MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoidinv Structured version   Visualization version   GIF version

Theorem grpoidinv 27332
Description: A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpoidinv (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
Distinct variable groups:   𝑥,𝑦,𝑢,𝐺   𝑢,𝑋,𝑥,𝑦

Proof of Theorem grpoidinv
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 473 . . . . . . . 8 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
21ralimi 2949 . . . . . . 7 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
3 oveq2 6643 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
53, 4eqeq12d 2635 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
65rspccva 3303 . . . . . . 7 ((∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
72, 6sylan 488 . . . . . 6 ((∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
87adantll 749 . . . . 5 (((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
98adantll 749 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
10 simpl 473 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
1110anim1i 591 . . . . . 6 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝐺 ∈ GrpOp ∧ 𝑥𝑋))
12 id 22 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝑢𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
1312adantrr 752 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
1413adantr 481 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
152adantl 482 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
1615ad2antlr 762 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
17 simpr 477 . . . . . . . . . . 11 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1817ralimi 2949 . . . . . . . . . 10 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1918adantl 482 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2019ad2antlr 762 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2114, 16, 20jca32 557 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ((𝐺 ∈ GrpOp ∧ 𝑢𝑋) ∧ (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)))
22 grpfo.1 . . . . . . . 8 𝑋 = ran 𝐺
23 biid 251 . . . . . . . 8 (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
24 biid 251 . . . . . . . 8 (∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2522, 23, 24grpoidinvlem3 27330 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝑢𝑋) ∧ (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2621, 25sylancom 700 . . . . . 6 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2722grpoidinvlem4 27331 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2811, 26, 27syl2anc 692 . . . . 5 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2928, 9eqtrd 2654 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = 𝑥)
309, 29, 26jca31 556 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3130ralrimiva 2963 . 2 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3222grpolidinv 27325 . 2 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))
3331, 32reximddv 3015 1 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  ran crn 5105  (class class class)co 6635  GrpOpcgr 27313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-ov 6638  df-grpo 27317
This theorem is referenced by:  grpoideu  27333  grpoidval  27337  grpoidinv2  27339  grpomndo  33645
  Copyright terms: Public domain W3C validator