Theorem ghmgrp 12833

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 12742	. . 3 |- ( ph -> G e. Mnd )
9	1, 2, 3, 4, 5, 6, 8	mhmmnd 12831	. 2 |- ( ph -> H e. Mnd )
10		fof 5422	. . . . . . . 8 |- ( F : X -onto-> Y -> F : X --> Y )
11	6, 10	syl 14	. . . . . . 7 |- ( ph -> F : X --> Y )
12	11	ad3antrrr 490	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> F : X --> Y )
13	7	ad3antrrr 490	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> G e. Grp )
14		simplr 526	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> i e. X )
15		eqid 2171	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
16	2, 15	grpinvcl 12773	. . . . . . 7 |- ( ( G e. Grp /\ i e. X ) -> ( ( invg ` G ) ` i ) e. X )
17	13, 14, 16	syl2anc 409	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( invg ` G ) ` i ) e. X )
18	12, 17	ffvelrnd 5636	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( invg ` G ) ` i ) ) e. Y )
19	1	3adant1r 1227	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
20	7, 16	sylan 281	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( ( invg ` G ) ` i ) e. X )
21		simpr 109	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> i e. X )
22	19, 20, 21	mhmlem 12829	. . . . . . 7 |- ( ( ph /\ i e. X ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ ( F ` i ) ) )
23	22	ad4ant13 511	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ ( F ` i ) ) )
24		eqid 2171	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
25	2, 4, 24, 15	grplinv 12774	. . . . . . . . 9 |- ( ( G e. Grp /\ i e. X ) -> ( ( ( invg ` G ) ` i ) .+ i ) = ( 0g ` G ) )
26	25	fveq2d 5503	. . . . . . . 8 |- ( ( G e. Grp /\ i e. X ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( F ` ( 0g ` G ) ) )
27	13, 14, 26	syl2anc 409	. . . . . . 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 12830	. . . . . . . 8 |- ( ph -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
29	28	ad3antrrr 490	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
30	27, 29	eqtrd 2204	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( 0g ` H ) )
31		simpr 109	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` i ) = a )
32	31	oveq2d 5873	. . . . . 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 2213	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) )
34		oveq1 5864	. . . . . . 7 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( f .+^ a ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) )
35	34	eqeq1d 2180	. . . . . 6 |- ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( ( f .+^ a ) = ( 0g ` H ) <-> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) ) )
36	35	rspcev 2835	. . . . 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 409	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
38		foelrni 5552	. . . . 5 |- ( ( F : X -onto-> Y /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
39	6, 38	sylan 281	. . . 4 |- ( ( ph /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
40	37, 39	r19.29a 2614	. . 3 |- ( ( ph /\ a e. Y ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
41	40	ralrimiva 2544	. 2 |- ( ph -> A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
42		eqid 2171	. . 3 |- ( 0g ` H ) = ( 0g ` H )
43	3, 5, 42	isgrp 12736	. 2 |- ( H e. Grp <-> ( H e. Mnd /\ A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) ) )
44	9, 41, 43	sylanbrc 415	1 |- ( ph -> H e. Grp )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 974   = wceq 1349   e. wcel 2142   A.wral 2449   E.wrex 2450   -->wf 5196   -onto->wfo 5198   ` cfv 5200  (class class class)co 5857   Basecbs 12420   +g cplusg 12484   0gc0g 12618   Mndcmnd 12674   Grpcgrp 12730   invgcminusg 12731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-cnex 7869  ax-resscn 7870  ax-1re 7872  ax-addrcl 7875

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-inn 8883  df-2 8941  df-ndx 12423  df-slot 12424  df-base 12426  df-plusg 12497  df-0g 12620  df-mgm 12632  df-sgrp 12665  df-mnd 12675  df-grp 12733  df-minusg 12734

This theorem is referenced by: (None)