Theorem ghmf1 13343

Description: Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)

Hypotheses

Ref	Expression
f1ghm0to0.a	|- A = ( Base ` R )
f1ghm0to0.b	|- B = ( Base ` S )
f1ghm0to0.n	|- N = ( 0g ` R )
f1ghm0to0.0	|- .0. = ( 0g ` S )

Assertion

Ref	Expression
ghmf1	|- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> A. x e. A ( ( F ` x ) = .0. -> x = N ) ) )

Distinct variable groups:   x, .0.   x, A   x, B   x, F   x, N   x, R   x, S

Proof of Theorem ghmf1

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ghm0to0.a	. . . . . 6 |- A = ( Base ` R )
2		f1ghm0to0.b	. . . . . 6 |- B = ( Base ` S )
3		f1ghm0to0.n	. . . . . 6 |- N = ( 0g ` R )
4		f1ghm0to0.0	. . . . . 6 |- .0. = ( 0g ` S )
5	1, 2, 3, 4	f1ghm0to0 13342	. . . . 5 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. <-> x = N ) )
6	5	3expa 1205	. . . 4 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. A ) -> ( ( F ` x ) = .0. <-> x = N ) )
7	6	biimpd 144	. . 3 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. A ) -> ( ( F ` x ) = .0. -> x = N ) )
8	7	ralrimiva 2567	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> A. x e. A ( ( F ` x ) = .0. -> x = N ) )
9	1, 2	ghmf 13317	. . . 4 |- ( F e. ( R GrpHom S ) -> F : A --> B )
10	9	adantr 276	. . 3 |- ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) -> F : A --> B )
11		eqid 2193	. . . . . . . . . 10 |- ( -g ` R ) = ( -g ` R )
12		eqid 2193	. . . . . . . . . 10 |- ( -g ` S ) = ( -g ` S )
13	1, 11, 12	ghmsub 13321	. . . . . . . . 9 |- ( ( F e. ( R GrpHom S ) /\ y e. A /\ z e. A ) -> ( F ` ( y ( -g ` R ) z ) ) = ( ( F ` y ) ( -g ` S ) ( F ` z ) ) )
14	13	3expb 1206	. . . . . . . 8 |- ( ( F e. ( R GrpHom S ) /\ ( y e. A /\ z e. A ) ) -> ( F ` ( y ( -g ` R ) z ) ) = ( ( F ` y ) ( -g ` S ) ( F ` z ) ) )
15	14	adantlr 477	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( F ` ( y ( -g ` R ) z ) ) = ( ( F ` y ) ( -g ` S ) ( F ` z ) ) )
16	15	eqeq1d 2202	. . . . . 6 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( ( F ` ( y ( -g ` R ) z ) ) = .0. <-> ( ( F ` y ) ( -g ` S ) ( F ` z ) ) = .0. ) )
17		fveqeq2 5563	. . . . . . . 8 |- ( x = ( y ( -g ` R ) z ) -> ( ( F ` x ) = .0. <-> ( F ` ( y ( -g ` R ) z ) ) = .0. ) )
18		eqeq1 2200	. . . . . . . 8 |- ( x = ( y ( -g ` R ) z ) -> ( x = N <-> ( y ( -g ` R ) z ) = N ) )
19	17, 18	imbi12d 234	. . . . . . 7 |- ( x = ( y ( -g ` R ) z ) -> ( ( ( F ` x ) = .0. -> x = N ) <-> ( ( F ` ( y ( -g ` R ) z ) ) = .0. -> ( y ( -g ` R ) z ) = N ) ) )
20		simplr 528	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> A. x e. A ( ( F ` x ) = .0. -> x = N ) )
21		ghmgrp1 13315	. . . . . . . . 9 |- ( F e. ( R GrpHom S ) -> R e. Grp )
22	21	adantr 276	. . . . . . . 8 |- ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) -> R e. Grp )
23	1, 11	grpsubcl 13152	. . . . . . . . 9 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( y ( -g ` R ) z ) e. A )
24	23	3expb 1206	. . . . . . . 8 |- ( ( R e. Grp /\ ( y e. A /\ z e. A ) ) -> ( y ( -g ` R ) z ) e. A )
25	22, 24	sylan 283	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( y ( -g ` R ) z ) e. A )
26	19, 20, 25	rspcdva 2869	. . . . . 6 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( ( F ` ( y ( -g ` R ) z ) ) = .0. -> ( y ( -g ` R ) z ) = N ) )
27	16, 26	sylbird 170	. . . . 5 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( ( ( F ` y ) ( -g ` S ) ( F ` z ) ) = .0. -> ( y ( -g ` R ) z ) = N ) )
28		ghmgrp2 13316	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> S e. Grp )
29	28	ad2antrr 488	. . . . . 6 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> S e. Grp )
30	9	ad2antrr 488	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> F : A --> B )
31		simprl 529	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> y e. A )
32	30, 31	ffvelcdmd 5694	. . . . . 6 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( F ` y ) e. B )
33		simprr 531	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> z e. A )
34	30, 33	ffvelcdmd 5694	. . . . . 6 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( F ` z ) e. B )
35	2, 4, 12	grpsubeq0 13158	. . . . . 6 |- ( ( S e. Grp /\ ( F ` y ) e. B /\ ( F ` z ) e. B ) -> ( ( ( F ` y ) ( -g ` S ) ( F ` z ) ) = .0. <-> ( F ` y ) = ( F ` z ) ) )
36	29, 32, 34, 35	syl3anc 1249	. . . . 5 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( ( ( F ` y ) ( -g ` S ) ( F ` z ) ) = .0. <-> ( F ` y ) = ( F ` z ) ) )
37	21	ad2antrr 488	. . . . . 6 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> R e. Grp )
38	1, 3, 11	grpsubeq0 13158	. . . . . 6 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( ( y ( -g ` R ) z ) = N <-> y = z ) )
39	37, 31, 33, 38	syl3anc 1249	. . . . 5 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( ( y ( -g ` R ) z ) = N <-> y = z ) )
40	27, 36, 39	3imtr3d 202	. . . 4 |- ( ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) /\ ( y e. A /\ z e. A ) ) -> ( ( F ` y ) = ( F ` z ) -> y = z ) )
41	40	ralrimivva 2576	. . 3 |- ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) -> A. y e. A A. z e. A ( ( F ` y ) = ( F ` z ) -> y = z ) )
42		dff13 5811	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. y e. A A. z e. A ( ( F ` y ) = ( F ` z ) -> y = z ) ) )
43	10, 41, 42	sylanbrc 417	. 2 |- ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) -> F : A -1-1-> B )
44	8, 43	impbida 596	1 |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> A. x e. A ( ( F ` x ) = .0. -> x = N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   -->wf 5250   -1-1->wf1 5251   ` cfv 5254  (class class class)co 5918   Basecbs 12618   0gc0g 12867   Grpcgrp 13072   -gcsg 13074   GrpHom cghm 13310

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-ghm 13311

This theorem is referenced by: (None)