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Mirrors > Home > MPE Home > Th. List > grpinvf | Structured version Visualization version GIF version |
Description: The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.) |
Ref | Expression |
---|---|
grpinvcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvcl.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvf | ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2797 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2797 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 1, 2, 3 | grpinveu 17769 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) |
5 | riotacl 6851 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺) → (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ 𝐵) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ 𝐵) |
7 | grpinvcl.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
8 | 1, 2, 3, 7 | grpinvfval 17773 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
9 | 6, 8 | fmptd 6608 | 1 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃!wreu 3089 ⟶wf 6095 ‘cfv 6099 ℩crio 6836 (class class class)co 6876 Basecbs 16181 +gcplusg 16264 0gc0g 16412 Grpcgrp 17735 invgcminusg 17736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-0g 16414 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-minusg 17739 |
This theorem is referenced by: grpinvcl 17780 isgrpinv 17785 grpinvcnv 17796 grpinvf1o 17798 grp1inv 17836 pwsinvg 17841 pwssub 17842 oppginv 18098 invoppggim 18099 symgtrinv 18201 invghm 18551 gsumzinv 18657 dprdfinv 18731 grpvlinv 20523 grpvrinv 20524 mdetralt 20737 istgp2 22220 symgtgp 22230 subgtgp 22234 tgpconncomp 22241 prdstgpd 22253 tsmssub 22277 tsmsxplem1 22281 tlmtgp 22324 nrginvrcn 22821 |
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