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Mirrors > Home > MPE Home > Th. List > vcgrp | Structured version Visualization version GIF version |
Description: Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vcabl.1 | ⊢ 𝐺 = (1st ‘𝑊) |
Ref | Expression |
---|---|
vcgrp | ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vcabl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | 1 | vcablo 28507 | . 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp) |
3 | ablogrpo 28485 | . 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 1st c1st 7715 GrpOpcgr 28427 AbelOpcablo 28482 CVecOLDcvc 28496 |
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 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7482 |
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 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-ov 7176 df-1st 7717 df-2nd 7718 df-ablo 28483 df-vc 28497 |
This theorem is referenced by: vclcan 28509 vczcl 28510 vc0rid 28511 vcm 28514 |
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