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Mirrors > Home > MPE Home > Th. List > vclcan | Structured version Visualization version GIF version |
Description: Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vclcan.1 | ⊢ 𝐺 = (1st ‘𝑊) |
vclcan.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
vclcan | ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vclcan.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | 1 | vcgrp 28139 | . 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp) |
3 | vclcan.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 3 | grpolcan 28099 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵)) |
5 | 2, 4 | sylan 572 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ran crn 5412 ‘cfv 6193 (class class class)co 6982 1st c1st 7505 GrpOpcgr 28058 CVecOLDcvc 28127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-1st 7507 df-2nd 7508 df-grpo 28062 df-gid 28063 df-ginv 28064 df-ablo 28114 df-vc 28128 |
This theorem is referenced by: vc0 28143 |
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