Theorem ghmf1 14053

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 14052	. . . . 5 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. <-> x = N ) )
6	5	3expa 1234	. . . 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 2623	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> A. x e. A ( ( F ` x ) = .0. -> x = N ) )
9	1, 2	ghmf 14027	. . . 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 2238	. . . . . . . . . 10 |- ( -g ` R ) = ( -g ` R )
12		eqid 2238	. . . . . . . . . 10 |- ( -g ` S ) = ( -g ` S )
13	1, 11, 12	ghmsub 14031	. . . . . . . . 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 1235	. . . . . . . 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 481	. . . . . . 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 2247	. . . . . 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 5699	. . . . . . . 8 |- ( x = ( y ( -g ` R ) z ) -> ( ( F ` x ) = .0. <-> ( F ` ( y ( -g ` R ) z ) ) = .0. ) )
18		eqeq1 2245	. . . . . . . 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 533	. . . . . . 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 14025	. . . . . . . . 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 13862	. . . . . . . . 9 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( y ( -g ` R ) z ) e. A )
24	23	3expb 1235	. . . . . . . 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 2934	. . . . . 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 14026	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> S e. Grp )
29	28	ad2antrr 492	. . . . . 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 492	. . . . . . 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 535	. . . . . . 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 5835	. . . . . 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 537	. . . . . . 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 5835	. . . . . 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 13868	. . . . . 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 1278	. . . . 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 492	. . . . . 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 13868	. . . . . 6 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( ( y ( -g ` R ) z ) = N <-> y = z ) )
39	37, 31, 33, 38	syl3anc 1278	. . . . 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 2632	. . 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 5964	. . 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 421	. 2 |- ( ( F e. ( R GrpHom S ) /\ A. x e. A ( ( F ` x ) = .0. -> x = N ) ) -> F : A -1-1-> B )
44	8, 43	impbida 604	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 1402   e. wcel 2209   A.wral 2528   -->wf 5368   -1-1->wf1 5369   ` cfv 5372  (class class class)co 6075   Basecbs 13330   0gc0g 13587   Grpcgrp 13782   -gcsg 13784   GrpHom cghm 14020

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787  df-ghm 14021

This theorem is referenced by: (None)