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Mirrors > Home > MPE Home > Th. List > grpinvf1o | Structured version Visualization version GIF version |
Description: The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
grpinvinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvinv.n | ⊢ 𝑁 = (invg‘𝐺) |
grpinv11.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Ref | Expression |
---|---|
grpinvf1o | ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinv11.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | grpinvinv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpinvinv.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
4 | 2, 3 | grpinvf 18280 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵) |
6 | 5 | ffnd 6515 | . 2 ⊢ (𝜑 → 𝑁 Fn 𝐵) |
7 | 2, 3 | grpinvcnv 18297 | . . . . 5 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ◡𝑁 = 𝑁) |
9 | 8 | fneq1d 6441 | . . 3 ⊢ (𝜑 → (◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵)) |
10 | 6, 9 | mpbird 260 | . 2 ⊢ (𝜑 → ◡𝑁 Fn 𝐵) |
11 | dff1o4 6638 | . 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ◡𝑁 Fn 𝐵)) | |
12 | 6, 10, 11 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ◡ccnv 5534 Fn wfn 6344 ⟶wf 6345 –1-1-onto→wf1o 6348 ‘cfv 6349 Basecbs 16598 Grpcgrp 18231 invgcminusg 18232 |
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 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
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 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-0g 16830 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-grp 18234 df-minusg 18235 |
This theorem is referenced by: invoppggim 18618 gsumsub 19199 dprdfsub 19274 psrnegcl 20787 psrlinv 20788 mdetleib2 21351 lflnegl 36745 |
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