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Theorem isgrpinv 18804
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 18791 . . . . . . . . 9 (𝑥𝐵 → (𝑁𝑥) = (𝑒𝐵 (𝑒 + 𝑥) = 0 ))
65ad2antlr 725 . . . . . . . 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 774 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑀:𝐵𝐵)
9 simplr 767 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → 𝑥𝐵)
108, 9ffvelcdmd 7036 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → (𝑀𝑥) ∈ 𝐵)
111, 2, 3grpinveu 18785 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
1211ad4ant13 749 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑀:𝐵𝐵) ∧ 𝑥𝐵) ∧ ((𝑀𝑥) + 𝑥) = 0 ) → ∃!𝑒𝐵 (𝑒 + 𝑥) = 0 )
13 oveq1 7364 . . . . . . . . . . . 12 (𝑒 = (𝑀𝑥) → (𝑒 + 𝑥) = ((𝑀𝑥) + 𝑥))
1413eqeq1d 2738 . . . . . . . . . . 11 (𝑒 = (𝑀𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
1514riota2 7339 . . . . . . . . . 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 2776 . . . . . . 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 3164 . . . . 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 18792 . . . . 5 𝑁 Fn 𝐵
23 ffn 6668 . . . . . 6 (𝑀:𝐵𝐵𝑀 Fn 𝐵)
2423ad2antrl 726 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
25 eqfnfv 6982 . . . . 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 18797 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
301, 2, 3, 4grplinv 18800 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((𝑁𝑥) + 𝑥) = 0 )
3130ralrimiva 3143 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 )
3229, 31jca 512 . . 3 (𝐺 ∈ Grp → (𝑁:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑁𝑥) + 𝑥) = 0 ))
33 feq1 6649 . . . 4 (𝑁 = 𝑀 → (𝑁:𝐵𝐵𝑀:𝐵𝐵))
34 fveq1 6841 . . . . . . 7 (𝑁 = 𝑀 → (𝑁𝑥) = (𝑀𝑥))
3534oveq1d 7372 . . . . . 6 (𝑁 = 𝑀 → ((𝑁𝑥) + 𝑥) = ((𝑀𝑥) + 𝑥))
3635eqeq1d 2738 . . . . 5 (𝑁 = 𝑀 → (((𝑁𝑥) + 𝑥) = 0 ↔ ((𝑀𝑥) + 𝑥) = 0 ))
3736ralbidv 3174 . . . 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 1541  wcel 2106  wral 3064  ∃!wreu 3351   Fn wfn 6491  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752
This theorem is referenced by:  oppginv  19140
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