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Theorem subginv 17911
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 17907 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantr 473 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐻 ∈ Grp)
41subgbas 17908 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
54eleq2d 2862 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆𝑋 ∈ (Base‘𝐻)))
65biimpa 469 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
7 eqid 2797 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
8 eqid 2797 . . . . 5 (+g𝐻) = (+g𝐻)
9 eqid 2797 . . . . 5 (0g𝐻) = (0g𝐻)
10 subginv.j . . . . 5 𝐽 = (invg𝐻)
117, 8, 9, 10grprinv 17782 . . . 4 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
123, 6, 11syl2anc 580 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
13 eqid 2797 . . . . . 6 (+g𝐺) = (+g𝐺)
141, 13ressplusg 16311 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
1514adantr 473 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
1615oveqd 6893 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (𝑋(+g𝐻)(𝐽𝑋)))
17 eqid 2797 . . . . 5 (0g𝐺) = (0g𝐺)
181, 17subg0 17910 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
1918adantr 473 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
2012, 16, 193eqtr4d 2841 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺))
21 subgrcl 17909 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2221adantr 473 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐺 ∈ Grp)
23 eqid 2797 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2423subgss 17905 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2524sselda 3796 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
267, 10grpinvcl 17780 . . . . . . . 8 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽𝑋) ∈ (Base‘𝐻))
2726ex 402 . . . . . . 7 (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
282, 27syl 17 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
294eleq2d 2862 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽𝑋) ∈ 𝑆 ↔ (𝐽𝑋) ∈ (Base‘𝐻)))
3028, 5, 293imtr4d 286 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆 → (𝐽𝑋) ∈ 𝑆))
3130imp 396 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ 𝑆)
3224sselda 3796 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽𝑋) ∈ 𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
3331, 32syldan 586 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
34 subginv.i . . . 4 𝐼 = (invg𝐺)
3523, 13, 17, 34grpinvid1 17783 . . 3 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ (𝐽𝑋) ∈ (Base‘𝐺)) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3622, 25, 33, 35syl3anc 1491 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3720, 36mpbird 249 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cfv 6099  (class class class)co 6876  Basecbs 16181  s cress 16182  +gcplusg 16264  0gc0g 16412  Grpcgrp 17735  invgcminusg 17736  SubGrpcsubg 17898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-ndx 16184  df-slot 16185  df-base 16187  df-sets 16188  df-ress 16189  df-plusg 16277  df-0g 16414  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-grp 17738  df-minusg 17739  df-subg 17901
This theorem is referenced by:  subginvcl  17913  subgsub  17916  subgmulg  17918  zringlpirlem1  20151  prmirred  20162  psgninv  20246  subgtgp  22234  clmneg  23205  qrngneg  25661
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