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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 10752 | . . . 4 ⊢ 1 ∈ ℂ | |
2 | addcl 10776 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
4 | adddi 10783 | . . . 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 583 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) |
7 | 3 | mulid1d 10815 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
9 | adddir 10789 | . . . . 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 | mulid2 10797 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
12 | 11 | adantl 485 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · 𝐵) = 𝐵) |
13 | 12 | oveq1d 7206 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 · 𝐵) + (𝐴 · 𝐵)) = (𝐵 + (𝐴 · 𝐵))) |
14 | 10, 13 | eqtrd 2771 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = (𝐵 + (𝐴 · 𝐵))) |
15 | 8, 14 | oveq12d 7209 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
16 | 6, 15 | eqtrd 2771 | 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 399 = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℂcc 10692 1c1 10695 + caddc 10697 · cmul 10699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-mulcl 10756 ax-mulcom 10758 ax-mulass 10760 ax-distr 10761 ax-1rid 10764 ax-cnre 10767 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 |
This theorem is referenced by: muladd11r 11010 bernneq 13761 |
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