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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 2738 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2738 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 1, 2, 3 | grpinveu 18256 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) |
5 | riotacl 7145 | . . 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 18260 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
9 | 6, 8 | fmptd 6888 | 1 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃!wreu 3055 ⟶wf 6335 ‘cfv 6339 ℩crio 7126 (class class class)co 7170 Basecbs 16586 +gcplusg 16668 0gc0g 16816 Grpcgrp 18219 invgcminusg 18220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-riota 7127 df-ov 7173 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-minusg 18223 |
This theorem is referenced by: grpinvcl 18269 isgrpinv 18274 grpinvcnv 18285 grpinvf1o 18287 grp1inv 18325 pwsinvg 18330 pwssub 18331 oppginv 18605 invoppggim 18606 symgtrinv 18718 invghm 19073 gsumzinv 19184 dprdfinv 19260 grpvlinv 21148 grpvrinv 21149 mdetralt 21359 istgp2 22842 subgtgp 22856 symgtgp 22857 tgpconncomp 22864 prdstgpd 22876 tsmssub 22900 tsmsxplem1 22904 tlmtgp 22947 nrginvrcn 23445 |
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