝑘 ∈ ∅ 𝐶))
4		prodeq1 11806	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
5		prodeq1 11806	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝑦 𝐶)
6	4, 5	breq12d 4056	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶))
7		prodeq1 11806	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
8		prodeq1 11806	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
9	7, 8	breq12d 4056	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
10		prodeq1 11806	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
11		prodeq1 11806	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝐴 𝐶)
12	10, 11	breq12d 4056	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶))
13		prod0 11838	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
14		prod0 11838	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
15	13, 14	eqtr4i 2228	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝐵 = ∏𝑘 ∈ ∅ 𝐶
16		1re 8070	. . . . 5 ⊢ 1 ∈ ℝ
17	13, 16	eqeltri 2277	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 ∈ ℝ
18	17	eqlei 8165	. . 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 1550	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑦 ∈ Fin
22	20, 21	nfan 1587	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ Fin)
23		nfv 1550	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))
24	22, 23	nfan 1587	. . . . . . 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 3193	. . . . . . . 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 11867	. . . . . 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 11867	. . . . . 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 3176	. . . . . . 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 2576	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℝ)
43		nfv 1550	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝐵 ∈ ℝ
44		nfcsb1v 3125	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
45	44	nfel1 2358	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ
46		csbeq1a 3101	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
47	46	eleq1d 2273	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ))
48	43, 45, 47	cbvral 2733	. . . . . . . . 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 2552	. . . . . . . 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 2576	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℝ)
56		nfv 1550	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝐶 ∈ ℝ
57		nfcsb1v 3125	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
58	57	nfel1 2358	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ
59		csbeq1a 3101	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
60	59	eleq1d 2273	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ))
61	56, 58, 60	cbvral 2733	. . . . . . . . 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 2552	. . . . . . . 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 11890	. . . . . 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 2576	. . . . . . . 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 2347	. . . . . . . . 9 ⊢ Ⅎ𝑘0
75		nfcv 2347	. . . . . . . . 9 ⊢ Ⅎ𝑘 ≤
76	74, 75, 44	nfbr 4089	. . . . . . . 8 ⊢ Ⅎ𝑘0 ≤ ⦋𝑧 / 𝑘⦌𝐵
77	46	breq2d 4055	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑧 / 𝑘⦌𝐵))
78	76, 77	rspc 2870	. . . . . . 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 2576	. . . . . . 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 4089	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶
88	46, 59	breq12d 4056	. . . . . . 7 ⊢ (𝑘 = 𝑧 → (𝐵 ≤ 𝐶 ↔ ⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶))
89	87, 88	rspc 2870	. . . . . 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 9014	. . . 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 3177	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
93	30	recnd 8100	. . . . . . . 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 8100	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
96	24, 44, 25, 39, 92, 94, 46, 95	fprodsplitsn 11886	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
97	35	recnd 8100	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐶 ∈ ℂ)
98	65	recnd 8100	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
99	24, 57, 25, 39, 92, 97, 59, 98	fprodsplitsn 11886	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (∏𝑘 ∈ 𝑦 𝐶 · ⦋𝑧 / 𝑘⦌𝐶))
100	96, 99	breq12d 4056	. . . . 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 6988	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372  Ⅎwnf 1482   ∈ wcel 2175  ∀wral 2483  ⦋csb 3092   ∖ cdif 3162   ∪ cun 3163   ⊆ wss 3165  ∅c0 3459  {csn 3632   class class class wbr 4043  (class class class)co 5943  Fincfn 6826  ℂcc 7922  ℝcr 7923  0cc0 7924  1c1 7925   · cmul 7929   ≤ cle 8107  ∏cprod 11803

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-frec 6476  df-1o 6501  df-oadd 6505  df-er 6619  df-en 6827  df-dom 6828  df-fin 6829  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-ico 10015  df-fz 10130  df-fzo 10264  df-seqfrec 10591  df-exp 10682  df-ihash 10919  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-clim 11532  df-proddc 11804

This theorem is referenced by: (None)