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Mirrors > Home > MPE Home > Th. List > invoppggim | Structured version Visualization version GIF version |
Description: The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
invoppggim.o | ⊢ 𝑂 = (oppg‘𝐺) |
invoppggim.i | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
invoppggim | ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | invoppggim.o | . . . 4 ⊢ 𝑂 = (oppg‘𝐺) | |
3 | 2, 1 | oppgbas 18473 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝑂) |
4 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | eqid 2821 | . . 3 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
6 | id 22 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
7 | 2 | oppggrp 18479 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑂 ∈ Grp) |
8 | invoppggim.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
9 | 1, 8 | grpinvf 18144 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
10 | 1, 4, 8 | grpinvadd 18171 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥))) |
11 | 10 | 3expb 1116 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥))) |
12 | 4, 2, 5 | oppgplus 18471 | . . . 4 ⊢ ((𝐼‘𝑥)(+g‘𝑂)(𝐼‘𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥)) |
13 | 11, 12 | syl6eqr 2874 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑥)(+g‘𝑂)(𝐼‘𝑦))) |
14 | 1, 3, 4, 5, 6, 7, 9, 13 | isghmd 18361 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpHom 𝑂)) |
15 | 1, 8, 6 | grpinvf1o 18163 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)–1-1-onto→(Base‘𝐺)) |
16 | 1, 3 | isgim 18396 | . 2 ⊢ (𝐼 ∈ (𝐺 GrpIso 𝑂) ↔ (𝐼 ∈ (𝐺 GrpHom 𝑂) ∧ 𝐼:(Base‘𝐺)–1-1-onto→(Base‘𝐺))) |
17 | 14, 15, 16 | sylanbrc 585 | 1 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Grpcgrp 18097 invgcminusg 18098 GrpHom cghm 18349 GrpIso cgim 18391 oppgcoppg 18467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-ghm 18350 df-gim 18393 df-oppg 18468 |
This theorem is referenced by: oppggic 18483 symgtrinv 18594 gsumzinv 19059 |
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