Theorem ghmf1 14026

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 14025	. . . . 5 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. <-> x = N ) )
6	5	3expa 1230	. . . 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 2617	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> A. x e. A ( ( F ` x ) = .0. -> x = N ) )
9	1, 2	ghmf 14000	. . . 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 2234	. . . . . . . . . 10 |- ( -g ` R ) = ( -g ` R )
12		eqid 2234	. . . . . . . . . 10 |- ( -g ` S ) = ( -g ` S )
13	1, 11, 12	ghmsub 14004	. . . . . . . . 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 1231	. . . . . . . 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 2243	. . . . . 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 5684	. . . . . . . 8 |- ( x = ( y ( -g ` R ) z ) -> ( ( F ` x ) = .0. <-> ( F ` ( y ( -g ` R ) z ) ) = .0. ) )
18		eqeq1 2241	. . . . . . . 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 529	. . . . . . 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 13998	. . . . . . . . 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 13835	. . . . . . . . 9 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( y ( -g ` R ) z ) e. A )
24	23	3expb 1231	. . . . . . . 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 2928	. . . . . 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 13999	. . . . . . 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 531	. . . . . . 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 5818	. . . . . 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 533	. . . . . . 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 5818	. . . . . 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 13841	. . . . . 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 1274	. . . . 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 13841	. . . . . 6 |- ( ( R e. Grp /\ y e. A /\ z e. A ) -> ( ( y ( -g ` R ) z ) = N <-> y = z ) )
39	37, 31, 33, 38	syl3anc 1274	. . . . 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 2626	. . 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 5947	. . 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 600	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 1398   e. wcel 2205   A.wral 2522   -->wf 5353   -1-1->wf1 5354   ` cfv 5357  (class class class)co 6058   Basecbs 13296   0gc0g 13553   Grpcgrp 13755   -gcsg 13757   GrpHom cghm 13993

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-sbg 13760  df-ghm 13994

This theorem is referenced by: (None)