Theorem ghmgrp 13248

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 13145	. . 3 |- ( ph -> G e. Mnd )
9	1, 2, 3, 4, 5, 6, 8	mhmmnd 13246	. 2 |- ( ph -> H e. Mnd )
10		fof 5480	. . . . . . . 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 2196	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
16	2, 15	grpinvcl 13180	. . . . . . 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 5698	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( invg ` G ) ` i ) ) e. Y )
19	1	3adant1r 1233	. . . . . . . 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 13244	. . . . . . 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 2196	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
25	2, 4, 24, 15	grplinv 13182	. . . . . . . . 9 |- ( ( G e. Grp /\ i e. X ) -> ( ( ( invg ` G ) ` i ) .+ i ) = ( 0g ` G ) )
26	25	fveq2d 5562	. . . . . . . 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 13245	. . . . . . . 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 2229	. . . . . 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 5938	. . . . . 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 2238	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) )
34		oveq1 5929	. . . . . . 7 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( f .+^ a ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) )
35	34	eqeq1d 2205	. . . . . 6 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( ( f .+^ a ) = ( 0g ` H ) <-> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) ) )
36	35	rspcev 2868	. . . . 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 5613	. . . . 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 2640	. . 3 |- ( ( ph /\ a e. Y ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
41	40	ralrimiva 2570	. 2 |- ( ph -> A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
42		eqid 2196	. . 3 |- ( 0g ` H ) = ( 0g ` H )
43	3, 5, 42	isgrp 13138	. 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 980   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   -->wf 5254   -onto->wfo 5256   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Mndcmnd 13057   Grpcgrp 13132   invgcminusg 13133

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136

This theorem is referenced by:  ghmfghm  13456  ghmabl  13458