Theorem ghmcmn 14080

Description: The image of a commutative monoid G under a group homomorphism F is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)

Hypotheses

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

Assertion

Ref	Expression
ghmcmn	|- ( ph -> H e. CMnd )

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

Proof of Theorem ghmcmn

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

Step	Hyp	Ref	Expression
1		ghmabl.f	. . 3 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
2		ghmabl.x	. . 3 |- X = ( Base ` G )
3		ghmabl.y	. . 3 |- Y = ( Base ` H )
4		ghmabl.p	. . 3 |- .+ = ( +g ` G )
5		ghmabl.q	. . 3 |- .+^ = ( +g ` H )
6		ghmabl.1	. . 3 |- ( ph -> F : X -onto-> Y )
7		ghmcmn.3	. . . 4 |- ( ph -> G e. CMnd )
8		cmnmnd 14054	. . . 4 |- ( G e. CMnd -> G e. Mnd )
9	7, 8	syl 14	. . 3 |- ( ph -> G e. Mnd )
10	1, 2, 3, 4, 5, 6, 9	mhmmnd 13869	. 2 |- ( ph -> H e. Mnd )
11		simp-6l 547	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ph )
12	11, 7	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> G e. CMnd )
13		simp-4r 544	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> a e. X )
14		simplr 529	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> b e. X )
15	2, 4	cmncom 14055	. . . . . . . . . 10 |- ( ( G e. CMnd /\ a e. X /\ b e. X ) -> ( a .+ b ) = ( b .+ a ) )
16	12, 13, 14, 15	syl3anc 1274	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( a .+ b ) = ( b .+ a ) )
17	16	fveq2d 5679	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( F ` ( a .+ b ) ) = ( F ` ( b .+ a ) ) )
18	11, 1	syl3an1 1307	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
19	18, 13, 14	mhmlem 13867	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) )
20	18, 14, 13	mhmlem 13867	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( F ` ( b .+ a ) ) = ( ( F ` b ) .+^ ( F ` a ) ) )
21	17, 19, 20	3eqtr3d 2275	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( ( F ` a ) .+^ ( F ` b ) ) = ( ( F ` b ) .+^ ( F ` a ) ) )
22		simpllr 536	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( F ` a ) = i )
23		simpr 110	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( F ` b ) = j )
24	22, 23	oveq12d 6076	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( ( F ` a ) .+^ ( F ` b ) ) = ( i .+^ j ) )
25	23, 22	oveq12d 6076	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( ( F ` b ) .+^ ( F ` a ) ) = ( j .+^ i ) )
26	21, 24, 25	3eqtr3d 2275	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) /\ b e. X ) /\ ( F ` b ) = j ) -> ( i .+^ j ) = ( j .+^ i ) )
27		foelcdmi 5734	. . . . . . . 8 |- ( ( F : X -onto-> Y /\ j e. Y ) -> E. b e. X ( F ` b ) = j )
28	6, 27	sylan 283	. . . . . . 7 |- ( ( ph /\ j e. Y ) -> E. b e. X ( F ` b ) = j )
29	28	ad5ant13 519	. . . . . 6 |- ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) -> E. b e. X ( F ` b ) = j )
30	26, 29	r19.29a 2688	. . . . 5 |- ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) -> ( i .+^ j ) = ( j .+^ i ) )
31		foelcdmi 5734	. . . . . . 7 |- ( ( F : X -onto-> Y /\ i e. Y ) -> E. a e. X ( F ` a ) = i )
32	6, 31	sylan 283	. . . . . 6 |- ( ( ph /\ i e. Y ) -> E. a e. X ( F ` a ) = i )
33	32	adantr 276	. . . . 5 |- ( ( ( ph /\ i e. Y ) /\ j e. Y ) -> E. a e. X ( F ` a ) = i )
34	30, 33	r19.29a 2688	. . . 4 |- ( ( ( ph /\ i e. Y ) /\ j e. Y ) -> ( i .+^ j ) = ( j .+^ i ) )
35	34	anasss 399	. . 3 |- ( ( ph /\ ( i e. Y /\ j e. Y ) ) -> ( i .+^ j ) = ( j .+^ i ) )
36	35	ralrimivva 2626	. 2 |- ( ph -> A. i e. Y A. j e. Y ( i .+^ j ) = ( j .+^ i ) )
37	3, 5	iscmn 14046	. 2 |- ( H e. CMnd <-> ( H e. Mnd /\ A. i e. Y A. j e. Y ( i .+^ j ) = ( j .+^ i ) ) )
38	10, 36, 37	sylanbrc 417	1 |- ( ph -> H e. CMnd )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   -onto->wfo 5355   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   Mndcmnd 13677  CMndccmn 14037

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-cmn 14039

This theorem is referenced by:  ghmabl  14081