Proof of Theorem mccllem
Step | Hyp | Ref
| Expression |
1 | | nfv 1922 |
. . . . 5
⊢
Ⅎ𝑘𝜑 |
2 | | nfcv 2904 |
. . . . 5
⊢
Ⅎ𝑘(!‘(𝐵‘𝐷)) |
3 | | mccllem.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ Fin) |
4 | | mccllem.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
5 | | ssfi 8851 |
. . . . . 6
⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ Fin) |
6 | 3, 4, 5 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ Fin) |
7 | | mccllem.d |
. . . . 5
⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶)) |
8 | | eldifn 4042 |
. . . . . 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 8530 |
. . . . . . . . . 10
⊢ (𝐵 ∈ (ℕ0
↑m (𝐶 ∪
{𝐷})) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ0) |
12 | 10, 11 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ0) |
13 | 12 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ0) |
14 | | elun1 4090 |
. . . . . . . . 9
⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷})) |
15 | 14 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷})) |
16 | 13, 15 | ffvelrnd 6905 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈
ℕ0) |
17 | 16 | faccld 13850 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ) |
18 | 17 | nncnd 11846 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ) |
19 | | 2fveq3 6722 |
. . . . 5
⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷))) |
20 | | snidg 4575 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷}) |
21 | 7, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ {𝐷}) |
22 | | elun2 4091 |
. . . . . . . . 9
⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷})) |
23 | 21, 22 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷})) |
24 | 12, 23 | ffvelrnd 6905 |
. . . . . . 7
⊢ (𝜑 → (𝐵‘𝐷) ∈
ℕ0) |
25 | 24 | faccld 13850 |
. . . . . 6
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ) |
26 | 25 | nncnd 11846 |
. . . . 5
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ) |
27 | 1, 2, 6, 7, 9, 18,
19, 26 | fprodsplitsn 15551 |
. . . 4
⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) |
28 | 27 | oveq2d 7229 |
. . 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 3878 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 ∈ 𝐴) |
30 | | snssi 4721 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝐷} ⊆ 𝐴) |
32 | 4, 31 | unssd 4100 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴) |
33 | | ssfi 8851 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin) |
34 | 3, 32, 33 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin) |
35 | 12 | ffvelrnda 6904 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈
ℕ0) |
36 | 34, 35 | fsumnn0cl 15300 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈
ℕ0) |
37 | 36 | faccld 13850 |
. . . . . . 7
⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ) |
38 | 37 | nncnd 11846 |
. . . . . 6
⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ) |
39 | 1, 6, 18 | fprodclf 15554 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ) |
40 | 39, 26 | mulcld 10853 |
. . . . . 6
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ) |
41 | 17 | nnne0d 11880 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0) |
42 | 6, 18, 41 | fprodn0 15541 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0) |
43 | 25 | nnne0d 11880 |
. . . . . . 7
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0) |
44 | 39, 26, 42, 43 | mulne0d 11484 |
. . . . . 6
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0) |
45 | 38, 40, 44 | divcld 11608 |
. . . . 5
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ) |
46 | 45 | mulid2d 10851 |
. . . 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 2743 |
. . 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 15300 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈
ℕ0) |
49 | 48 | faccld 13850 |
. . . . . . . 8
⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
50 | 49 | nncnd 11846 |
. . . . . . 7
⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ) |
51 | | nnne0 11864 |
. . . . . . . 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 11606 |
. . . . . 6
⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = 1) |
54 | 53 | eqcomd 2743 |
. . . . 5
⊢ (𝜑 → 1 = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
55 | 39, 26 | mulcomd 10854 |
. . . . . . 7
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
56 | 55 | oveq2d 7229 |
. . . . . 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 11639 |
. . . . . . 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 2743 |
. . . . . 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 2777 |
. . . . 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 7231 |
. . . 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 11608 |
. . . . 5
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ) |
62 | 50, 50, 61, 39, 52, 42 | divmul13d 11650 |
. . . 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 2777 |
. . 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 2781 |
. 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 11639 |
. . . . 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 3836 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘) |
