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Mirrors > Home > MPE Home > Th. List > subginv | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
subg0.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
subginv.i | ⊢ 𝐼 = (invg‘𝐺) |
subginv.j | ⊢ 𝐽 = (invg‘𝐻) |
Ref | Expression |
---|---|
subginv | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subg0.h | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
2 | 1 | subggrp 19053 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
3 | 1 | subgbas 19054 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
4 | 3 | eleq2d 2813 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (Base‘𝐻))) |
5 | 4 | biimpa 476 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻)) |
6 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
7 | eqid 2726 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
8 | eqid 2726 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
9 | subginv.j | . . . . 5 ⊢ 𝐽 = (invg‘𝐻) | |
10 | 6, 7, 8, 9 | grprinv 18917 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻)) |
11 | 2, 5, 10 | syl2an2r 682 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻)) |
12 | eqid 2726 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
13 | 1, 12 | ressplusg 17241 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻)) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻)) |
15 | 14 | oveqd 7421 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (𝑋(+g‘𝐻)(𝐽‘𝑋))) |
16 | eqid 2726 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 1, 16 | subg0 19056 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻)) |
18 | 17 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻)) |
19 | 11, 15, 18 | 3eqtr4d 2776 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺)) |
20 | subgrcl 19055 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
21 | 20 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝐺 ∈ Grp) |
22 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
23 | 22 | subgss 19051 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
24 | 23 | sselda 3977 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺)) |
25 | 6, 9 | grpinvcl 18914 | . . . . . . . 8 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽‘𝑋) ∈ (Base‘𝐻)) |
26 | 25 | ex 412 | . . . . . . 7 ⊢ (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽‘𝑋) ∈ (Base‘𝐻))) |
27 | 2, 26 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽‘𝑋) ∈ (Base‘𝐻))) |
28 | 3 | eleq2d 2813 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽‘𝑋) ∈ 𝑆 ↔ (𝐽‘𝑋) ∈ (Base‘𝐻))) |
29 | 27, 4, 28 | 3imtr4d 294 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 → (𝐽‘𝑋) ∈ 𝑆)) |
30 | 29 | imp 406 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ 𝑆) |
31 | 23 | sselda 3977 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽‘𝑋) ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺)) |
32 | 30, 31 | syldan 590 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺)) |
33 | subginv.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
34 | 22, 12, 16, 33 | grpinvid1 18918 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ (𝐽‘𝑋) ∈ (Base‘𝐺)) → ((𝐼‘𝑋) = (𝐽‘𝑋) ↔ (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺))) |
35 | 21, 24, 32, 34 | syl3anc 1368 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → ((𝐼‘𝑋) = (𝐽‘𝑋) ↔ (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺))) |
36 | 19, 35 | mpbird 257 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 ↾s cress 17179 +gcplusg 17203 0gc0g 17391 Grpcgrp 18860 invgcminusg 18861 SubGrpcsubg 19044 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-subg 19047 |
This theorem is referenced by: subginvcl 19059 subgsub 19062 subgmulg 19064 zringlpirlem1 21344 prmirred 21356 psgninv 21470 mplneg 21906 subgtgp 23959 clmneg 24958 qrngneg 27506 |
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