Proof of Theorem flcidc
Step | Hyp | Ref
| Expression |
1 | | flcidc.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))) |
2 | 1 | fveq1d 6193 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑖) = ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖)) |
3 | 2 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) = ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖)) |
4 | | flcidc.k |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ 𝑆) |
5 | 4 | snssd 4340 |
. . . . . . . . 9
⊢ (𝜑 → {𝐾} ⊆ 𝑆) |
6 | 5 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → 𝑖 ∈ 𝑆) |
7 | | eqeq1 2626 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 → (𝑗 = 𝐾 ↔ 𝑖 = 𝐾)) |
8 | 7 | ifbid 4108 |
. . . . . . . . 9
⊢ (𝑗 = 𝑖 → if(𝑗 = 𝐾, 1, 0) = if(𝑖 = 𝐾, 1, 0)) |
9 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0)) = (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0)) |
10 | | 1ex 10035 |
. . . . . . . . . 10
⊢ 1 ∈
V |
11 | | c0ex 10034 |
. . . . . . . . . 10
⊢ 0 ∈
V |
12 | 10, 11 | ifex 4156 |
. . . . . . . . 9
⊢ if(𝑖 = 𝐾, 1, 0) ∈ V |
13 | 8, 9, 12 | fvmpt 6282 |
. . . . . . . 8
⊢ (𝑖 ∈ 𝑆 → ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖) = if(𝑖 = 𝐾, 1, 0)) |
14 | 6, 13 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖) = if(𝑖 = 𝐾, 1, 0)) |
15 | 3, 14 | eqtrd 2656 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) = if(𝑖 = 𝐾, 1, 0)) |
16 | | elsni 4194 |
. . . . . . . 8
⊢ (𝑖 ∈ {𝐾} → 𝑖 = 𝐾) |
17 | 16 | iftrued 4094 |
. . . . . . 7
⊢ (𝑖 ∈ {𝐾} → if(𝑖 = 𝐾, 1, 0) = 1) |
18 | 17 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → if(𝑖 = 𝐾, 1, 0) = 1) |
19 | 15, 18 | eqtrd 2656 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) = 1) |
20 | 19 | oveq1d 6665 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → ((𝐹‘𝑖) · 𝐵) = (1 · 𝐵)) |
21 | | flcidc.b |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑆) → 𝐵 ∈ ℂ) |
22 | 6, 21 | syldan 487 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → 𝐵 ∈ ℂ) |
23 | 22 | mulid2d 10058 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (1 · 𝐵) = 𝐵) |
24 | 20, 23 | eqtrd 2656 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → ((𝐹‘𝑖) · 𝐵) = 𝐵) |
25 | 24 | sumeq2dv 14433 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ {𝐾} ((𝐹‘𝑖) · 𝐵) = Σ𝑖 ∈ {𝐾}𝐵) |
26 | | ax-1cn 9994 |
. . . . . 6
⊢ 1 ∈
ℂ |
27 | | 0cn 10032 |
. . . . . 6
⊢ 0 ∈
ℂ |
28 | 26, 27 | keepel 4155 |
. . . . 5
⊢ if(𝑖 = 𝐾, 1, 0) ∈ ℂ |
29 | 15, 28 | syl6eqel 2709 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) ∈ ℂ) |
30 | 29, 22 | mulcld 10060 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → ((𝐹‘𝑖) · 𝐵) ∈ ℂ) |
31 | 2 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (𝐹‘𝑖) = ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖)) |
32 | | eldifi 3732 |
. . . . . . . . 9
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → 𝑖 ∈ 𝑆) |
33 | 32 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → 𝑖 ∈ 𝑆) |
34 | 33, 13 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖) = if(𝑖 = 𝐾, 1, 0)) |
35 | 31, 34 | eqtrd 2656 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (𝐹‘𝑖) = if(𝑖 = 𝐾, 1, 0)) |
36 | | eldifn 3733 |
. . . . . . . . 9
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → ¬ 𝑖 ∈ {𝐾}) |
