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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 2821 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2821 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 1, 2, 3 | grpinveu 18138 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) |
5 | riotacl 7131 | . . 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 18142 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
9 | 6, 8 | fmptd 6878 | 1 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃!wreu 3140 ⟶wf 6351 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 invgcminusg 18104 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 |
This theorem is referenced by: grpinvcl 18151 isgrpinv 18156 grpinvcnv 18167 grpinvf1o 18169 grp1inv 18207 pwsinvg 18212 pwssub 18213 oppginv 18487 invoppggim 18488 symgtrinv 18600 invghm 18954 gsumzinv 19065 dprdfinv 19141 mhpinvcl 20339 grpvlinv 21006 grpvrinv 21007 mdetralt 21217 istgp2 22699 subgtgp 22713 symgtgp 22714 tgpconncomp 22721 prdstgpd 22733 tsmssub 22757 tsmsxplem1 22761 tlmtgp 22804 nrginvrcn 23301 |
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