Theorem ghmf1 13810

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 13809	. . . . 5 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. <-> x = N ) )
6	5	3expa 1227	. . . 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 2603	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> A. x e. A ( ( F ` x ) = .0. -> x = N ) )
9	1, 2	ghmf 13784	. . . 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 2229	. . . . . . . . . 10 |- ( -g ` R ) = ( -g ` R )
12		eqid 2229	. . . . . . . . . 10 |- ( -g ` S ) = ( -g ` S )
13	1, 11, 12	ghmsub 13788	. . . . . . . . 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 1228	. . . . . . . 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 2238	. . . . . 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 5636	. . . . . . . 8 |- ( x = ( y ( -g ` R ) z ) -> ( ( F ` x ) = .0. <-> ( F ` ( y ( -g ` R ) z ) ) = .0. ) )
18		eqeq1 2236	. . . . . . . 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 13782	. . . . . . . . 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 13613	. . . . . . . . 9 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( y ( -g ` R ) z ) e. A )
24	23	3expb 1228	. . . . . . . 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 2912	. . . . . 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 13783	. . . . . . 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 5771	. . . . . 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 5771	. . . . . 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 13619	. . . . . 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 1271	. . . . 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 13619	. . . . . 6 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( ( y ( -g ` R ) z ) = N <-> y = z ) )
39	37, 31, 33, 38	syl3anc 1271	. . . . 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 2612	. . 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 5892	. . 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 598	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 1395   e. wcel 2200   A.wral 2508   -->wf 5314   -1-1->wf1 5315   ` cfv 5318  (class class class)co 6001   Basecbs 13032   0gc0g 13289   Grpcgrp 13533   -gcsg 13535   GrpHom cghm 13777

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-sbg 13538  df-ghm 13778

This theorem is referenced by: (None)