37 | | velsn 4193 |
. . . . . . . . 9
⊢ (𝑖 ∈ {𝐾} ↔ 𝑖 = 𝐾) |
38 | 36, 37 | sylnib 318 |
. . . . . . . 8
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → ¬ 𝑖 = 𝐾) |
39 | 38 | iffalsed 4097 |
. . . . . . 7
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → if(𝑖 = 𝐾, 1, 0) = 0) |
40 | 39 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → if(𝑖 = 𝐾, 1, 0) = 0) |
41 | 35, 40 | eqtrd 2656 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (𝐹‘𝑖) = 0) |
42 | 41 | oveq1d 6665 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → ((𝐹‘𝑖) · 𝐵) = (0 · 𝐵)) |
43 | 33, 21 | syldan 487 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → 𝐵 ∈ ℂ) |
44 | 43 | mul02d 10234 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (0 · 𝐵) = 0) |
45 | 42, 44 | eqtrd 2656 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → ((𝐹‘𝑖) · 𝐵) = 0) |
46 | | flcidc.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ Fin) |
47 | 5, 30, 45, 46 | fsumss 14456 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ {𝐾} ((𝐹‘𝑖) · 𝐵) = Σ𝑖 ∈ 𝑆 ((𝐹‘𝑖) · 𝐵)) |
48 | | eleq1 2689 |
. . . . . . . 8
⊢ (𝑗 = 𝐾 → (𝑗 ∈ 𝑆 ↔ 𝐾 ∈ 𝑆)) |
49 | 48 | anbi2d 740 |
. . . . . . 7
⊢ (𝑗 = 𝐾 → ((𝜑 ∧ 𝑗 ∈ 𝑆) ↔ (𝜑 ∧ 𝐾 ∈ 𝑆))) |
50 | | csbeq1 3536 |
. . . . . . . 8
⊢ (𝑗 = 𝐾 → ⦋𝑗 / 𝑖⦌𝐵 = ⦋𝐾 / 𝑖⦌𝐵) |
51 | 50 | eleq1d 2686 |
. . . . . . 7
⊢ (𝑗 = 𝐾 → (⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ ↔ ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ)) |
52 | 49, 51 | imbi12d 334 |
. . . . . 6
⊢ (𝑗 = 𝐾 → (((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐾 ∈ 𝑆) → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ))) |
53 | | nfv 1843 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝑆) |
54 | | nfcsb1v 3549 |
. . . . . . . . 9
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐵 |
55 | 54 | nfel1 2779 |
. . . . . . . 8
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ |
56 | 53, 55 | nfim 1825 |
. . . . . . 7
⊢
Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ) |
57 | | eleq1 2689 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝑆 ↔ 𝑗 ∈ 𝑆)) |
58 | 57 | anbi2d 740 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ 𝑆) ↔ (𝜑 ∧ 𝑗 ∈ 𝑆))) |
59 | | csbeq1a 3542 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑖⦌𝐵) |
60 | 59 | eleq1d 2686 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ)) |
61 | 58, 60 | imbi12d 334 |
. . . . . . 7
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ 𝑆) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ))) |
62 | 56, 61, 21 | chvar 2262 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ) |
63 | 52, 62 | vtoclg 3266 |
. . . . 5
⊢ (𝐾 ∈ 𝑆 → ((𝜑 ∧ 𝐾 ∈ 𝑆) → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ)) |
64 | 63 | anabsi7 860 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆) → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ) |
65 | 4, 64 | mpdan 702 |
. . 3
⊢ (𝜑 → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ) |
66 | | sumsns 14479 |
. . 3
⊢ ((𝐾 ∈ 𝑆 ∧ ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ) → Σ𝑖 ∈ {𝐾}𝐵 = ⦋𝐾 / 𝑖⦌𝐵) |
67 | 4, 65, 66 | syl2anc 693 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ {𝐾}𝐵 = ⦋𝐾 / 𝑖⦌𝐵) |
68 | 25, 47, 67 | 3eqtr3d 2664 |
1
⊢ (𝜑 → Σ𝑖 ∈ 𝑆 ((𝐹‘𝑖) · 𝐵) = ⦋𝐾 / 𝑖⦌𝐵) |