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Theorem isgrpinv 17519
Description: Properties showing that a function 𝑀 is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
grpinv.b 𝐵 = (Base‘𝐺)
grpinv.p + = (+g𝐺)
grpinv.u 0 = (0g𝐺)
grpinv.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
isgrpinv (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥, 0   𝑥, +   𝑥,𝑀   𝑥,𝑁

Proof of Theorem isgrpinv
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 grpinv.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
2 grpinv.p . . . . . . . . . 10 + = (+g𝐺)
3 grpinv.u . . . . . . . . . 10 0 = (0g𝐺)
4 grpinv.n . . . . . . . . . 10 𝑁 = (invg𝐺)
51, 2, 3, 4grpinvval 17508 . . . . . . . . 9 (𝑥𝐵 → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
65ad2antlr 763 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
7 simpr 476 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ((𝑀𝑥) + 𝑥) = 0 )
8 simpllr 815 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑀:𝐵𝐵)
9 simplr 807 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑥𝐵)
108, 9ffvelrnd 6400 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑀𝑥) ∈ 𝐵)
11 simplll 813 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝐺 ∈ Grp)
121, 2, 3grpinveu 17503 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
1311, 9, 12syl2anc 694 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
14 oveq1 6697 . . . . . . . . . . . 12 (𝑒 = (𝑀𝑥) → (𝑒 + 𝑥) = ((𝑀𝑥) + 𝑥))
1514eqeq1d 2653 . . . . . . . . . . 11 (𝑒 = (𝑀𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
1615riota2 6673 . . . . . . . . . 10 (((𝑀𝑥) ∈ 𝐵 ∧ ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 ) → (((𝑀𝑥) + 𝑥) = 0 ↔ (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥)))
1710, 13, 16syl2anc 694 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (((𝑀𝑥) + 𝑥) = 0 ↔ (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥)))
187, 17mpbid 222 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥))
196, 18eqtrd 2685 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑁𝑥) = (𝑀𝑥))
2019ex 449 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) → (((𝑀𝑥) + 𝑥) = 0 → (𝑁𝑥) = (𝑀𝑥)))
2120ralimdva 2991 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) → (∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 → ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥)))
2221impr 648 . . . 4 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥))
231, 4grpinvfn 17509 . . . . 5 𝑁 Fn 𝐵
24 ffn 6083 . . . . . 6 (𝑀:𝐵𝐵𝑀 Fn 𝐵)
2524ad2antrl 764 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
26 eqfnfv 6351 . . . . 5 ((𝑁 Fn 𝐵𝑀 Fn 𝐵) → (𝑁 = 𝑀 ↔ ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥)))
2723, 25, 26sylancr 696 . . . 4 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → (𝑁 = 𝑀 ↔ ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥)))
2822, 27mpbird 247 . . 3 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → 𝑁 = 𝑀)
2928ex 449 . 2 (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) → 𝑁 = 𝑀))
301, 4grpinvf 17513 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
311, 2, 3, 4grplinv 17515 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((𝑁𝑥) + 𝑥) = 0 )
3231ralrimiva 2995 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 )
3330, 32jca 553 . . 3 (𝐺 ∈ Grp → (𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ))
34 feq1 6064 . . . 4 (𝑁 = 𝑀 → (𝑁:𝐵𝐵𝑀:𝐵𝐵))
35 fveq1 6228 . . . . . . 7 (𝑁 = 𝑀 → (𝑁𝑥) = (𝑀𝑥))
3635oveq1d 6705 . . . . . 6 (𝑁 = 𝑀 → ((𝑁𝑥) + 𝑥) = ((𝑀𝑥) + 𝑥))
3736eqeq1d 2653 . . . . 5 (𝑁 = 𝑀 → (((𝑁𝑥) + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
3837ralbidv 3015 . . . 4 (𝑁 = 𝑀 → (∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ↔ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ))
3934, 38anbi12d 747 . . 3 (𝑁 = 𝑀 → ((𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ) ↔ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )))
4033, 39syl5ibcom 235 . 2 (𝐺 ∈ Grp → (𝑁 = 𝑀 → (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )))
4129, 40impbid 202 1 (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  ∃!wreu 2943   Fn wfn 5921  wf 5922  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473
This theorem is referenced by:  oppginv  17835
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