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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 29393 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺)) |
4 | 3 | oveq1d 7366 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴)) |
5 | 4 | adantr 481 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴)) |
6 | 1 | nvgrp 29404 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
7 | nv0id.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 1 | bafval 29391 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
9 | eqid 2736 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
10 | 8, 9 | grpolid 29303 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴) |
11 | 6, 10 | sylan 580 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴) |
12 | 5, 11 | eqtrd 2776 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 GrpOpcgr 29276 GIdcgi 29277 NrmCVeccnv 29371 +𝑣 cpv 29372 BaseSetcba 29373 0veccn0v 29375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-1st 7913 df-2nd 7914 df-grpo 29280 df-gid 29281 df-ablo 29332 df-vc 29346 df-nv 29379 df-va 29382 df-ba 29383 df-sm 29384 df-0v 29385 df-nmcv 29387 |
This theorem is referenced by: nvpncan2 29440 nvmeq0 29445 imsmetlem 29477 ipdirilem 29616 |
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