Theorem subginv 13046

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 = (𝐺 ↾s 𝑆)
subginv.i	⊢ 𝐼 = (invg‘𝐺)
subginv.j	⊢ 𝐽 = (invg‘𝐻)

Assertion

Ref	Expression
subginv	⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋))

Proof of Theorem subginv

Step	Hyp	Ref	Expression
1		subg0.h	. . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
2	1	subggrp 13042	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3	1	subgbas 13043	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
4	3	eleq2d 2247	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (Base‘𝐻)))
5	4	biimpa 296	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻))
6		eqid 2177	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
7		eqid 2177	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
8		eqid 2177	. . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻)
9		subginv.j	. . . . 5 ⊢ 𝐽 = (invg‘𝐻)
10	6, 7, 8, 9	grprinv 12928	. . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻))
11	2, 5, 10	syl2an2r 595	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻))
12	1	a1i 9	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺 ↾s 𝑆))
13		eqidd 2178	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
14		id 19	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
15		subgrcl 13044	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
16	12, 13, 14, 15	ressplusgd 12589	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
17	16	adantr 276	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻))
18	17	oveqd 5894	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (𝑋(+g‘𝐻)(𝐽‘𝑋)))
19		eqid 2177	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
20	1, 19	subg0 13045	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
21	20	adantr 276	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻))
22	11, 18, 21	3eqtr4d 2220	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺))
23	15	adantr 276	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝐺 ∈ Grp)
24		eqid 2177	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
25	24	subgss 13039	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
26	25	sselda 3157	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺))
27	6, 9	grpinvcl 12926	. . . . . . . 8 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽‘𝑋) ∈ (Base‘𝐻))
28	27	ex 115	. . . . . . 7 ⊢ (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽‘𝑋) ∈ (Base‘𝐻)))
29	2, 28	syl 14	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽‘𝑋) ∈ (Base‘𝐻)))
30	3	eleq2d 2247	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽‘𝑋) ∈ 𝑆 ↔ (𝐽‘𝑋) ∈ (Base‘𝐻)))
31	29, 4, 30	3imtr4d 203	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 → (𝐽‘𝑋) ∈ 𝑆))
32	31	imp 124	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ 𝑆)
33	25	sselda 3157	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽‘𝑋) ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺))
34	32, 33	syldan 282	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺))
35		eqid 2177	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
36		subginv.i	. . . 4 ⊢ 𝐼 = (invg‘𝐺)
37	24, 35, 19, 36	grpinvid1 12929	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ (𝐽‘𝑋) ∈ (Base‘𝐺)) → ((𝐼‘𝑋) = (𝐽‘𝑋) ↔ (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺)))
38	23, 26, 34, 37	syl3anc 1238	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → ((𝐼‘𝑋) = (𝐽‘𝑋) ↔ (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺)))
39	22, 38	mpbird 167	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ‘cfv 5218  (class class class)co 5877  Basecbs 12464   ↾_s cress 12465  +_gcplusg 12538  0_gc0g 12710  Grpcgrp 12882  inv_gcminusg 12883  SubGrpcsubg 13032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-subg 13035

This theorem is referenced by:  subginvcl  13048  subgsub  13051  subgmulg  13053