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Mirrors > Home > HSE Home > Th. List > hvmulcom | Structured version Visualization version GIF version |
Description: Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvmulcom | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcom 10617 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | |
2 | 1 | oveq1d 7165 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = ((𝐵 · 𝐴) ·ℎ 𝐶)) |
3 | 2 | 3adant3 1128 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = ((𝐵 · 𝐴) ·ℎ 𝐶)) |
4 | ax-hvmulass 28778 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | |
5 | ax-hvmulass 28778 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐴) ·ℎ 𝐶) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) | |
6 | 5 | 3com12 1119 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐴) ·ℎ 𝐶) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) |
7 | 3, 4, 6 | 3eqtr3d 2864 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 · cmul 10536 ℋchba 28690 ·ℎ csm 28692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-mulcom 10595 ax-hvmulass 28778 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 |
This theorem is referenced by: hvmulcomi 28818 hvsubdistr1 28820 lnopmi 29771 |
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