Theorem ghmf1 13212

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 13211	. . . . 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 2563	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> A. x e. A ( ( F ` x ) = .0. -> x = N ) )
9	1, 2	ghmf 13186	. . . 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 2189	. . . . . . . . . 10 |- ( -g ` R ) = ( -g ` R )
12		eqid 2189	. . . . . . . . . 10 |- ( -g ` S ) = ( -g ` S )
13	1, 11, 12	ghmsub 13190	. . . . . . . . 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 2198	. . . . . 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 5543	. . . . . . . 8 |- ( x = ( y ( -g ` R ) z ) -> ( ( F ` x ) = .0. <-> ( F ` ( y ( -g ` R ) z ) ) = .0. ) )
18		eqeq1 2196	. . . . . . . 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 13184	. . . . . . . . 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 13024	. . . . . . . . 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 2861	. . . . . 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 13185	. . . . . . 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 5673	. . . . . 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 5673	. . . . . 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 13030	. . . . . 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 13030	. . . . . 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 2572	. . 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 5790	. . 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 2160   A.wral 2468   -->wf 5231   -1-1->wf1 5232   ` cfv 5235  (class class class)co 5896   Basecbs 12512   0gc0g 12761   Grpcgrp 12945   -gcsg 12947   GrpHom cghm 13179

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1re 7935  ax-addrcl 7938

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-inn 8950  df-2 9008  df-ndx 12515  df-slot 12516  df-base 12518  df-plusg 12602  df-0g 12763  df-mgm 12832  df-sgrp 12865  df-mnd 12878  df-grp 12948  df-minusg 12949  df-sbg 12950  df-ghm 13180

This theorem is referenced by: (None)