Theorem grplcan 19240

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 7451	. . . . . 6 ⊢ ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
2	1	adantl 487	. . . . 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 2763	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6		eqid 2763	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	3, 4, 5, 6	grplinv 19229	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
8	7	adantlr 728	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
9	8	oveq1d 7458	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g‘𝐺) + 𝑋))
10	3, 6	grpinvcl 19227	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
11	10	adantrl 729	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
12		simprr 785	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
13		simprl 783	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
14	11, 12, 13	3jca 1146	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
15	3, 4	grpass 19182	. . . . . . . . . 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 603	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
17	16	anassrs 473	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
18	3, 4, 5	grplid 19207	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
19	18	adantr 486	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
20	9, 17, 19	3eqtr3d 2806	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
21	20	adantrl 729	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
22	21	adantr 486	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
23	7	adantrl 729	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
24	23	oveq1d 7458	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g‘𝐺) + 𝑌))
25	10	adantrl 729	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
26		simprr 785	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
27		simprl 783	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
28	25, 26, 27	3jca 1146	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
29	3, 4	grpass 19182	. . . . . . . . 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 603	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
31	3, 4, 5	grplid 19207	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
32	31	adantrr 730	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((0g‘𝐺) + 𝑌) = 𝑌)
33	24, 30, 32	3eqtr3d 2806	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
34	33	adantlr 728	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
35	34	adantr 486	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
36	2, 22, 35	3eqtr3d 2806	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
37	36	exp53 453	. . 3 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑍 ∈ 𝐵 → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌)))))
38	37	3imp2 1368	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39		oveq2 7451	. 2 ⊢ (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
40	38, 39	impbid1 228	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  +_gcplusg 17457  0_gc0g 17639  Grpcgrp 19173  inv_gcminusg 19174

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-riota 7400  df-ov 7446  df-0g 17641  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-grp 19176  df-minusg 19177

This theorem is used by:  grpidrcan  19243  grpinvinv  19245  grplmulf1o  19252  grplactcnv  19282  conjghm  19492  conjnmzb  19496  sylow3lem2  19872  gex2abl  20095  rnglz  20417  ringcom  20539  lmodlcan  21182  lmodfopne  21205  r1peuqusdeg1  36575  isnumbasgrplem2  44298  grptcmon  50936