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Mirrors > Home > HSE Home > Th. List > hvmulcomi | Structured version Visualization version GIF version |
Description: Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvmulcom.1 | ⊢ 𝐴 ∈ ℂ |
hvmulcom.2 | ⊢ 𝐵 ∈ ℂ |
hvmulcom.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
hvmulcomi | ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvmulcom.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | hvmulcom.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | hvmulcom.3 | . 2 ⊢ 𝐶 ∈ ℋ | |
4 | hvmulcom 28822 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1457 | 1 ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 ℋchba 28698 ·ℎ csm 28700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-mulcom 10603 ax-hvmulass 28786 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: (None) |
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