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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 18846 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵) |
6 | 5 | ffnd 6705 | . 2 ⊢ (𝜑 → 𝑁 Fn 𝐵) |
7 | 2, 3 | grpinvcnv 18865 | . . . . 5 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ◡𝑁 = 𝑁) |
9 | 8 | fneq1d 6631 | . . 3 ⊢ (𝜑 → (◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵)) |
10 | 6, 9 | mpbird 256 | . 2 ⊢ (𝜑 → ◡𝑁 Fn 𝐵) |
11 | dff1o4 6828 | . 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 1541 ∈ wcel 2106 ◡ccnv 5668 Fn wfn 6527 ⟶wf 6528 –1-1-onto→wf1o 6531 ‘cfv 6532 Basecbs 17126 Grpcgrp 18794 invgcminusg 18795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 |
This theorem is referenced by: invoppggim 19191 gsumsub 19775 dprdfsub 19850 psrnegcl 21446 psrlinv 21447 mdetleib2 22019 lflnegl 37751 |
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