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Theorem isgrpinv 18094
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 18082 . . . . . . . . 9 (𝑥𝐵 → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
65ad2antlr 723 . . . . . . . 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 772 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑀:𝐵𝐵)
9 simplr 765 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑥𝐵)
108, 9ffvelrnd 6844 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑀𝑥) ∈ 𝐵)
111, 2, 3grpinveu 18076 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
1211ad4ant13 747 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
13 oveq1 7152 . . . . . . . . . . . 12 (𝑒 = (𝑀𝑥) → (𝑒 + 𝑥) = ((𝑀𝑥) + 𝑥))
1413eqeq1d 2820 . . . . . . . . . . 11 (𝑒 = (𝑀𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
1514riota2 7128 . . . . . . . . . 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 233 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑒𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀𝑥))
186, 17eqtrd 2853 . . . . . . 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 3174 . . . . 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 18083 . . . . 5 𝑁 Fn 𝐵
23 ffn 6507 . . . . . 6 (𝑀:𝐵𝐵𝑀 Fn 𝐵)
2423ad2antrl 724 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
25 eqfnfv 6794 . . . . 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 258 . . 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 18088 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
301, 2, 3, 4grplinv 18090 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((𝑁𝑥) + 𝑥) = 0 )
3130ralrimiva 3179 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 )
3229, 31jca 512 . . 3 (𝐺 ∈ Grp → (𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ))
33 feq1 6488 . . . 4 (𝑁 = 𝑀 → (𝑁:𝐵𝐵𝑀:𝐵𝐵))
34 fveq1 6662 . . . . . . 7 (𝑁 = 𝑀 → (𝑁𝑥) = (𝑀𝑥))
3534oveq1d 7160 . . . . . 6 (𝑁 = 𝑀 → ((𝑁𝑥) + 𝑥) = ((𝑀𝑥) + 𝑥))
3635eqeq1d 2820 . . . . 5 (𝑁 = 𝑀 → (((𝑁𝑥) + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
3736ralbidv 3194 . . . 4 (𝑁 = 𝑀 → (∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ↔ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ))
3833, 37anbi12d 630 . . 3 (𝑁 = 𝑀 → ((𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ) ↔ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )))
3932, 38syl5ibcom 246 . 2 (𝐺 ∈ Grp → (𝑁 = 𝑀 → (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )))
4028, 39impbid 213 1 (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  ∃!wreu 3137   Fn wfn 6343  wf 6344  cfv 6348  crio 7102  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041  invgcminusg 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045
This theorem is referenced by:  oppginv  18425
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