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Mirrors > Home > ILE Home > Th. List > mul31 | GIF version |
Description: Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mul31 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcom 7667 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) | |
2 | 1 | oveq2d 5742 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐴 · (𝐶 · 𝐵))) |
3 | 2 | 3adant1 980 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐴 · (𝐶 · 𝐵))) |
4 | mulass 7669 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | |
5 | mulcl 7665 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 · 𝐵) ∈ ℂ) | |
6 | 5 | ancoms 266 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶 · 𝐵) ∈ ℂ) |
7 | 6 | 3adant1 980 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶 · 𝐵) ∈ ℂ) |
8 | simp1 962 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ) | |
9 | 7, 8 | mulcomd 7705 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐵) · 𝐴) = (𝐴 · (𝐶 · 𝐵))) |
10 | 3, 4, 9 | 3eqtr4d 2155 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 943 = wceq 1312 ∈ wcel 1461 (class class class)co 5726 ℂcc 7539 · cmul 7546 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-mulcl 7637 ax-mulcom 7640 ax-mulass 7642 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rex 2394 df-v 2657 df-un 3039 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-iota 5044 df-fv 5087 df-ov 5729 |
This theorem is referenced by: mul31d 7833 |
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