Theorem grplcan 18942

Description: Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)

Hypotheses

Ref	Expression
grplcan.b	⊢ 𝐵 = (Base‘𝐺)
grplcan.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
grplcan	⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Proof of Theorem grplcan

Step	Hyp	Ref	Expression
1		oveq2 7376	. . . . . 6 ⊢ ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
2	1	adantl 481	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
3		grplcan.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
4		grplcan.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
5		eqid 2737	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6		eqid 2737	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	3, 4, 5, 6	grplinv 18931	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
8	7	adantlr 716	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
9	8	oveq1d 7383	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g‘𝐺) + 𝑋))
10	3, 6	grpinvcl 18929	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
11	10	adantrl 717	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
12		simprr 773	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
13		simprl 771	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
14	11, 12, 13	3jca 1129	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
15	3, 4	grpass 18884	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
16	14, 15	syldan 592	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
17	16	anassrs 467	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
18	3, 4, 5	grplid 18909	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
19	18	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
20	9, 17, 19	3eqtr3d 2780	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
21	20	adantrl 717	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
22	21	adantr 480	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
23	7	adantrl 717	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
24	23	oveq1d 7383	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g‘𝐺) + 𝑌))
25	10	adantrl 717	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
26		simprr 773	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
27		simprl 771	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
28	25, 26, 27	3jca 1129	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
29	3, 4	grpass 18884	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
30	28, 29	syldan 592	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
31	3, 4, 5	grplid 18909	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
32	31	adantrr 718	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((0g‘𝐺) + 𝑌) = 𝑌)
33	24, 30, 32	3eqtr3d 2780	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
34	33	adantlr 716	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
35	34	adantr 480	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
36	2, 22, 35	3eqtr3d 2780	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
37	36	exp53 447	. . 3 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑍 ∈ 𝐵 → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌)))))
38	37	3imp2 1351	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39		oveq2 7376	. 2 ⊢ (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
40	38, 39	impbid1 225	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  Grpcgrp 18875  inv_gcminusg 18876

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-riota 7325  df-ov 7371  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879

This theorem is referenced by:  grpidrcan  18945  grpinvinv  18947  grplmulf1o  18955  grplactcnv  18985  conjghm  19190  conjnmzb  19194  sylow3lem2  19569  gex2abl  19792  rnglz  20112  ringcom  20227  lmodlcan  20840  lmodfopne  20863  r1peuqusdeg1  35856  isnumbasgrplem2  43450  grptcmon  49941