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Mirrors > Home > MPE Home > Th. List > nv0rid | Structured version Visualization version GIF version |
Description: The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nv0id.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nv0id.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nv0id.6 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
nv0rid | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nv0id.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | nv0id.6 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | 0vfval 28377 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺)) |
4 | 3 | oveq2d 7166 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺))) |
5 | 4 | adantr 483 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺))) |
6 | 1 | nvgrp 28388 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
7 | nv0id.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 1 | bafval 28375 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
9 | eqid 2821 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
10 | 8, 9 | grporid 28288 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴) |
11 | 6, 10 | sylan 582 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴) |
12 | 5, 11 | eqtrd 2856 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 GrpOpcgr 28260 GIdcgi 28261 NrmCVeccnv 28355 +𝑣 cpv 28356 BaseSetcba 28357 0veccn0v 28359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 |
This theorem is referenced by: nvabs 28443 nvnd 28459 imsmetlem 28461 lnomul 28531 0lno 28561 ipdirilem 28600 hladdid 28674 |
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