Proof of Theorem mccllem
Step | Hyp | Ref
| Expression |
1 | | nfv 1896 |
. . . . 5
⊢
Ⅎ𝑘𝜑 |
2 | | nfcv 2951 |
. . . . 5
⊢
Ⅎ𝑘(!‘(𝐵‘𝐷)) |
3 | | mccllem.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ Fin) |
4 | | mccllem.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
5 | | ssfi 8591 |
. . . . . 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 4031 |
. . . . . 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 8285 |
. . . . . . . . . 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 4079 |
. . . . . . . . 9
⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷})) |
15 | 14 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷})) |
16 | 13, 15 | ffvelrnd 6724 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈
ℕ0) |
17 | 16 | faccld 13498 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ) |
18 | 17 | nncnd 11508 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ) |
19 | | 2fveq3 6550 |
. . . . 5
⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷))) |
20 | | snidg 4510 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷}) |
21 | 7, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ {𝐷}) |
22 | | elun2 4080 |
. . . . . . . . 9
⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷})) |
23 | 21, 22 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷})) |
24 | 12, 23 | ffvelrnd 6724 |
. . . . . . 7
⊢ (𝜑 → (𝐵‘𝐷) ∈
ℕ0) |
25 | 24 | faccld 13498 |
. . . . . 6
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ) |
26 | 25 | nncnd 11508 |
. . . . 5
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ) |
27 | 1, 2, 6, 7, 9, 18,
19, 26 | fprodsplitsn 15180 |
. . . 4
⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) |
28 | 27 | oveq2d 7039 |
. . 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 3877 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 ∈ 𝐴) |
30 | | snssi 4654 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝐷} ⊆ 𝐴) |
32 | 4, 31 | unssd 4089 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴) |
33 | | ssfi 8591 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin) |
34 | 3, 32, 33 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin) |
35 | 12 | ffvelrnda 6723 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈
ℕ0) |
36 | 34, 35 | fsumnn0cl 14930 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈
ℕ0) |
37 | 36 | faccld 13498 |
. . . . . . 7
⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ) |
38 | 37 | nncnd 11508 |
. . . . . 6
⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ) |
39 | 1, 6, 18 | fprodclf 15183 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ) |
40 | 39, 26 | mulcld 10514 |
. . . . . 6
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ) |
41 | 17 | nnne0d 11541 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0) |
42 | 6, 18, 41 | fprodn0 15170 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0) |
43 | 25 | nnne0d 11541 |
. . . . . . 7
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0) |
44 | 39, 26, 42, 43 | mulne0d 11146 |
. . . . . 6
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0) |
45 | 38, 40, 44 | divcld 11270 |
. . . . 5
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ) |
46 | 45 | mulid2d 10512 |
. . . 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 2803 |
. . 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 14930 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈
ℕ0) |
49 | 48 | faccld 13498 |
. . . . . . . 8
⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
50 | 49 | nncnd 11508 |
. . . . . . 7
⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ) |
51 | | nnne0 11525 |
. . . . . . . 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 11268 |
. . . . . 6
⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = 1) |
54 | 53 | eqcomd 2803 |
. . . . 5
⊢ (𝜑 → 1 = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
55 | 39, 26 | mulcomd 10515 |
. . . . . . 7
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
56 | 55 | oveq2d 7039 |
. . . . . 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 11301 |
. . . . . . 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 2803 |
. . . . . 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 2833 |
. . . . 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 7041 |
. . . 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 11270 |
. . . . 5
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ) |
62 | 50, 50, 61, 39, 52, 42 | divmul13d 11312 |
. . . 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 2833 |
. . 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 2837 |
. 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 11301 |
. . . . 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 3839 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘) |
67 | 16 | nn0cnd 11811 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ) |
68 | | csbeq1a 3830 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
69 | | csbfv 6590 |
. . . . . . . . . . . . 13
⊢
⦋𝐷 /
𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷) |
70 | 69 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)) |
71 | 24 | nn0cnd 11811 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ) |
72 | 70, 71 | eqeltrd 2885 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ) |
