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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 19091 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
3 | 1 | subgbas 19092 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
4 | 3 | eleq2d 2815 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (Base‘𝐻))) |
5 | 4 | biimpa 475 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻)) |
6 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
7 | eqid 2728 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
8 | eqid 2728 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
9 | subginv.j | . . . . 5 ⊢ 𝐽 = (invg‘𝐻) | |
10 | 6, 7, 8, 9 | grprinv 18954 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻)) |
11 | 2, 5, 10 | syl2an2r 683 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻)) |
12 | eqid 2728 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
13 | 1, 12 | ressplusg 17278 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻)) |
14 | 13 | adantr 479 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻)) |
15 | 14 | oveqd 7443 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (𝑋(+g‘𝐻)(𝐽‘𝑋))) |
16 | eqid 2728 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 1, 16 | subg0 19094 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻)) |
18 | 17 | adantr 479 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻)) |
19 | 11, 15, 18 | 3eqtr4d 2778 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺)) |
20 | subgrcl 19093 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
21 | 20 | adantr 479 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝐺 ∈ Grp) |
22 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
23 | 22 | subgss 19089 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
24 | 23 | sselda 3982 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺)) |
25 | 6, 9 | grpinvcl 18951 | . . . . . . . 8 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽‘𝑋) ∈ (Base‘𝐻)) |
26 | 25 | ex 411 | . . . . . . 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 2815 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽‘𝑋) ∈ 𝑆 ↔ (𝐽‘𝑋) ∈ (Base‘𝐻))) |
29 | 27, 4, 28 | 3imtr4d 293 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 → (𝐽‘𝑋) ∈ 𝑆)) |
30 | 29 | imp 405 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ 𝑆) |
31 | 23 | sselda 3982 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽‘𝑋) ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺)) |
32 | 30, 31 | syldan 589 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺)) |
33 | subginv.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
34 | 22, 12, 16, 33 | grpinvid1 18955 | . . 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 256 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 ↾s cress 17216 +gcplusg 17240 0gc0g 17428 Grpcgrp 18897 invgcminusg 18898 SubGrpcsubg 19082 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-subg 19085 |
This theorem is referenced by: subginvcl 19097 subgsub 19100 subgmulg 19102 zringlpirlem1 21395 prmirred 21407 psgninv 21521 mplneg 21959 subgtgp 24029 clmneg 25028 qrngneg 27576 |
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