67 | 16 | nn0cnd 12152 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ) |
68 | | csbeq1a 3825 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
69 | | csbfv 6762 |
. . . . . . . . . . . . 13
⊢
⦋𝐷 /
𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷) |
70 | 69 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)) |
71 | 24 | nn0cnd 12152 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ) |
72 | 70, 71 | eqeltrd 2838 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ) |
73 | 1, 66, 6, 29, 9, 67, 68, 72 | fsumsplitsn 15308 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))) |
74 | 73 | oveq1d 7228 |
. . . . . . . . 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 12152 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ) |
76 | 75, 72 | pncan2d 11191 |
. . . . . . . . 9
⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
77 | 74, 76, 70 | 3eqtrrd 2782 |
. . . . . . . 8
⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
78 | 77 | fveq2d 6721 |
. . . . . . 7
⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
79 | 78 | oveq1d 7228 |
. . . . . 6
⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
80 | 79 | oveq2d 7229 |
. . . . 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 12188 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℤ) |
82 | 36 | nn0zd 12280 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ) |
83 | 48 | nn0zd 12280 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ) |
84 | 48 | nn0ge0d 12153 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) |
85 | 24 | nn0ge0d 12153 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 ≤ (𝐵‘𝐷)) |
86 | 70 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
87 | 85, 86 | breqtrd 5079 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ≤
⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
88 | 48 | nn0red 12151 |
. . . . . . . . . . 11
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ) |
89 | 24 | nn0red 12151 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ) |
90 | 70, 89 | eqeltrd 2838 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ) |
91 | 88, 90 | addge01d 11420 |
. . . . . . . . . 10
⊢ (𝜑 → (0 ≤
⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ↔ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))) |
92 | 87, 91 | mpbid 235 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))) |
93 | 73 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) |
94 | 92, 93 | breqtrd 5079 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) |
95 | 81, 82, 83, 84, 94 | elfzd 13103 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))) |
96 | | bcval2 13871 |
. . . . . . 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))))) |
97 | 95, 96 | 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))))) |
98 | 97 | eqcomd 2743 |
. . . . 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))) |
99 | 65, 80, 98 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
100 | | bccl2 13889 |
. . . . 5
⊢
(Σ𝑘 ∈
𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
101 | 95, 100 | syl 17 |
. . . 4
⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
102 | 99, 101 | eqeltrd 2838 |
. . 3
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) ∈ ℕ) |
103 | | mccllem.6 |
. . . 4
⊢ (𝜑 → ∀𝑏 ∈ (ℕ0
↑m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ) |
104 | | ssun1 4086 |
. . . . . 6
⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷}) |
105 | 104 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷})) |
106 | | elmapssres 8548 |
. . . . 5
⊢ ((𝐵 ∈ (ℕ0
↑m (𝐶 ∪
{𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ0
↑m 𝐶)) |
107 | 10, 105, 106 | syl2anc 587 |
. . . 4
⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ0
↑m 𝐶)) |
108 | | fveq1 6716 |
. . . . . . . . . . 11
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘)) |
109 | 108 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘)) |
110 | | fvres 6736 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘)) |
111 | 110 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘)) |
112 | 109, 111 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘)) |
113 | 112 | sumeq2dv 15267 |
. . . . . . . 8
⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) |
114 | 113 | fveq2d 6721 |
. . . . . . 7
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
115 | 112 | fveq2d 6721 |
. . . . . . . 8
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘))) |
116 | 115 | prodeq2dv 15485 |
. . . . . . 7
⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) |
117 | 114, 116 | oveq12d 7231 |
. . . . . 6
⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
118 | 117 | eleq1d 2822 |
. . . . 5
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ ↔
((!‘Σ𝑘 ∈
𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)) |
119 | 118 | rspccva 3536 |
. . . 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) |
120 | 103, 107,
119 | syl2anc 587 |
. . 3
⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ) |
121 | 102, 120 | nnmulcld 11883 |
. 2
⊢ (𝜑 → ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) ∈ ℕ) |
122 | 64, 121 | eqeltrd 2838 |
1
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ) |