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Mirrors > Home > MPE Home > Th. List > vcablo | Structured version Visualization version GIF version |
Description: Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vcabl.1 | ⊢ 𝐺 = (1st ‘𝑊) |
Ref | Expression |
---|---|
vcablo | ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vcabl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | eqid 2823 | . . 3 ⊢ (2nd ‘𝑊) = (2nd ‘𝑊) | |
3 | eqid 2823 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
4 | 1, 2, 3 | vciOLD 28340 | . 2 ⊢ (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ (2nd ‘𝑊):(ℂ × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺((1(2nd ‘𝑊)𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝐺(𝑦(2nd ‘𝑊)(𝑥𝐺𝑧)) = ((𝑦(2nd ‘𝑊)𝑥)𝐺(𝑦(2nd ‘𝑊)𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)(2nd ‘𝑊)𝑥) = ((𝑦(2nd ‘𝑊)𝑥)𝐺(𝑧(2nd ‘𝑊)𝑥)) ∧ ((𝑦 · 𝑧)(2nd ‘𝑊)𝑥) = (𝑦(2nd ‘𝑊)(𝑧(2nd ‘𝑊)𝑥))))))) |
5 | 4 | simp1d 1138 | 1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 × cxp 5555 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 ℂcc 10537 1c1 10540 + caddc 10542 · cmul 10544 AbelOpcablo 28323 CVecOLDcvc 28337 |
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-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 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-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-1st 7691 df-2nd 7692 df-vc 28338 |
This theorem is referenced by: vcgrp 28349 nvablo 28395 ip0i 28604 ipdirilem 28608 |
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