Theorem ghmgrp 13641

Description: The image of a group G under a group homomorphism F is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator O in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)

Hypotheses

Ref	Expression
ghmgrp.f	|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
ghmgrp.x	|- X = ( Base ` G )
ghmgrp.y	|- Y = ( Base ` H )
ghmgrp.p	|- .+ = ( +g ` G )
ghmgrp.q	|- .+^ = ( +g ` H )
ghmgrp.1	|- ( ph -> F : X -onto-> Y )
ghmgrp.3	|- ( ph -> G e. Grp )

Assertion

Ref	Expression
ghmgrp	|- ( ph -> H e. Grp )

Distinct variable groups:   x, F, y   x, G, y   x, .+ , y   x, H, y   x, X, y   x, Y, y   x, .+^ , y   ph, x, y

Proof of Theorem ghmgrp

Dummy variables a f i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmgrp.f	. . 3 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
2		ghmgrp.x	. . 3 |- X = ( Base ` G )
3		ghmgrp.y	. . 3 |- Y = ( Base ` H )
4		ghmgrp.p	. . 3 |- .+ = ( +g ` G )
5		ghmgrp.q	. . 3 |- .+^ = ( +g ` H )
6		ghmgrp.1	. . 3 |- ( ph -> F : X -onto-> Y )
7		ghmgrp.3	. . . 4 |- ( ph -> G e. Grp )
8	7	grpmndd 13532	. . 3 |- ( ph -> G e. Mnd )
9	1, 2, 3, 4, 5, 6, 8	mhmmnd 13639	. 2 |- ( ph -> H e. Mnd )
10		fof 5544	. . . . . . . 8 |- ( F : X -onto-> Y -> F : X --> Y )
11	6, 10	syl 14	. . . . . . 7 |- ( ph -> F : X --> Y )
12	11	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> F : X --> Y )
13	7	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> G e. Grp )
14		simplr 528	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> i e. X )
15		eqid 2229	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
16	2, 15	grpinvcl 13567	. . . . . . 7 |- ( ( G e. Grp /\ i e. X ) -> ( ( invg ` G ) ` i ) e. X )
17	13, 14, 16	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( invg ` G ) ` i ) e. X )
18	12, 17	ffvelcdmd 5764	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( invg ` G ) ` i ) ) e. Y )
19	1	3adant1r 1255	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
20	7, 16	sylan 283	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( ( invg ` G ) ` i ) e. X )
21		simpr 110	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> i e. X )
22	19, 20, 21	mhmlem 13637	. . . . . . 7 |- ( ( ph /\ i e. X ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ ( F ` i ) ) )
23	22	ad4ant13 513	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ ( F ` i ) ) )
24		eqid 2229	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
25	2, 4, 24, 15	grplinv 13569	. . . . . . . . 9 |- ( ( G e. Grp /\ i e. X ) -> ( ( ( invg ` G ) ` i ) .+ i ) = ( 0g ` G ) )
26	25	fveq2d 5627	. . . . . . . 8 |- ( ( G e. Grp /\ i e. X ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( F ` ( 0g ` G ) ) )
27	13, 14, 26	syl2anc 411	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( F ` ( 0g ` G ) ) )
28	1, 2, 3, 4, 5, 6, 8, 24	mhmid 13638	. . . . . . . 8 |- ( ph -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
29	28	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
30	27, 29	eqtrd 2262	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( 0g ` H ) )
31		simpr 110	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` i ) = a )
32	31	oveq2d 6010	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ ( F ` i ) ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) )
33	23, 30, 32	3eqtr3rd 2271	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) )
34		oveq1 6001	. . . . . . 7 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( f .+^ a ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) )
35	34	eqeq1d 2238	. . . . . 6 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( ( f .+^ a ) = ( 0g ` H ) <-> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) ) )
36	35	rspcev 2907	. . . . 5 |- ( ( ( F ` ( ( invg ` G ) ` i ) ) e. Y /\ ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
37	18, 33, 36	syl2anc 411	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
38		foelcdmi 5679	. . . . 5 |- ( ( F : X -onto-> Y /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
39	6, 38	sylan 283	. . . 4 |- ( ( ph /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
40	37, 39	r19.29a 2674	. . 3 |- ( ( ph /\ a e. Y ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
41	40	ralrimiva 2603	. 2 |- ( ph -> A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
42		eqid 2229	. . 3 |- ( 0g ` H ) = ( 0g ` H )
43	3, 5, 42	isgrp 13525	. 2 |- ( H e. Grp <-> ( H e. Mnd /\ A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) ) )
44	9, 41, 43	sylanbrc 417	1 |- ( ph -> H e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   -->wf 5310   -onto->wfo 5312   ` cfv 5314  (class class class)co 5994   Basecbs 13018   +g cplusg 13096   0gc0g 13275   Mndcmnd 13435   Grpcgrp 13519   invgcminusg 13520

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-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 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-minusg 13523

This theorem is referenced by:  ghmfghm  13849  ghmabl  13851