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Mirrors > Home > MPE Home > Th. List > nvcom | Structured version Visualization version GIF version |
Description: The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvgcl.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvgcl.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvcom | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvgcl.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | 1 | nvablo 28551 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp) |
3 | nvgcl.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 28539 | . . 3 ⊢ 𝑋 = ran 𝐺 |
5 | 4 | ablocom 28483 | . 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
6 | 2, 5 | syl3an1 1164 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 (class class class)co 7170 AbelOpcablo 28479 NrmCVeccnv 28519 +𝑣 cpv 28520 BaseSetcba 28521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-1st 7714 df-2nd 7715 df-ablo 28480 df-vc 28494 df-nv 28527 df-va 28530 df-ba 28531 df-sm 28532 df-0v 28533 df-nmcv 28535 |
This theorem is referenced by: nvmval2 28578 nvpncan 28589 nvdif 28601 nvpi 28602 nvabs 28607 dipcj 28649 hlcom 28835 |
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