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Theorem subginv 18277
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 18273 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
31subgbas 18274 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
43eleq2d 2899 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆𝑋 ∈ (Base‘𝐻)))
54biimpa 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
6 eqid 2822 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
7 eqid 2822 . . . . 5 (+g𝐻) = (+g𝐻)
8 eqid 2822 . . . . 5 (0g𝐻) = (0g𝐻)
9 subginv.j . . . . 5 𝐽 = (invg𝐻)
106, 7, 8, 9grprinv 18144 . . . 4 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
112, 5, 10syl2an2r 684 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
12 eqid 2822 . . . . . 6 (+g𝐺) = (+g𝐺)
131, 12ressplusg 16603 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
1413adantr 484 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
1514oveqd 7157 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (𝑋(+g𝐻)(𝐽𝑋)))
16 eqid 2822 . . . . 5 (0g𝐺) = (0g𝐺)
171, 16subg0 18276 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
1817adantr 484 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
1911, 15, 183eqtr4d 2867 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺))
20 subgrcl 18275 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2120adantr 484 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐺 ∈ Grp)
22 eqid 2822 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2322subgss 18271 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2423sselda 3942 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
256, 9grpinvcl 18142 . . . . . . . 8 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽𝑋) ∈ (Base‘𝐻))
2625ex 416 . . . . . . 7 (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
272, 26syl 17 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
283eleq2d 2899 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽𝑋) ∈ 𝑆 ↔ (𝐽𝑋) ∈ (Base‘𝐻)))
2927, 4, 283imtr4d 297 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆 → (𝐽𝑋) ∈ 𝑆))
3029imp 410 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ 𝑆)
3123sselda 3942 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽𝑋) ∈ 𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
3230, 31syldan 594 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
33 subginv.i . . . 4 𝐼 = (invg𝐺)
3422, 12, 16, 33grpinvid1 18145 . . 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 260 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140  Basecbs 16474  s cress 16475  +gcplusg 16556  0gc0g 16704  Grpcgrp 18094  invgcminusg 18095  SubGrpcsubg 18264
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-minusg 18098  df-subg 18267
This theorem is referenced by:  subginvcl  18279  subgsub  18282  subgmulg  18284  zringlpirlem1  20175  prmirred  20186  psgninv  20269  mplneg  20679  subgtgp  22708  clmneg  23684  qrngneg  26205
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