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Theorem subginv 17582
Description: The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
subg0.h 𝐻 = (𝐺s 𝑆)
subginv.i 𝐼 = (invg𝐺)
subginv.j 𝐽 = (invg𝐻)
Assertion
Ref Expression
subginv ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))

Proof of Theorem subginv
StepHypRef Expression
1 subg0.h . . . . . 6 𝐻 = (𝐺s 𝑆)
21subggrp 17578 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantr 481 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐻 ∈ Grp)
41subgbas 17579 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
54eleq2d 2685 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆𝑋 ∈ (Base‘𝐻)))
65biimpa 501 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
7 eqid 2620 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
8 eqid 2620 . . . . 5 (+g𝐻) = (+g𝐻)
9 eqid 2620 . . . . 5 (0g𝐻) = (0g𝐻)
10 subginv.j . . . . 5 𝐽 = (invg𝐻)
117, 8, 9, 10grprinv 17450 . . . 4 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
123, 6, 11syl2anc 692 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
13 eqid 2620 . . . . . 6 (+g𝐺) = (+g𝐺)
141, 13ressplusg 15974 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
1514adantr 481 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
1615oveqd 6652 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (𝑋(+g𝐻)(𝐽𝑋)))
17 eqid 2620 . . . . 5 (0g𝐺) = (0g𝐺)
181, 17subg0 17581 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
1918adantr 481 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
2012, 16, 193eqtr4d 2664 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺))
21 subgrcl 17580 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2221adantr 481 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐺 ∈ Grp)
23 eqid 2620 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2423subgss 17576 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2524sselda 3595 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
267, 10grpinvcl 17448 . . . . . . . 8 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽𝑋) ∈ (Base‘𝐻))
2726ex 450 . . . . . . 7 (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
282, 27syl 17 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
294eleq2d 2685 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽𝑋) ∈ 𝑆 ↔ (𝐽𝑋) ∈ (Base‘𝐻)))
3028, 5, 293imtr4d 283 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆 → (𝐽𝑋) ∈ 𝑆))
3130imp 445 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ 𝑆)
3224sselda 3595 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽𝑋) ∈ 𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
3331, 32syldan 487 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
34 subginv.i . . . 4 𝐼 = (invg𝐺)
3523, 13, 17, 34grpinvid1 17451 . . 3 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ (𝐽𝑋) ∈ (Base‘𝐺)) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3622, 25, 33, 35syl3anc 1324 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3720, 36mpbird 247 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  cfv 5876  (class class class)co 6635  Basecbs 15838  s cress 15839  +gcplusg 15922  0gc0g 16081  Grpcgrp 17403  invgcminusg 17404  SubGrpcsubg 17569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-subg 17572
This theorem is referenced by:  subginvcl  17584  subgsub  17587  subgmulg  17589  zringlpirlem1  19813  prmirred  19824  psgninv  19909  subgtgp  21890  clmneg  22862  qrngneg  25293
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