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Mirrors > Home > ILE Home > Th. List > mul12d | GIF version |
Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
muld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcomd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mul12d.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
mul12d | ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addcomd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mul12d.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | mul12 7859 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) | |
5 | 1, 2, 3, 4 | syl3anc 1201 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 · cmul 7593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-mulcom 7689 ax-mulass 7691 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: mulreim 8334 divrecap 8416 remullem 10611 cvgratnnlemnexp 11261 cvgratnnlemmn 11262 tanval3ap 11348 sinadd 11370 dvdscmulr 11449 bezoutlemnewy 11611 dvdsmulgcd 11640 lcmgcdlem 11685 cncongr1 11711 tangtx 12846 |
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