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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 2798 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | invoppggim.o | . . . 4 ⊢ 𝑂 = (oppg‘𝐺) | |
3 | 2, 1 | oppgbas 18471 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝑂) |
4 | eqid 2798 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | eqid 2798 | . . 3 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
6 | id 22 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
7 | 2 | oppggrp 18477 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑂 ∈ Grp) |
8 | invoppggim.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
9 | 1, 8 | grpinvf 18142 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
10 | 1, 4, 8 | grpinvadd 18169 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥))) |
11 | 10 | 3expb 1117 | . . . 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 18469 | . . . 4 ⊢ ((𝐼‘𝑥)(+g‘𝑂)(𝐼‘𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥)) |
13 | 11, 12 | eqtr4di 2851 | . . 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 18359 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpHom 𝑂)) |
15 | 1, 8, 6 | grpinvf1o 18161 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)–1-1-onto→(Base‘𝐺)) |
16 | 1, 3 | isgim 18394 | . 2 ⊢ (𝐼 ∈ (𝐺 GrpIso 𝑂) ↔ (𝐼 ∈ (𝐺 GrpHom 𝑂) ∧ 𝐼:(Base‘𝐺)–1-1-onto→(Base‘𝐺))) |
17 | 14, 15, 16 | sylanbrc 586 | 1 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Grpcgrp 18095 invgcminusg 18096 GrpHom cghm 18347 GrpIso cgim 18389 oppgcoppg 18465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-ghm 18348 df-gim 18391 df-oppg 18466 |
This theorem is referenced by: oppggic 18481 symgtrinv 18592 gsumzinv 19058 |
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