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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 18150 | . . . 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 18167 | . . . . 5 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ◡𝑁 = 𝑁) |
9 | 8 | fneq1d 6446 | . . 3 ⊢ (𝜑 → (◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵)) |
10 | 6, 9 | mpbird 259 | . 2 ⊢ (𝜑 → ◡𝑁 Fn 𝐵) |
11 | dff1o4 6623 | . 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ◡𝑁 Fn 𝐵)) | |
12 | 6, 10, 11 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ◡ccnv 5554 Fn wfn 6350 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 Basecbs 16483 Grpcgrp 18103 invgcminusg 18104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 |
This theorem is referenced by: invoppggim 18488 gsumsub 19068 dprdfsub 19143 psrnegcl 20176 psrlinv 20177 mdetleib2 21197 lflnegl 36227 |
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