Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  subginv Structured version   Visualization version   GIF version

Theorem subginv 17424
 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 17420 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantr 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐻 ∈ Grp)
41subgbas 17421 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
54eleq2d 2673 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆𝑋 ∈ (Base‘𝐻)))
65biimpa 500 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
7 eqid 2610 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
8 eqid 2610 . . . . 5 (+g𝐻) = (+g𝐻)
9 eqid 2610 . . . . 5 (0g𝐻) = (0g𝐻)
10 subginv.j . . . . 5 𝐽 = (invg𝐻)
117, 8, 9, 10grprinv 17292 . . . 4 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
123, 6, 11syl2anc 691 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐻)(𝐽𝑋)) = (0g𝐻))
13 eqid 2610 . . . . . 6 (+g𝐺) = (+g𝐺)
141, 13ressplusg 15818 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
1514adantr 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
1615oveqd 6566 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (𝑋(+g𝐻)(𝐽𝑋)))
17 eqid 2610 . . . . 5 (0g𝐺) = (0g𝐺)
181, 17subg0 17423 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
1918adantr 480 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
2012, 16, 193eqtr4d 2654 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺))
21 subgrcl 17422 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2221adantr 480 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝐺 ∈ Grp)
23 eqid 2610 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2423subgss 17418 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2524sselda 3568 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
267, 10grpinvcl 17290 . . . . . . . 8 ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽𝑋) ∈ (Base‘𝐻))
2726ex 449 . . . . . . 7 (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
282, 27syl 17 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽𝑋) ∈ (Base‘𝐻)))
294eleq2d 2673 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽𝑋) ∈ 𝑆 ↔ (𝐽𝑋) ∈ (Base‘𝐻)))
3028, 5, 293imtr4d 282 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (𝑋𝑆 → (𝐽𝑋) ∈ 𝑆))
3130imp 444 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ 𝑆)
3224sselda 3568 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽𝑋) ∈ 𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
3331, 32syldan 486 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐽𝑋) ∈ (Base‘𝐺))
34 subginv.i . . . 4 𝐼 = (invg𝐺)
3523, 13, 17, 34grpinvid1 17293 . . 3 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ (𝐽𝑋) ∈ (Base‘𝐺)) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3622, 25, 33, 35syl3anc 1318 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → ((𝐼𝑋) = (𝐽𝑋) ↔ (𝑋(+g𝐺)(𝐽𝑋)) = (0g𝐺)))
3720, 36mpbird 246 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  SubGrpcsubg 17411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414 This theorem is referenced by:  subginvcl  17426  subgsub  17429  subgmulg  17431  zringlpirlem1  19651  prmirred  19662  psgninv  19747  subgtgp  21719  clmneg  22689  qrngneg  25112
 Copyright terms: Public domain W3C validator