Theorem grporcan 31002

Description: Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grprcan.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grporcan	⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Proof of Theorem grporcan

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grprcan.1	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
2		eqid 2760	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
3	1, 2	grpoidinv2 30999	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))))
4		simpr 490	. . . . . . . . 9 ⊢ (((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → (𝐶𝐺𝑦) = (GId‘𝐺))
5	4	reximi 3100	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
6	5	adantl 487	. . . . . . 7 ⊢ (((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
7	3, 6	syl 18	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
8	7	ad2ant2rl 762	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
9		oveq1 7421	. . . . . . . . . . . 12 ⊢ ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
10	9	ad2antll 742	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
11	1	grpoass 30987	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
12	11	3anassrs 1381	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
13	12	adantlrl 733	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
14	13	adantrr 730	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
15	1	grpoass 30987	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
16	15	3exp2 1373	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ GrpOp → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → (𝑦 ∈ 𝑋 → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦))))))
17	16	imp42 432	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
18	17	adantllr 732	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
19	18	adantrr 730	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
20	10, 14, 19	3eqtr3d 2803	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
21	20	adantrrl 737	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
22		oveq2 7422	. . . . . . . . . . 11 ⊢ ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
23	22	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
24	23	adantl 487	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
25		oveq2 7422	. . . . . . . . . . 11 ⊢ ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
26	25	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
27	26	adantl 487	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
28	21, 24, 27	3eqtr3d 2803	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = (𝐵𝐺(GId‘𝐺)))
29	1, 2	grporid 31001	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
30	29	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
31	1, 2	grporid 31001	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
32	31	ad2ant2r 760	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
33	32	adantr 486	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
34	28, 30, 33	3eqtr3d 2803	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → 𝐴 = 𝐵)
35	34	exp45 444	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑦 ∈ 𝑋 → ((𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))))
36	35	rexlimdv 3161	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵)))
37	8, 36	mpd 16	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))
38		oveq1 7421	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝐺𝐶) = (𝐵𝐺𝐶))
39	37, 38	impbid1 228	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
40	39	exp43 442	. 2 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)))))
41	40	3imp2 1368	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  ran crn 5656  ‘cfv 6533  (class class class)co 7414  GrpOpcgr 30973  GIdcgi 30974

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-riota 7371  df-ov 7417  df-grpo 30977  df-gid 30978

This theorem is used by:  grpoinveu  31003  grpoid  31004  nvrcan  31108  ghomdiv  38645  rngorcan  38670  rngorz  38676