73 | 1, 66, 6, 29, 9, 67, 68, 72 | fsumsplitsn 14937 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))) |
74 | 73 | oveq1d 7038 |
. . . . . . . . 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 11811 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ) |
76 | 75, 72 | pncan2d 10853 |
. . . . . . . . 9
⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
77 | 74, 76, 70 | 3eqtrrd 2838 |
. . . . . . . 8
⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
78 | 77 | fveq2d 6549 |
. . . . . . 7
⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
79 | 78 | oveq1d 7038 |
. . . . . 6
⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
80 | 79 | oveq2d 7039 |
. . . . 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 11847 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℤ) |
82 | 36 | nn0zd 11939 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ) |
83 | 48 | nn0zd 11939 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ) |
84 | 81, 82, 83 | 3jca 1121 |
. . . . . . . . 9
⊢ (𝜑 → (0 ∈ ℤ ∧
Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ)) |
85 | 48 | nn0ge0d 11812 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) |
86 | 24 | nn0ge0d 11812 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝐵‘𝐷)) |
87 | 70 | eqcomd 2803 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
88 | 86, 87 | breqtrd 4994 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 ≤
⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
89 | 48 | nn0red 11810 |
. . . . . . . . . . . 12
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ) |
90 | 24 | nn0red 11810 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ) |
91 | 70, 90 | eqeltrd 2885 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ) |
92 | 89, 91 | addge01d 11082 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 ≤
⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ↔ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))) |
93 | 88, 92 | mpbid 233 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))) |
94 | 73 | eqcomd 2803 |
. . . . . . . . . 10
⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) |
95 | 93, 94 | breqtrd 4994 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) |
96 | 84, 85, 95 | jca32 516 |
. . . . . . . 8
⊢ (𝜑 → ((0 ∈ ℤ ∧
Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ) ∧ (0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)))) |
97 | | elfz2 12753 |
. . . . . . . 8
⊢
(Σ𝑘 ∈
𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ↔ ((0 ∈ ℤ ∧
Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ) ∧ (0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)))) |
98 | 96, 97 | sylibr 235 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))) |
99 | | bcval2 13519 |
. . . . . . 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))))) |
100 | 98, 99 | 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))))) |
101 | 100 | eqcomd 2803 |
. . . . 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))) |
102 | 65, 80, 101 | 3eqtrd 2837 |
. . . 4
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
103 | | bccl2 13537 |
. . . . 5
⊢
(Σ𝑘 ∈
𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
104 | 98, 103 | syl 17 |
. . . 4
⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
105 | 102, 104 | eqeltrd 2885 |
. . 3
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) ∈ ℕ) |
106 | | mccllem.6 |
. . . 4
⊢ (𝜑 → ∀𝑏 ∈ (ℕ0
↑𝑚 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ) |
107 | | ssun1 4075 |
. . . . . 6
⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷}) |
108 | 107 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷})) |
109 | | elmapssres 8288 |
. . . . 5
⊢ ((𝐵 ∈ (ℕ0
↑𝑚 (𝐶 ∪ {𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ0
↑𝑚 𝐶)) |
110 | 10, 108, 109 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ0
↑𝑚 𝐶)) |
111 | | fveq1 6544 |
. . . . . . . . . . 11
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘)) |
112 | 111 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘)) |
113 | | fvres 6564 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘)) |
114 | 113 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘)) |
115 | 112, 114 | eqtrd 2833 |
. . . . . . . . 9
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘)) |
116 | 115 | sumeq2dv 14897 |
. . . . . . . 8
⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) |
117 | 116 | fveq2d 6549 |
. . . . . . 7
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
118 | 115 | fveq2d 6549 |
. . . . . . . 8
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘))) |
119 | 118 | prodeq2dv 15114 |
. . . . . . 7
⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) |
120 | 117, 119 | oveq12d 7041 |
. . . . . 6
⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
121 | 120 | eleq1d 2869 |
. . . . 5
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ ↔
((!‘Σ𝑘 ∈
𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)) |
122 | 121 | rspccva 3560 |
. . . 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) |
123 | 106, 110,
122 | syl2anc 584 |
. . 3
⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ) |
124 | 105, 123 | nnmulcld 11544 |
. 2
⊢ (𝜑 → ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) ∈ ℕ) |
125 | 64, 124 | eqeltrd 2885 |
1
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ) |