Theorem ghmgrp 13504

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 13395	. . 3 |- ( ph -> G e. Mnd )
9	1, 2, 3, 4, 5, 6, 8	mhmmnd 13502	. 2 |- ( ph -> H e. Mnd )
10		fof 5507	. . . . . . . 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 2206	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
16	2, 15	grpinvcl 13430	. . . . . . 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 5726	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( invg ` G ) ` i ) ) e. Y )
19	1	3adant1r 1234	. . . . . . . 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 13500	. . . . . . 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 2206	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
25	2, 4, 24, 15	grplinv 13432	. . . . . . . . 9 |- ( ( G e. Grp /\ i e. X ) -> ( ( ( invg ` G ) ` i ) .+ i ) = ( 0g ` G ) )
26	25	fveq2d 5590	. . . . . . . 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 13501	. . . . . . . 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 2239	. . . . . 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 5970	. . . . . 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 2248	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) )
34		oveq1 5961	. . . . . . 7 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( f .+^ a ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) )
35	34	eqeq1d 2215	. . . . . 6 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( ( f .+^ a ) = ( 0g ` H ) <-> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) ) )
36	35	rspcev 2879	. . . . 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 5641	. . . . 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 2650	. . 3 |- ( ( ph /\ a e. Y ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
41	40	ralrimiva 2580	. 2 |- ( ph -> A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
42		eqid 2206	. . 3 |- ( 0g ` H ) = ( 0g ` H )
43	3, 5, 42	isgrp 13388	. 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 981   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   -->wf 5273   -onto->wfo 5275   ` cfv 5277  (class class class)co 5954   Basecbs 12882   +g cplusg 12959   0gc0g 13138   Mndcmnd 13298   Grpcgrp 13382   invgcminusg 13383

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-cnex 8029  ax-resscn 8030  ax-1re 8032  ax-addrcl 8035

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-inn 9050  df-2 9108  df-ndx 12885  df-slot 12886  df-base 12888  df-plusg 12972  df-0g 13140  df-mgm 13238  df-sgrp 13284  df-mnd 13299  df-grp 13385  df-minusg 13386

This theorem is referenced by:  ghmfghm  13712  ghmabl  13714