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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 2726 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2726 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 1, 2, 3 | grpinveu 18969 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) |
5 | riotacl 7398 | . . 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 18973 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
9 | 6, 8 | fmptd 7128 | 1 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃!wreu 3362 ⟶wf 6550 ‘cfv 6554 ℩crio 7379 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 0gc0g 17454 Grpcgrp 18928 invgcminusg 18929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-riota 7380 df-ov 7427 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 |
This theorem is referenced by: grpinvcl 18982 isgrpinv 18988 grpinvcnv 19001 grpinvf1o 19003 grp1inv 19042 pwsinvg 19047 pwssub 19048 oppginv 19356 invoppggim 19357 symgtrinv 19470 invghm 19831 gsumzinv 19943 dprdfinv 20019 grpvlinv 22389 grpvrinv 22390 mdetralt 22601 istgp2 24086 subgtgp 24100 symgtgp 24101 tgpconncomp 24108 prdstgpd 24120 tsmssub 24144 tsmsxplem1 24148 tlmtgp 24191 nrginvrcn 24700 |
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