Proof of Theorem flcidc
| Step | Hyp | Ref
| Expression |
| 1 | | flcidc.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))) |
| 2 | 1 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑖) = ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖)) |
| 3 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) = ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖)) |
| 4 | | flcidc.k |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ 𝑆) |
| 5 | 4 | snssd 4809 |
. . . . . . . . 9
⊢ (𝜑 → {𝐾} ⊆ 𝑆) |
| 6 | 5 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → 𝑖 ∈ 𝑆) |
| 7 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 → (𝑗 = 𝐾 ↔ 𝑖 = 𝐾)) |
| 8 | 7 | ifbid 4549 |
. . . . . . . . 9
⊢ (𝑗 = 𝑖 → if(𝑗 = 𝐾, 1, 0) = if(𝑖 = 𝐾, 1, 0)) |
| 9 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0)) = (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0)) |
| 10 | | 1ex 11257 |
. . . . . . . . . 10
⊢ 1 ∈
V |
| 11 | | c0ex 11255 |
. . . . . . . . . 10
⊢ 0 ∈
V |
| 12 | 10, 11 | ifex 4576 |
. . . . . . . . 9
⊢ if(𝑖 = 𝐾, 1, 0) ∈ V |
| 13 | 8, 9, 12 | fvmpt 7016 |
. . . . . . . 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 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) = if(𝑖 = 𝐾, 1, 0)) |
| 16 | | elsni 4643 |
. . . . . . . 8
⊢ (𝑖 ∈ {𝐾} → 𝑖 = 𝐾) |
| 17 | 16 | iftrued 4533 |
. . . . . . 7
⊢ (𝑖 ∈ {𝐾} → if(𝑖 = 𝐾, 1, 0) = 1) |
| 18 | 17 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → if(𝑖 = 𝐾, 1, 0) = 1) |
| 19 | 15, 18 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) = 1) |
| 20 | 19 | oveq1d 7446 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → ((𝐹‘𝑖) · 𝐵) = (1 · 𝐵)) |
| 21 | | flcidc.b |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑆) → 𝐵 ∈ ℂ) |
| 22 | 6, 21 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → 𝐵 ∈ ℂ) |
| 23 | 22 | mullidd 11279 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (1 · 𝐵) = 𝐵) |
| 24 | 20, 23 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → ((𝐹‘𝑖) · 𝐵) = 𝐵) |
| 25 | 24 | sumeq2dv 15738 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ {𝐾} ((𝐹‘𝑖) · 𝐵) = Σ𝑖 ∈ {𝐾}𝐵) |
| 26 | | ax-1cn 11213 |
. . . . . 6
⊢ 1 ∈
ℂ |
| 27 | | 0cn 11253 |
. . . . . 6
⊢ 0 ∈
ℂ |
| 28 | 26, 27 | ifcli 4573 |
. . . . 5
⊢ if(𝑖 = 𝐾, 1, 0) ∈ ℂ |
| 29 | 15, 28 | eqeltrdi 2849 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → (𝐹‘𝑖) ∈ ℂ) |
| 30 | 29, 22 | mulcld 11281 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ {𝐾}) → ((𝐹‘𝑖) · 𝐵) ∈ ℂ) |
| 31 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (𝐹‘𝑖) = ((𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))‘𝑖)) |
| 32 | | eldifi 4131 |
. . . . . . . . 9
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → 𝑖 ∈ 𝑆) |
| 33 | 32 | adantl 481 |
. . . . . . . 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 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (𝐹‘𝑖) = if(𝑖 = 𝐾, 1, 0)) |
| 36 | | eldifn 4132 |
. . . . . . . . 9
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → ¬ 𝑖 ∈ {𝐾}) |
| 37 | | velsn 4642 |
. . . . . . . . 9
⊢ (𝑖 ∈ {𝐾} ↔ 𝑖 = 𝐾) |
| 38 | 36, 37 | sylnib 328 |
. . . . . . . 8
