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Theorem isgrpinv 18701
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 18689 . . . . . . . . 9 (𝑥𝐵 → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
65ad2antlr 724 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
7 simpr 485 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ((𝑀𝑥) + 𝑥) = 0 )
8 simpllr 773 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑀:𝐵𝐵)
9 simplr 766 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑥𝐵)
108, 9ffvelcdmd 7001 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑀𝑥) ∈ 𝐵)
111, 2, 3grpinveu 18683 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
1211ad4ant13 748 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
13 oveq1 7322 . . . . . . . . . . . 12 (𝑒 = (𝑀𝑥) → (𝑒 + 𝑥) = ((𝑀𝑥) + 𝑥))
1413eqeq1d 2739 . . . . . . . . . . 11 (𝑒 = (𝑀𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
1514riota2 7298 . . . . . . . . . 10 (((𝑀𝑥) ∈ 𝐵 ∧ ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 ) → (((𝑀𝑥) + 𝑥) = 0 ↔ (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥)))
1610, 12, 15syl2anc 584 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (((𝑀𝑥) + 𝑥) = 0 ↔ (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥)))
177, 16mpbid 231 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥))
186, 17eqtrd 2777 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑁𝑥) = (𝑀𝑥))
1918ex 413 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) → (((𝑀𝑥) + 𝑥) = 0 → (𝑁𝑥) = (𝑀𝑥)))
2019ralimdva 3161 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) → (∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 → ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥)))
2120impr 455 . . . 4 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥))
221, 4grpinvfn 18690 . . . . 5 𝑁 Fn 𝐵
23 ffn 6637 . . . . . 6 (𝑀:𝐵𝐵𝑀 Fn 𝐵)
2423ad2antrl 725 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
25 eqfnfv 6948 . . . . 5 ((𝑁 Fn 𝐵𝑀 Fn 𝐵) → (𝑁 = 𝑀 ↔ ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥)))
2622, 24, 25sylancr 587 . . . 4 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → (𝑁 = 𝑀 ↔ ∀𝑥𝐵 (𝑁𝑥) = (𝑀𝑥)))
2721, 26mpbird 256 . . 3 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → 𝑁 = 𝑀)
2827ex 413 . 2 (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) → 𝑁 = 𝑀))
291, 4grpinvf 18695 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
301, 2, 3, 4grplinv 18697 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((𝑁𝑥) + 𝑥) = 0 )
3130ralrimiva 3140 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 )
3229, 31jca 512 . . 3 (𝐺 ∈ Grp → (𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ))
33 feq1 6618 . . . 4 (𝑁 = 𝑀 → (𝑁:𝐵𝐵𝑀:𝐵𝐵))
34 fveq1 6810 . . . . . . 7 (𝑁 = 𝑀 → (𝑁𝑥) = (𝑀𝑥))
3534oveq1d 7330 . . . . . 6 (𝑁 = 𝑀 → ((𝑁𝑥) + 𝑥) = ((𝑀𝑥) + 𝑥))
3635eqeq1d 2739 . . . . 5 (𝑁 = 𝑀 → (((𝑁𝑥) + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
3736ralbidv 3171 . . . 4 (𝑁 = 𝑀 → (∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ↔ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ))
3833, 37anbi12d 631 . . 3 (𝑁 = 𝑀 → ((𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ) ↔ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )))
3932, 38syl5ibcom 244 . 2 (𝐺 ∈ Grp → (𝑁 = 𝑀 → (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )))
4028, 39impbid 211 1 (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3062  ∃!wreu 3348   Fn wfn 6460  wf 6461  cfv 6465  crio 7271  (class class class)co 7315  Basecbs 16982  +gcplusg 17032  0gc0g 17220  Grpcgrp 18646  invgcminusg 18647
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 2708  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fv 6473  df-riota 7272  df-ov 7318  df-0g 17222  df-mgm 18396  df-sgrp 18445  df-mnd 18456  df-grp 18649  df-minusg 18650
This theorem is referenced by:  oppginv  19035
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