Proof of Theorem mccllem
Step | Hyp | Ref
| Expression |
1 | | nfv 1830 |
. . . . 5
⊢
Ⅎ𝑘𝜑 |
2 | | nfcv 2751 |
. . . . 5
⊢
Ⅎ𝑘(!‘(𝐵‘𝐷)) |
3 | | mccllem.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ Fin) |
4 | | mccllem.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
5 | | ssfi 8065 |
. . . . . 6
⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ Fin) |
6 | 3, 4, 5 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ Fin) |
7 | | mccllem.d |
. . . . 5
⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶)) |
8 | | eldifn 3695 |
. . . . . 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 7765 |
. . . . . . . . . 10
⊢ (𝐵 ∈ (ℕ0
↑𝑚 (𝐶 ∪ {𝐷})) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ0) |
12 | 10, 11 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ0) |
13 | 12 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵:(𝐶 ∪ {𝐷})⟶ℕ0) |
14 | | elun1 3742 |
. . . . . . . . 9
⊢ (𝑘 ∈ 𝐶 → 𝑘 ∈ (𝐶 ∪ {𝐷})) |
15 | 14 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐶 ∪ {𝐷})) |
16 | 13, 15 | ffvelrnd 6268 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈
ℕ0) |
17 | 16 | faccld 12933 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℕ) |
18 | 17 | nncnd 10913 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ∈ ℂ) |
19 | | fveq2 6103 |
. . . . . 6
⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = (𝐵‘𝐷)) |
20 | 19 | fveq2d 6107 |
. . . . 5
⊢ (𝑘 = 𝐷 → (!‘(𝐵‘𝑘)) = (!‘(𝐵‘𝐷))) |
21 | | snidg 4153 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ {𝐷}) |
22 | 7, 21 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ {𝐷}) |
23 | | elun2 3743 |
. . . . . . . . 9
⊢ (𝐷 ∈ {𝐷} → 𝐷 ∈ (𝐶 ∪ {𝐷})) |
24 | 22, 23 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ (𝐶 ∪ {𝐷})) |
25 | 12, 24 | ffvelrnd 6268 |
. . . . . . 7
⊢ (𝜑 → (𝐵‘𝐷) ∈
ℕ0) |
26 | 25 | faccld 12933 |
. . . . . 6
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℕ) |
27 | 26 | nncnd 10913 |
. . . . 5
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ∈ ℂ) |
28 | 1, 2, 6, 7, 9, 18,
20, 27 | fprodsplitsn 14559 |
. . . 4
⊢ (𝜑 → ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘)) = (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) |
29 | 28 | oveq2d 6565 |
. . 3
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))))) |
30 | 7 | eldifad 3552 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 ∈ 𝐴) |
31 | | snssi 4280 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴) |
32 | 30, 31 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝐷} ⊆ 𝐴) |
33 | 4, 32 | unssd 3751 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∪ {𝐷}) ⊆ 𝐴) |
34 | | ssfi 8065 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Fin ∧ (𝐶 ∪ {𝐷}) ⊆ 𝐴) → (𝐶 ∪ {𝐷}) ∈ Fin) |
35 | 3, 33, 34 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → (𝐶 ∪ {𝐷}) ∈ Fin) |
36 | 12 | ffvelrnda 6267 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∪ {𝐷})) → (𝐵‘𝑘) ∈
ℕ0) |
37 | 35, 36 | fsumnn0cl 14314 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈
ℕ0) |
38 | 37 | faccld 12933 |
. . . . . . 7
⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℕ) |
39 | 38 | nncnd 10913 |
. . . . . 6
⊢ (𝜑 → (!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) ∈ ℂ) |
40 | 1, 6, 18 | fprodclf 14562 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ∈ ℂ) |
41 | 40, 27 | mulcld 9939 |
. . . . . 6
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ∈ ℂ) |
42 | 17 | nnne0d 10942 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (!‘(𝐵‘𝑘)) ≠ 0) |
43 | 6, 18, 42 | fprodn0 14548 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) ≠ 0) |
44 | 26 | nnne0d 10942 |
. . . . . . 7
⊢ (𝜑 → (!‘(𝐵‘𝐷)) ≠ 0) |
45 | 40, 27, 43, 44 | mulne0d 10558 |
. . . . . 6
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) ≠ 0) |
46 | 39, 41, 45 | divcld 10680 |
. . . . 5
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) ∈ ℂ) |
47 | 46 | mulid2d 9937 |
. . . 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))))) |
48 | 47 | eqcomd 2616 |
. . 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)))))) |
49 | 6, 16 | fsumnn0cl 14314 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈
ℕ0) |
50 | 49 | faccld 12933 |
. . . . . . . 8
⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
51 | 50 | nncnd 10913 |
. . . . . . 7
⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℂ) |
