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Mirrors > Home > MPE Home > Th. List > nv0lid | Structured version Visualization version GIF version |
Description: The zero vector is a left 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 |
---|---|
nv0lid | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nv0id.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | nv0id.6 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | 0vfval 28375 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺)) |
4 | 3 | oveq1d 7163 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴)) |
5 | 4 | adantr 483 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴)) |
6 | 1 | nvgrp 28386 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
7 | nv0id.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 1 | bafval 28373 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
9 | eqid 2819 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
10 | 8, 9 | grpolid 28285 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴) |
11 | 6, 10 | sylan 582 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴) |
12 | 5, 11 | eqtrd 2854 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 GrpOpcgr 28258 GIdcgi 28259 NrmCVeccnv 28353 +𝑣 cpv 28354 BaseSetcba 28355 0veccn0v 28357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-1st 7681 df-2nd 7682 df-grpo 28262 df-gid 28263 df-ablo 28314 df-vc 28328 df-nv 28361 df-va 28364 df-ba 28365 df-sm 28366 df-0v 28367 df-nmcv 28369 |
This theorem is referenced by: nvpncan2 28422 nvmeq0 28427 imsmetlem 28459 ipdirilem 28598 |
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