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Theorem isgrpinv 17393
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 17382 . . . . . . . . 9 (𝑥𝐵 → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
65ad2antlr 762 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
7 simpr 477 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ((𝑀𝑥) + 𝑥) = 0 )
8 simpllr 798 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑀:𝐵𝐵)
9 simplr 791 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑥𝐵)
108, 9ffvelrnd 6316 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑀𝑥) ∈ 𝐵)
11 simplll 797 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝐺 ∈ Grp)
121, 2, 3grpinveu 17377 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
1311, 9, 12syl2anc 692 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
14 oveq1 6611 . . . . . . . . . . . 12 (𝑒 = (𝑀𝑥) → (𝑒 + 𝑥) = ((𝑀𝑥) + 𝑥))
1514eqeq1d 2623 . . . . . . . . . . 11 (𝑒 = (𝑀𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
1615riota2 6587 . . . . . . . . . 10 (((𝑀𝑥) ∈ 𝐵 ∧ ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 ) → (((𝑀𝑥) + 𝑥) = 0 ↔ (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥)))
1710, 13, 16syl2anc 692 . . . . . . . . 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 2655 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑁𝑥) = (𝑀𝑥))
2019ex 450 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) → (((𝑀𝑥) + 𝑥) = 0 → (𝑁𝑥) = (𝑀𝑥)))
2120ralimdva 2956 . . . . 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 17383 . . . . 5 𝑁 Fn 𝐵
24 ffn 6002 . . . . . 6 (𝑀:𝐵𝐵𝑀 Fn 𝐵)
2524ad2antrl 763 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
26 eqfnfv 6267 . . . . 5 ((𝑁 Fn 𝐵𝑀 Fn 𝐵) → (𝑁 = 𝑀 ↔ ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥)))
2723, 25, 26sylancr 694 . . . 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 450 . 2 (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) → 𝑁 = 𝑀))
301, 4grpinvf 17387 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
311, 2, 3, 4grplinv 17389 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((𝑁𝑥) + 𝑥) = 0 )
3231ralrimiva 2960 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 )
3330, 32jca 554 . . 3 (𝐺 ∈ Grp → (𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ))
34 feq1 5983 . . . 4 (𝑁 = 𝑀 → (𝑁:𝐵𝐵𝑀:𝐵𝐵))
35 fveq1 6147 . . . . . . 7 (𝑁 = 𝑀 → (𝑁𝑥) = (𝑀𝑥))
3635oveq1d 6619 . . . . . 6 (𝑁 = 𝑀 → ((𝑁𝑥) + 𝑥) = ((𝑀𝑥) + 𝑥))
3736eqeq1d 2623 . . . . 5 (𝑁 = 𝑀 → (((𝑁𝑥) + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
3837ralbidv 2980 . . . 4 (𝑁 = 𝑀 → (∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ↔ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ))
3934, 38anbi12d 746 . . 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 384   = wceq 1480  wcel 1987  wral 2907  ∃!wreu 2909   Fn wfn 5842  wf 5843  cfv 5847  crio 6564  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343  invgcminusg 17344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347
This theorem is referenced by:  oppginv  17710
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