grpinvf1o - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > grpinvf1o		Structured version Visualization version GIF version

Theorem grpinvf1o 18645

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.)

**Hypotheses**
Ref	Expression
grpinvinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinvinv.n	⊢ 𝑁 = (inv_g‘𝐺)
grpinv11.g	⊢ (𝜑 → 𝐺 ∈ Grp)

**Assertion**
Ref	Expression
grpinvf1o	⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵)

**Proof of Theorem grpinvf1o**
Step	Hyp	Ref	Expression
1		grpinv11.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grpinvinv.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		grpinvinv.n	. . . . 5 ⊢ 𝑁 = (inv_g‘𝐺)
4	2, 3	grpinvf 18626	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵)
6	5	ffnd 6601	. 2 ⊢ (𝜑 → 𝑁 Fn 𝐵)
7	2, 3	grpinvcnv 18643	. . . . 5 ⊢ (𝐺 ∈ Grp → ^◡𝑁 = 𝑁)
8	1, 7	syl 17	. . . 4 ⊢ (𝜑 → ^◡𝑁 = 𝑁)
9	8	fneq1d 6526	. . 3 ⊢ (𝜑 → (^◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵))
10	6, 9	mpbird 256	. 2 ⊢ (𝜑 → ^◡𝑁 Fn 𝐵)
11		dff1o4 6724	. 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 1539 ∈ wcel 2106 ^◡ccnv 5588 Fn wfn 6428 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 Basecbs 16912 Grpcgrp 18577 inv_gcminusg 18578

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581

This theorem is referenced by: invoppggim 18967 gsumsub 19549 dprdfsub 19624 psrnegcl 21165 psrlinv 21166 mdetleib2 21737 lflnegl 37090

W3C validator