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Theorem grpoidinv 28289
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 486 . . . . . . . 8 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
21ralimi 3152 . . . . . . 7 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
3 oveq2 7148 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
53, 4eqeq12d 2838 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
65rspccva 3597 . . . . . . 7 ((∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
72, 6sylan 583 . . . . . 6 ((∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
87adantll 713 . . . . 5 (((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
98adantll 713 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
10 simpl 486 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
1110anim1i 617 . . . . . 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 716 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
1413adantr 484 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
152adantl 485 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
1615ad2antlr 726 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
17 simpr 488 . . . . . . . . . . 11 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1817ralimi 3152 . . . . . . . . . 10 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1918adantl 485 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2019ad2antlr 726 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2114, 16, 20jca32 519 . . . . . . 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 264 . . . . . . . 8 (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
24 biid 264 . . . . . . . 8 (∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2522, 23, 24grpoidinvlem3 28287 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝑢𝑋) ∧ (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2621, 25sylancom 591 . . . . . 6 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2722grpoidinvlem4 28288 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2811, 26, 27syl2anc 587 . . . . 5 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2928, 9eqtrd 2857 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = 𝑥)
309, 29, 26jca31 518 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3130ralrimiva 3174 . 2 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3222grpolidinv 28282 . 2 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))
3331, 32reximddv 3261 1 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  wrex 3131  ran crn 5533  (class class class)co 7140  GrpOpcgr 28270
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fo 6340  df-fv 6342  df-ov 7143  df-grpo 28274
This theorem is referenced by:  grpoideu  28290  grpoidval  28294  grpoidinv2  28296  grpomndo  35271
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