Step | Hyp | Ref
| Expression |
1 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵) |
2 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ ∅ 𝐶) |
3 | 1, 2 | breq12d 4002 |
. 2
⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ ∅ 𝐵 ≤ ∏𝑘 ∈ ∅ 𝐶)) |
4 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵) |
5 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝑦 𝐶) |
6 | 4, 5 | breq12d 4002 |
. 2
⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶)) |
7 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) |
8 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) |
9 | 7, 8 | breq12d 4002 |
. 2
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ≤ ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)) |
10 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵) |
11 | | prodeq1 11516 |
. . 3
⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝐴 𝐶) |
12 | 10, 11 | breq12d 4002 |
. 2
⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 𝐵 ≤ ∏𝑘 ∈ 𝑤 𝐶 ↔ ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶)) |
13 | | prod0 11548 |
. . . 4
⊢
∏𝑘 ∈
∅ 𝐵 =
1 |
14 | | prod0 11548 |
. . . 4
⊢
∏𝑘 ∈
∅ 𝐶 =
1 |
15 | 13, 14 | eqtr4i 2194 |
. . 3
⊢
∏𝑘 ∈
∅ 𝐵 = ∏𝑘 ∈ ∅ 𝐶 |
16 | | 1re 7919 |
. . . . 5
⊢ 1 ∈
ℝ |
17 | 13, 16 | eqeltri 2243 |
. . . 4
⊢
∏𝑘 ∈
∅ 𝐵 ∈
ℝ |
18 | 17 | eqlei 8013 |
. . 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 1521 |
. . . . . . . . 9
⊢
Ⅎ𝑘 𝑦 ∈ Fin |
22 | 20, 21 | nfan 1558 |
. . . . . . . 8
⊢
Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ Fin) |
23 | | nfv 1521 |
. . . . . . . 8
⊢
Ⅎ𝑘(𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) |
24 | 22, 23 | nfan 1558 |
. . . . . . 7
⊢
Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) |
25 | | simplr 525 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin) |
26 | | simplll 528 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑) |
27 | | simplrl 530 |
. . . . . . . . 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 3148 |
. . . . . . . 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 11577 |
. . . . . 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 11577 |
. . . . . 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 524 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑) |
39 | | simprr 527 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦)) |
40 | 39 | eldifad 3132 |
. . . . . . 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 2541 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
43 | | nfv 1521 |
. . . . . . . . . 10
⊢
Ⅎ𝑧 𝐵 ∈ ℝ |
44 | | nfcsb1v 3082 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 |
45 | 44 | nfel1 2323 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ |
46 | | csbeq1a 3058 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵) |
47 | 46 | eleq1d 2239 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℝ)) |
48 | 43, 45, 47 | cbvral 2692 |
. . . . . . . . 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 2517 |
. . . . . . . 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 2541 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℝ) |
56 | | nfv 1521 |
. . . . . . . . . 10
⊢
Ⅎ𝑧 𝐶 ∈ ℝ |
57 | | nfcsb1v 3082 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 |
58 | 57 | nfel1 2323 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ |
59 | | csbeq1a 3058 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶) |
60 | 59 | eleq1d 2239 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℝ)) |
61 | 56, 58, 60 | cbvral 2692 |
. . . . . . . . 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 2517 |
. . . . . . . 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 11600 |
. . . . . 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 2541 |
. . . . . . . 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 2312 |
. . . . . . . . 9
⊢
Ⅎ𝑘0 |
75 | | nfcv 2312 |
. . . . . . . . 9
⊢
Ⅎ𝑘
≤ |
76 | 74, 75, 44 | nfbr 4035 |
. . . . . . . 8
⊢
Ⅎ𝑘0 ≤
⦋𝑧 / 𝑘⦌𝐵 |
77 | 46 | breq2d 4001 |
. . . . . . . 8
⊢ (𝑘 = 𝑧 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑧 / 𝑘⦌𝐵)) |
78 | 76, 77 | rspc 2828 |
. . . . . . 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 2541 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ≤ 𝐶) |
86 | 85 | ad3antrrr 489 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 𝐵 ≤ ∏𝑘 ∈ 𝑦 𝐶) → ∀𝑘 ∈ 𝐴 𝐵 ≤ 𝐶) |
87 | 44, 75, 57 | nfbr 4035 |
. . . . . . 7
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶 |
88 | 46, 59 | breq12d 4002 |
. . . . . . 7
⊢ (𝑘 = 𝑧 → (𝐵 ≤ 𝐶 ↔ ⦋𝑧 / 𝑘⦌𝐵 ≤ ⦋𝑧 / 𝑘⦌𝐶)) |
89 | 87, 88 | rspc 2828 |
. . . . . 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 8858 |
. . . 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 3133 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦) |
93 | 30 | recnd 7948 |
. . . . . . . 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 7948 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) |
96 | 24, 44, 25, 39, 92, 94, 46, 95 | fprodsplitsn 11596 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) |
97 | 35 | recnd 7948 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐶 ∈ ℂ) |
98 | 65 | recnd 7948 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) |
99 | 24, 57, 25, 39, 92, 97, 59, 98 | fprodsplitsn 11596 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (∏𝑘 ∈ 𝑦 𝐶 · ⦋𝑧 / 𝑘⦌𝐶)) |
100 | 96, 99 | breq12d 4002 |
. . . . 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 6870 |
1
⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶) |