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Mirrors > Home > MPE Home > Th. List > nvrcan | Structured version Visualization version GIF version |
Description: Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvgcl.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvgcl.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvrcan | ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvgcl.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | 1 | nvgrp 28388 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvgcl.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 28375 | . . 3 ⊢ 𝑋 = ran 𝐺 |
5 | 4 | grporcan 28289 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)) |
6 | 2, 5 | sylan 582 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 GrpOpcgr 28260 NrmCVeccnv 28355 +𝑣 cpv 28356 BaseSetcba 28357 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 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: nvmeq0 28429 imsmetlem 28461 |
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