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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 2739 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | invoppggim.o | . . . 4 ⊢ 𝑂 = (oppg‘𝐺) | |
3 | 2, 1 | oppgbas 18775 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝑂) |
4 | eqid 2739 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | eqid 2739 | . . 3 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
6 | id 22 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
7 | 2 | oppggrp 18781 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑂 ∈ Grp) |
8 | invoppggim.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
9 | 1, 8 | grpinvf 18446 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
10 | 1, 4, 8 | grpinvadd 18473 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥))) |
11 | 10 | 3expb 1122 | . . . 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 18773 | . . . 4 ⊢ ((𝐼‘𝑥)(+g‘𝑂)(𝐼‘𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥)) |
13 | 11, 12 | eqtr4di 2798 | . . 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 18663 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpHom 𝑂)) |
15 | 1, 8, 6 | grpinvf1o 18465 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)–1-1-onto→(Base‘𝐺)) |
16 | 1, 3 | isgim 18698 | . 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 1543 ∈ wcel 2112 –1-1-onto→wf1o 6399 ‘cfv 6400 (class class class)co 7234 Basecbs 16792 +gcplusg 16834 Grpcgrp 18397 invgcminusg 18398 GrpHom cghm 18651 GrpIso cgim 18693 oppgcoppg 18769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-tpos 7991 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-nn 11860 df-2 11922 df-sets 16749 df-slot 16767 df-ndx 16777 df-base 16793 df-plusg 16847 df-0g 16978 df-mgm 18146 df-sgrp 18195 df-mnd 18206 df-grp 18400 df-minusg 18401 df-ghm 18652 df-gim 18695 df-oppg 18770 |
This theorem is referenced by: oppggic 18785 symgtrinv 18896 gsumzinv 19362 |
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