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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 19017 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵) |
6 | 5 | ffnd 6738 | . 2 ⊢ (𝜑 → 𝑁 Fn 𝐵) |
7 | 2, 3 | grpinvcnv 19037 | . . . . 5 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ◡𝑁 = 𝑁) |
9 | 8 | fneq1d 6662 | . . 3 ⊢ (𝜑 → (◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵)) |
10 | 6, 9 | mpbird 257 | . 2 ⊢ (𝜑 → ◡𝑁 Fn 𝐵) |
11 | dff1o4 6857 | . 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ◡𝑁 Fn 𝐵)) | |
12 | 6, 10, 11 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ◡ccnv 5688 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 Basecbs 17245 Grpcgrp 18964 invgcminusg 18965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 |
This theorem is referenced by: invoppggim 19394 gsumsub 19981 dprdfsub 20056 psrnegcl 21992 psrlinv 21993 mdetleib2 22610 ply1divalg3 35627 lflnegl 39058 |
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