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Mirrors > Home > ILE Home > Th. List > muladd11 | 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 7901 | . . . 4 ⊢ 1 ∈ ℂ | |
2 | addcl 7933 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 424 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
4 | adddi 7940 | . . . 4 ⊢ (((1 + 𝐴) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) | |
5 | 1, 4 | mp3an2 1325 | . . 3 ⊢ (((1 + 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) |
6 | 3, 5 | sylan 283 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) |
7 | 3 | mulridd 7971 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
8 | 7 | adantr 276 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
9 | adddir 7945 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = ((1 · 𝐵) + (𝐴 · 𝐵))) | |
10 | 1, 9 | mp3an1 1324 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = ((1 · 𝐵) + (𝐴 · 𝐵))) |
11 | mullid 7952 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
12 | 11 | adantl 277 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · 𝐵) = 𝐵) |
13 | 12 | oveq1d 5887 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 · 𝐵) + (𝐴 · 𝐵)) = (𝐵 + (𝐴 · 𝐵))) |
14 | 10, 13 | eqtrd 2210 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = (𝐵 + (𝐴 · 𝐵))) |
15 | 8, 14 | oveq12d 5890 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
16 | 6, 15 | eqtrd 2210 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 (class class class)co 5872 ℂcc 7806 1c1 7809 + caddc 7811 · cmul 7813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-resscn 7900 ax-1cn 7901 ax-icn 7903 ax-addcl 7904 ax-mulcl 7906 ax-mulcom 7909 ax-mulass 7911 ax-distr 7912 ax-1rid 7915 ax-cnre 7919 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-iota 5177 df-fv 5223 df-ov 5875 |
This theorem is referenced by: muladd11r 8109 bernneq 10635 |
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