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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 2736 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | invoppggim.o | . . . 4 ⊢ 𝑂 = (oppg‘𝐺) | |
3 | 2, 1 | oppgbas 19052 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝑂) |
4 | eqid 2736 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | eqid 2736 | . . 3 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
6 | id 22 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
7 | 2 | oppggrp 19060 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑂 ∈ Grp) |
8 | invoppggim.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
9 | 1, 8 | grpinvf 18722 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
10 | 1, 4, 8 | grpinvadd 18749 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥))) |
11 | 10 | 3expb 1119 | . . . 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 19049 | . . . 4 ⊢ ((𝐼‘𝑥)(+g‘𝑂)(𝐼‘𝑦)) = ((𝐼‘𝑦)(+g‘𝐺)(𝐼‘𝑥)) |
13 | 11, 12 | eqtr4di 2794 | . . 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 18939 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpHom 𝑂)) |
15 | 1, 8, 6 | grpinvf1o 18741 | . 2 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)–1-1-onto→(Base‘𝐺)) |
16 | 1, 3 | isgim 18974 | . 2 ⊢ (𝐼 ∈ (𝐺 GrpIso 𝑂) ↔ (𝐼 ∈ (𝐺 GrpHom 𝑂) ∧ 𝐼:(Base‘𝐺)–1-1-onto→(Base‘𝐺))) |
17 | 14, 15, 16 | sylanbrc 583 | 1 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 –1-1-onto→wf1o 6478 ‘cfv 6479 (class class class)co 7337 Basecbs 17009 +gcplusg 17059 Grpcgrp 18673 invgcminusg 18674 GrpHom cghm 18927 GrpIso cgim 18969 oppgcoppg 19045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-tpos 8112 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-plusg 17072 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-grp 18676 df-minusg 18677 df-ghm 18928 df-gim 18971 df-oppg 19046 |
This theorem is referenced by: oppggic 19064 symgtrinv 19176 gsumzinv 19641 |
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