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Theorem subginv 19095
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 . . . . 5 𝐻 = (𝐺s 𝑆)
21subggrp 19091 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
31subgbas 19092 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
43eleq2d 2815 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆𝑋 ∈ (Base‘𝐻)))
54biimpa 475 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
6 eqid 2728 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
7 eqid 2728 . . . . 5 (+g𝐻) = (+g𝐻)
8 eqid 2728 . . . . 5 (0g𝐻) = (0g𝐻)
9 subginv.j . . . . 5 𝐽 = (invg𝐻)
106, 7, 8, 9grprinv 18954 . . . 4 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
112, 5, 10syl2an2r 683 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
12 eqid 2728 . . . . . 6 (+g𝐺) = (+g𝐺)
131, 12ressplusg 17278 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
1413adantr 479 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
1514oveqd 7443 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (𝑋(+g𝐻)(𝐽𝑋)))
16 eqid 2728 . . . . 5 (0g𝐺) = (0g𝐺)
171, 16subg0 19094 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
1817adantr 479 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
1911, 15, 183eqtr4d 2778 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺))
20 subgrcl 19093 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2120adantr 479 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐺 ∈ Grp)
22 eqid 2728 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2322subgss 19089 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2423sselda 3982 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
256, 9grpinvcl 18951 . . . . . . . 8 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽𝑋) ∈ (Base‘𝐻))
2625ex 411 . . . . . . 7 (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
272, 26syl 17 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
283eleq2d 2815 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽𝑋) ∈ 𝑆 ↔ (𝐽𝑋) ∈ (Base‘𝐻)))
2927, 4, 283imtr4d 293 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆 → (𝐽𝑋) ∈ 𝑆))
3029imp 405 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ 𝑆)
3123sselda 3982 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽𝑋) ∈ 𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
3230, 31syldan 589 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
33 subginv.i . . . 4 𝐼 = (invg𝐺)
3422, 12, 16, 33grpinvid1 18955 . . 3 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ (𝐽𝑋) ∈ (Base‘𝐺)) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3521, 24, 32, 34syl3anc 1368 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3619, 35mpbird 256 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cfv 6553  (class class class)co 7426  Basecbs 17187  s cress 17216  +gcplusg 17240  0gc0g 17428  Grpcgrp 18897  invgcminusg 18898  SubGrpcsubg 19082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-subg 19085
This theorem is referenced by:  subginvcl  19097  subgsub  19100  subgmulg  19102  zringlpirlem1  21395  prmirred  21407  psgninv  21521  mplneg  21959  subgtgp  24029  clmneg  25028  qrngneg  27576
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