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → ¬ 𝑖 = 𝐾) |
| 39 | 38 | iffalsed 4536 |
. . . . . . 7
⊢ (𝑖 ∈ (𝑆 ∖ {𝐾}) → if(𝑖 = 𝐾, 1, 0) = 0) |
| 40 | 39 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → if(𝑖 = 𝐾, 1, 0) = 0) |
| 41 | 35, 40 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (𝐹‘𝑖) = 0) |
| 42 | 41 | oveq1d 7446 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → ((𝐹‘𝑖) · 𝐵) = (0 · 𝐵)) |
| 43 | 33, 21 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → 𝐵 ∈ ℂ) |
| 44 | 43 | mul02d 11459 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → (0 · 𝐵) = 0) |
| 45 | 42, 44 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑆 ∖ {𝐾})) → ((𝐹‘𝑖) · 𝐵) = 0) |
| 46 | | flcidc.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ Fin) |
| 47 | 5, 30, 45, 46 | fsumss 15761 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ {𝐾} ((𝐹‘𝑖) · 𝐵) = Σ𝑖 ∈ 𝑆 ((𝐹‘𝑖) · 𝐵)) |
| 48 | | eleq1 2829 |
. . . . . . . 8
⊢ (𝑗 = 𝐾 → (𝑗 ∈ 𝑆 ↔ 𝐾 ∈ 𝑆)) |
| 49 | 48 | anbi2d 630 |
. . . . . . 7
⊢ (𝑗 = 𝐾 → ((𝜑 ∧ 𝑗 ∈ 𝑆) ↔ (𝜑 ∧ 𝐾 ∈ 𝑆))) |
| 50 | | csbeq1 3902 |
. . . . . . . 8
⊢ (𝑗 = 𝐾 → ⦋𝑗 / 𝑖⦌𝐵 = ⦋𝐾 / 𝑖⦌𝐵) |
| 51 | 50 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑗 = 𝐾 → (⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ ↔ ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ)) |
| 52 | 49, 51 | imbi12d 344 |
. . . . . 6
⊢ (𝑗 = 𝐾 → (((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐾 ∈ 𝑆) → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ))) |
| 53 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝑆) |
| 54 | | nfcsb1v 3923 |
. . . . . . . . 9
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐵 |
| 55 | 54 | nfel1 2922 |
. . . . . . . 8
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ |
| 56 | 53, 55 | nfim 1896 |
. . . . . . 7
⊢
Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ) |
| 57 | | eleq1 2829 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝑆 ↔ 𝑗 ∈ 𝑆)) |
| 58 | 57 | anbi2d 630 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ 𝑆) ↔ (𝜑 ∧ 𝑗 ∈ 𝑆))) |
| 59 | | csbeq1a 3913 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑖⦌𝐵) |
| 60 | 59 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ)) |
| 61 | 58, 60 | imbi12d 344 |
. . . . . . 7
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ 𝑆) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ))) |
| 62 | 56, 61, 21 | chvarfv 2240 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑆) → ⦋𝑗 / 𝑖⦌𝐵 ∈ ℂ) |
| 63 | 52, 62 | vtoclg 3554 |
. . . . 5
⊢ (𝐾 ∈ 𝑆 → ((𝜑 ∧ 𝐾 ∈ 𝑆) → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ)) |
| 64 | 63 | anabsi7 671 |
. . . 4
⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆) → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ) |
| 65 | 4, 64 | mpdan 687 |
. . 3
⊢ (𝜑 → ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ) |
| 66 | | sumsns 15786 |
. . 3
⊢ ((𝐾 ∈ 𝑆 ∧ ⦋𝐾 / 𝑖⦌𝐵 ∈ ℂ) → Σ𝑖 ∈ {𝐾}𝐵 = ⦋𝐾 / 𝑖⦌𝐵) |
| 67 | 4, 65, 66 | syl2anc 584 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ {𝐾}𝐵 = ⦋𝐾 / 𝑖⦌𝐵) |
| 68 | 25, 47, 67 | 3eqtr3d 2785 |
1
⊢ (𝜑 → Σ𝑖 ∈ 𝑆 ((𝐹‘𝑖) · 𝐵) = ⦋𝐾 / 𝑖⦌𝐵) |