Theorem vclcan 30828

Description: Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vclcan.1	⊢ 𝐺 = (1st ‘𝑊)
vclcan.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
vclcan	⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Proof of Theorem vclcan

Step	Hyp	Ref	Expression
1		vclcan.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2	1	vcgrp 30827	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3		vclcan.2	. . 3 ⊢ 𝑋 = ran 𝐺
4	3	grpolcan 30787	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
5	2, 4	sylan 591	1 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ran crn 5652  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  GrpOpcgr 30746  CVec_OLDcvc 30815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-1st 7974  df-2nd 7975  df-grpo 30750  df-gid 30751  df-ginv 30752  df-ablo 30802  df-vc 30816

This theorem is referenced by:  vc0  30831