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Metamath Proof Explorer

Theorem ghmgrp

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)

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

Proof

Step Hyp Ref Expression
1 ghmgrp.f 
 |-  ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
2 ghmgrp.x 
 |-  X = ( Base ` G )
3 ghmgrp.y 
 |-  Y = ( Base ` H )
4 ghmgrp.p 
 |-  .+ = ( +g ` G )
5 ghmgrp.q 
 |-  .+^ = ( +g ` H )
6 ghmgrp.1 
 |-  ( ph -> F : X -onto-> Y )
7 ghmgrp.3 
 |-  ( ph -> G e. Grp )
8 7 grpmndd 
 |-  ( ph -> G e. Mnd )
9 1 2 3 4 5 6 8 mhmmnd 
 |-  ( ph -> H e. Mnd )
10 fof 
 |-  ( F : X -onto-> Y -> F : X --> Y )
11 6 10 syl 
 |-  ( ph -> F : X --> Y )
12 11 ad3antrrr 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> F : X --> Y )
13 7 ad3antrrr 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> G e. Grp )
14 simplr 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> i e. X )
15 eqid 
 |-  ( invg ` G ) = ( invg ` G )
16 2 15 grpinvcl 
 |-  ( ( G e. Grp /\ i e. X ) -> ( ( invg ` G ) ` i ) e. X )
17 13 14 16 syl2anc 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( invg ` G ) ` i ) e. X )
18 12 17 ffvelcdmd 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( invg ` G ) ` i ) ) e. Y )
19 1 3adant1r 
 |-  ( ( ( ph /\ i e. X ) /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
20 7 16 sylan 
 |-  ( ( ph /\ i e. X ) -> ( ( invg ` G ) ` i ) e. X )
21 simpr 
 |-  ( ( ph /\ i e. X ) -> i e. X )
22 19 20 21 mhmlem 
 |-  ( ( ph /\ i e. X ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ ( F ` i ) ) )
23 22 ad4ant13 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ ( F ` i ) ) )
24 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
25 2 4 24 15 grplinv 
 |-  ( ( G e. Grp /\ i e. X ) -> ( ( ( invg ` G ) ` i ) .+ i ) = ( 0g ` G ) )
26 25 fveq2d 
 |-  ( ( G e. Grp /\ i e. X ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( F ` ( 0g ` G ) ) )
27 13 14 26 syl2anc 
 |-  ( ( ( ( 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 
 |-  ( ph -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
29 28 ad3antrrr 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
30 27 29 eqtrd 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( ( ( invg ` G ) ` i ) .+ i ) ) = ( 0g ` H ) )
31 simpr 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` i ) = a )
32 31 oveq2d 
 |-  ( ( ( ( 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 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) )
34 oveq1 
 |-  ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( f .+^ a ) = ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) )
35 34 eqeq1d 
 |-  ( f = ( F ` ( ( invg ` G ) ` i ) ) -> ( ( f .+^ a ) = ( 0g ` H ) <-> ( ( F ` ( ( invg ` G ) ` i ) ) .+^ a ) = ( 0g ` H ) ) )
36 35 rspcev 
 |-  ( ( ( 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 
 |-  ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
38 foelcdmi 
 |-  ( ( F : X -onto-> Y /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
39 6 38 sylan 
 |-  ( ( ph /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
40 37 39 r19.29a 
 |-  ( ( ph /\ a e. Y ) -> E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
41 40 ralrimiva 
 |-  ( ph -> A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) )
42 eqid 
 |-  ( 0g ` H ) = ( 0g ` H )
43 3 5 42 isgrp 
 |-  ( H e. Grp <-> ( H e. Mnd /\ A. a e. Y E. f e. Y ( f .+^ a ) = ( 0g ` H ) ) )
44 9 41 43 sylanbrc 
 |-  ( ph -> H e. Grp )