Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6977 | . . . 4 ⊢ 1 ∈ ℂ | |
2 | addcl 7006 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 400 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
4 | adddi 7013 | . . . 4 ⊢ (((1 + 𝐴) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) | |
5 | 1, 4 | mp3an2 1220 | . . 3 ⊢ (((1 + 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) |
6 | 3, 5 | sylan 267 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵))) |
7 | 3 | mulid1d 7044 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
8 | 7 | adantr 261 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 1) = (1 + 𝐴)) |
9 | adddir 7018 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = ((1 · 𝐵) + (𝐴 · 𝐵))) | |
10 | 1, 9 | mp3an1 1219 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = ((1 · 𝐵) + (𝐴 · 𝐵))) |
11 | mulid2 7025 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
12 | 11 | adantl 262 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · 𝐵) = 𝐵) |
13 | 12 | oveq1d 5527 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 · 𝐵) + (𝐴 · 𝐵)) = (𝐵 + (𝐴 · 𝐵))) |
14 | 10, 13 | eqtrd 2072 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · 𝐵) = (𝐵 + (𝐴 · 𝐵))) |
15 | 8, 14 | oveq12d 5530 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((1 + 𝐴) · 1) + ((1 + 𝐴) · 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
16 | 6, 15 | eqtrd 2072 | 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 97 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 ℂcc 6887 1c1 6890 + caddc 6892 · cmul 6894 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-mulcl 6982 ax-mulcom 6985 ax-mulass 6987 ax-distr 6988 ax-1rid 6991 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: bernneq 9369 |
Copyright terms: Public domain | W3C validator |