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Mirrors > Home > ILE Home > Th. List > grpinvf1o | 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 12797 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
5 | 1, 4 | syl 14 | . . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵) |
6 | 5 | ffnd 5361 | . 2 ⊢ (𝜑 → 𝑁 Fn 𝐵) |
7 | 2, 3 | grpinvcnv 12814 | . . . . 5 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) |
8 | 1, 7 | syl 14 | . . . 4 ⊢ (𝜑 → ◡𝑁 = 𝑁) |
9 | 8 | fneq1d 5301 | . . 3 ⊢ (𝜑 → (◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵)) |
10 | 6, 9 | mpbird 167 | . 2 ⊢ (𝜑 → ◡𝑁 Fn 𝐵) |
11 | dff1o4 5464 | . 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ◡𝑁 Fn 𝐵)) | |
12 | 6, 10, 11 | sylanbrc 417 | 1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ◡ccnv 4621 Fn wfn 5206 ⟶wf 5207 –1-1-onto→wf1o 5210 ‘cfv 5211 Basecbs 12432 Grpcgrp 12754 invgcminusg 12755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 df-minusg 12758 |
This theorem is referenced by: (None) |
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