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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 31075 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1461 | 1 ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 (class class class)co 7448 ℂcc 11182 ℋchba 30951 ·ℎ csm 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-mulcom 11248 ax-hvmulass 31039 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-iota 6525 df-fv 6581 df-ov 7451 |
This theorem is referenced by: (None) |
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