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Theorem grpoidinv 30025
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 482 . . . . . . . 8 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
21ralimi 3082 . . . . . . 7 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
3 oveq2 7420 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
53, 4eqeq12d 2747 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
65rspccva 3612 . . . . . . 7 ((∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
72, 6sylan 579 . . . . . 6 ((∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
87adantll 711 . . . . 5 (((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
98adantll 711 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
10 simpl 482 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
1110anim1i 614 . . . . . 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 714 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
1413adantr 480 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
152adantl 481 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
1615ad2antlr 724 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
17 simpr 484 . . . . . . . . . . 11 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1817ralimi 3082 . . . . . . . . . 10 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1918adantl 481 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2019ad2antlr 724 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2114, 16, 20jca32 515 . . . . . . 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 260 . . . . . . . 8 (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
24 biid 260 . . . . . . . 8 (∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2522, 23, 24grpoidinvlem3 30023 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝑢𝑋) ∧ (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2621, 25sylancom 587 . . . . . 6 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2722grpoidinvlem4 30024 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2811, 26, 27syl2anc 583 . . . . 5 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2928, 9eqtrd 2771 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = 𝑥)
309, 29, 26jca31 514 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3130ralrimiva 3145 . 2 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3222grpolidinv 30018 . 2 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))
3331, 32reximddv 3170 1 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  wrex 3069  ran crn 5678  (class class class)co 7412  GrpOpcgr 30006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7415  df-grpo 30010
This theorem is referenced by:  grpoideu  30026  grpoidval  30030  grpoidinv2  30032  grpomndo  37047
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