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Mirrors > Home > MPE Home > Th. List > nvrinv | Structured version Visualization version GIF version |
Description: A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvrinv.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvrinv.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvrinv.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvrinv.6 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
nvrinv | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvrinv.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | 1 | nvgrp 30421 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvrinv.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 30408 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
5 | eqid 2728 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
6 | eqid 2728 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
7 | 4, 5, 6 | grporinv 30331 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = (GId‘𝐺)) |
8 | 2, 7 | sylan 579 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = (GId‘𝐺)) |
9 | nvrinv.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
10 | 3, 1, 9, 6 | nvinv 30443 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘𝐺)‘𝐴)) |
11 | 10 | oveq2d 7431 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (𝐴𝐺((inv‘𝐺)‘𝐴))) |
12 | nvrinv.6 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
13 | 1, 12 | 0vfval 30410 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺)) |
14 | 13 | adantr 480 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘𝐺)) |
15 | 8, 11, 14 | 3eqtr4d 2778 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7415 1c1 11134 -cneg 11470 GrpOpcgr 30293 GIdcgi 30294 invcgn 30295 NrmCVeccnv 30388 +𝑣 cpv 30389 BaseSetcba 30390 ·𝑠OLD cns 30391 0veccn0v 30392 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7988 df-2nd 7989 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-ltxr 11278 df-sub 11471 df-neg 11472 df-grpo 30297 df-gid 30298 df-ginv 30299 df-ablo 30349 df-vc 30363 df-nv 30396 df-va 30399 df-ba 30400 df-sm 30401 df-0v 30402 df-nmcv 30404 |
This theorem is referenced by: nvpncan2 30457 ipidsq 30514 ip2i 30632 ipdirilem 30633 ipasslem2 30636 |
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