52 | | nnne0 10930 |
. . . . . . . 8
⊢
((!‘Σ𝑘
∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ →
(!‘Σ𝑘 ∈
𝐶 (𝐵‘𝑘)) ≠ 0) |
53 | 50, 52 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ≠ 0) |
54 | 51, 53 | dividd 10678 |
. . . . . 6
⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = 1) |
55 | 54 | eqcomd 2616 |
. . . . 5
⊢ (𝜑 → 1 = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
56 | 40, 27 | mulcomd 9940 |
. . . . . . 7
⊢ (𝜑 → (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷))) = ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
57 | 56 | oveq2d 6565 |
. . . . . 6
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))) |
58 | 39, 27, 40, 44, 43 | divdiv1d 10711 |
. . . . . . 7
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))))) |
59 | 58 | eqcomd 2616 |
. . . . . 6
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) = (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
60 | 57, 59 | eqtrd 2644 |
. . . . 5
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)) · (!‘(𝐵‘𝐷)))) = (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
61 | 55, 60 | oveq12d 6567 |
. . . 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))))) |
62 | 39, 27, 44 | divcld 10680 |
. . . . 5
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) ∈ ℂ) |
63 | 51, 51, 62, 40, 53, 43 | divmul13d 10722 |
. . . 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))))) |
64 | 61, 63 | eqtrd 2644 |
. . 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))))) |
65 | 29, 48, 64 | 3eqtrd 2648 |
. 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))))) |
66 | 39, 27, 51, 44, 53 | divdiv1d 10711 |
. . . . 5
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))))) |
67 | | nfcsb1v 3515 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝐷 / 𝑘⦌(𝐵‘𝑘) |
68 | 16 | nn0cnd 11230 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵‘𝑘) ∈ ℂ) |
69 | | csbeq1a 3508 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐷 → (𝐵‘𝑘) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
70 | | csbfv 6143 |
. . . . . . . . . . . . 13
⊢
⦋𝐷 /
𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷) |
71 | 70 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) = (𝐵‘𝐷)) |
72 | 25 | nn0cnd 11230 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵‘𝐷) ∈ ℂ) |
73 | 71, 72 | eqeltrd 2688 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℂ) |
74 | 1, 67, 6, 30, 9, 68, 69, 73 | fsumsplitsn 38637 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) = (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))) |
75 | 74 | oveq1d 6564 |
. . . . . . . . 9
⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
76 | 49 | nn0cnd 11230 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℂ) |
77 | 76, 73 | pncan2d 10273 |
. . . . . . . . 9
⊢ (𝜑 → ((Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
78 | 75, 77, 71 | 3eqtrrd 2649 |
. . . . . . . 8
⊢ (𝜑 → (𝐵‘𝐷) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
79 | 78 | fveq2d 6107 |
. . . . . . 7
⊢ (𝜑 → (!‘(𝐵‘𝐷)) = (!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
80 | 79 | oveq1d 6564 |
. . . . . 6
⊢ (𝜑 → ((!‘(𝐵‘𝐷)) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = ((!‘(Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) − Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)))) |
81 | 80 | oveq2d 6565 |
. . . . 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))))) |
82 | | 0zd 11266 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℤ) |
83 | 37 | nn0zd 11356 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ) |
84 | 49 | nn0zd 11356 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ) |
85 | 82, 83, 84 | 3jca 1235 |
. . . . . . . . 9
⊢ (𝜑 → (0 ∈ ℤ ∧
Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘) ∈ ℤ ∧ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℤ)) |
86 | 49 | nn0ge0d 11231 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) |
87 | 25 | nn0ge0d 11231 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝐵‘𝐷)) |
88 | 71 | eqcomd 2616 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵‘𝐷) = ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
89 | 87, 88 | breqtrd 4609 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 ≤
⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) |
90 | 49 | nn0red 11229 |
