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| Mirrors > Home > MPE Home > Th. List > muladd11 | Structured version Visualization version GIF version | ||
| Description: A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.) |
| Ref | Expression |
|---|---|
| muladd11 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 2 | addcl 11237 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
| 4 | adddi 11244 | . . . 4 ⊢ (((1 + 𝐴) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) | |
| 5 | 1, 4 | mp3an2 1451 | . . 3 ⊢ (((1 + 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) |
| 6 | 3, 5 | sylan 580 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) |
| 7 | 3 | mulridd 11278 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
| 9 | adddir 11252 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = ((1 · 𝐵) + (𝐴 · 𝐵))) | |
| 10 | 1, 9 | mp3an1 1450 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = ((1 · 𝐵) + (𝐴 · 𝐵))) |
| 11 | mullid 11260 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
| 12 | 11 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · 𝐵) = 𝐵) |
| 13 | 12 | oveq1d 7446 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 · 𝐵) + (𝐴 · 𝐵)) = (𝐵 + (𝐴 · 𝐵))) |
| 14 | 10, 13 | eqtrd 2777 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = (𝐵 + (𝐴 · 𝐵))) |
| 15 | 8, 14 | oveq12d 7449 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
| 16 | 6, 15 | eqtrd 2777 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulcom 11219 ax-mulass 11221 ax-distr 11222 ax-1rid 11225 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: muladd11r 11474 bernneq 14268 |
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