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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 2622 | . . 3 ⊢ (2nd ‘𝑊) = (2nd ‘𝑊) | |
3 | eqid 2622 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
4 | 1, 2, 3 | vciOLD 27416 | . 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 1073 | 1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 × cxp 5112 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 AbelOpcablo 27398 CVecOLDcvc 27413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 df-vc 27414 |
This theorem is referenced by: vcgrp 27425 nvablo 27471 ip0i 27680 ipdirilem 27684 |
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