. . . . . . . . . . . 12
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ ℝ) |
91 | 25 | nn0red 11229 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐵‘𝐷) ∈ ℝ) |
92 | 71, 91 | eqeltrd 2688 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ∈ ℝ) |
93 | 90, 92 | addge01d 10494 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 ≤
⦋𝐷 / 𝑘⦌(𝐵‘𝑘) ↔ Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)))) |
94 | 89, 93 | mpbid 221 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘))) |
95 | 74 | eqcomd 2616 |
. . . . . . . . . 10
⊢ (𝜑 → (Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) + ⦋𝐷 / 𝑘⦌(𝐵‘𝑘)) = Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) |
96 | 94, 95 | breqtrd 4609 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ≤ Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) |
97 | 85, 86, 96 | jca32 556 |
. . . . . . . 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)))) |
98 | | elfz2 12204 |
. . . . . . . 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)))) |
99 | 97, 98 | sylibr 223 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ 𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘))) |
100 | | bcval2 12954 |
. . . . . . 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))))) |
101 | 99, 100 | 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))))) |
102 | 101 | eqcomd 2616 |
. . . . 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))) |
103 | 66, 81, 102 | 3eqtrd 2648 |
. . . 4
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) = (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
104 | | bccl2 12972 |
. . . . 5
⊢
(Σ𝑘 ∈
𝐶 (𝐵‘𝑘) ∈ (0...Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
105 | 99, 104 | syl 17 |
. . . 4
⊢ (𝜑 → (Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)CΣ𝑘 ∈ 𝐶 (𝐵‘𝑘)) ∈ ℕ) |
106 | 103, 105 | eqeltrd 2688 |
. . 3
⊢ (𝜑 → (((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) ∈ ℕ) |
107 | | mccllem.6 |
. . . 4
⊢ (𝜑 → ∀𝑏 ∈ (ℕ0
↑𝑚 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ) |
108 | | ssun1 3738 |
. . . . . 6
⊢ 𝐶 ⊆ (𝐶 ∪ {𝐷}) |
109 | 108 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ (𝐶 ∪ {𝐷})) |
110 | | elmapssres 7768 |
. . . . 5
⊢ ((𝐵 ∈ (ℕ0
↑𝑚 (𝐶 ∪ {𝐷})) ∧ 𝐶 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ↾ 𝐶) ∈ (ℕ0
↑𝑚 𝐶)) |
111 | 10, 109, 110 | syl2anc 691 |
. . . 4
⊢ (𝜑 → (𝐵 ↾ 𝐶) ∈ (ℕ0
↑𝑚 𝐶)) |
112 | | fveq1 6102 |
. . . . . . . . . . 11
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘)) |
113 | 112 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = ((𝐵 ↾ 𝐶)‘𝑘)) |
114 | | fvres 6117 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ 𝐶 → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘)) |
115 | 114 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → ((𝐵 ↾ 𝐶)‘𝑘) = (𝐵‘𝑘)) |
116 | 113, 115 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (𝑏‘𝑘) = (𝐵‘𝑘)) |
117 | 116 | sumeq2dv 14281 |
. . . . . . . 8
⊢ (𝑏 = (𝐵 ↾ 𝐶) → Σ𝑘 ∈ 𝐶 (𝑏‘𝑘) = Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) |
118 | 117 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) = (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) |
119 | 116 | fveq2d 6107 |
. . . . . . . 8
⊢ ((𝑏 = (𝐵 ↾ 𝐶) ∧ 𝑘 ∈ 𝐶) → (!‘(𝑏‘𝑘)) = (!‘(𝐵‘𝑘))) |
120 | 119 | prodeq2dv 14492 |
. . . . . . 7
⊢ (𝑏 = (𝐵 ↾ 𝐶) → ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘)) = ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) |
121 | 118, 120 | oveq12d 6567 |
. . . . . 6
⊢ (𝑏 = (𝐵 ↾ 𝐶) → ((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) = ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) |
122 | 121 | eleq1d 2672 |
. . . . 5
⊢ (𝑏 = (𝐵 ↾ 𝐶) → (((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ ↔
((!‘Σ𝑘 ∈
𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ)) |
123 | 122 | rspccva 3281 |
. . . 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) |
124 | 107, 111,
123 | syl2anc 691 |
. . 3
⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘))) ∈ ℕ) |
125 | 106, 124 | nnmulcld 10945 |
. 2
⊢ (𝜑 → ((((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / (!‘(𝐵‘𝐷))) / (!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘))) · ((!‘Σ𝑘 ∈ 𝐶 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝐵‘𝑘)))) ∈ ℕ) |
126 | 65, 125 | eqeltrd 2688 |
1
⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ) |