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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 | . . . . . 6 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
2 | 1 | subggrp 17420 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝐻 ∈ Grp) |
4 | 1 | subgbas 17421 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
5 | 4 | eleq2d 2673 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (Base‘𝐻))) |
6 | 5 | biimpa 500 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻)) |
7 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
8 | eqid 2610 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
9 | eqid 2610 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
10 | subginv.j | . . . . 5 ⊢ 𝐽 = (invg‘𝐻) | |
11 | 7, 8, 9, 10 | grprinv 17292 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻)) |
12 | 3, 6, 11 | syl2anc 691 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐻)(𝐽‘𝑋)) = (0g‘𝐻)) |
13 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
14 | 1, 13 | ressplusg 15818 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻)) |
15 | 14 | adantr 480 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻)) |
16 | 15 | oveqd 6566 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (𝑋(+g‘𝐻)(𝐽‘𝑋))) |
17 | eqid 2610 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
18 | 1, 17 | subg0 17423 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻)) |
19 | 18 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻)) |
20 | 12, 16, 19 | 3eqtr4d 2654 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺)) |
21 | subgrcl 17422 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
22 | 21 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝐺 ∈ Grp) |
23 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
24 | 23 | subgss 17418 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
25 | 24 | sselda 3568 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺)) |
26 | 7, 10 | grpinvcl 17290 | . . . . . . . 8 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐽‘𝑋) ∈ (Base‘𝐻)) |
27 | 26 | ex 449 | . . . . . . 7 ⊢ (𝐻 ∈ Grp → (𝑋 ∈ (Base‘𝐻) → (𝐽‘𝑋) ∈ (Base‘𝐻))) |
28 | 2, 27 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ (Base‘𝐻) → (𝐽‘𝑋) ∈ (Base‘𝐻))) |
29 | 4 | eleq2d 2673 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝐽‘𝑋) ∈ 𝑆 ↔ (𝐽‘𝑋) ∈ (Base‘𝐻))) |
30 | 28, 5, 29 | 3imtr4d 282 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑋 ∈ 𝑆 → (𝐽‘𝑋) ∈ 𝑆)) |
31 | 30 | imp 444 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ 𝑆) |
32 | 24 | sselda 3568 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐽‘𝑋) ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺)) |
33 | 31, 32 | syldan 486 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐽‘𝑋) ∈ (Base‘𝐺)) |
34 | subginv.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
35 | 23, 13, 17, 34 | grpinvid1 17293 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ (𝐽‘𝑋) ∈ (Base‘𝐺)) → ((𝐼‘𝑋) = (𝐽‘𝑋) ↔ (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺))) |
36 | 22, 25, 33, 35 | syl3anc 1318 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → ((𝐼‘𝑋) = (𝐽‘𝑋) ↔ (𝑋(+g‘𝐺)(𝐽‘𝑋)) = (0g‘𝐺))) |
37 | 20, 36 | mpbird 246 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋)) |
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 |
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