Theorem ghmcmn 14108

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 14081	. . . 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 13896	. 2 |- ( ph -> H e. Mnd )
11		simp-6l 551	. . . . . . . . . . 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 548	. . . . . . . . . 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 533	. . . . . . . . . 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 14082	. . . . . . . . . 10 |- ( ( G e. CMnd /\ a e. X /\ b e. X ) -> ( a .+ b ) = ( b .+ a ) )
16	12, 13, 14, 15	syl3anc 1278	. . . . . . . . 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 5694	. . . . . . . 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 1311	. . . . . . . . 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 13894	. . . . . . . 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 13894	. . . . . . . 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 2279	. . . . . . 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 540	. . . . . . . 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 6093	. . . . . . 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 6093	. . . . . . 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 2279	. . . . . 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 5749	. . . . . . . 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 523	. . . . . 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 2694	. . . . 5 |- ( ( ( ( ( ph /\ i e. Y ) /\ j e. Y ) /\ a e. X ) /\ ( F ` a ) = i ) -> ( i .+^ j ) = ( j .+^ i ) )
31		foelcdmi 5749	. . . . . . 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 2694	. . . 4 |- ( ( ( ph /\ i e. Y ) /\ j e. Y ) -> ( i .+^ j ) = ( j .+^ i ) )
35	34	anasss 403	. . 3 |- ( ( ph /\ ( i e. Y /\ j e. Y ) ) -> ( i .+^ j ) = ( j .+^ i ) )
36	35	ralrimivva 2632	. 2 |- ( ph -> A. i e. Y A. j e. Y ( i .+^ j ) = ( j .+^ i ) )
37	3, 5	iscmn 14073	. 2 |- ( H e. CMnd <-> ( H e. Mnd /\ A. i e. Y A. j e. Y ( i .+^ j ) = ( j .+^ i ) ) )
38	10, 36, 37	sylanbrc 421	1 |- ( ph -> H e. CMnd )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   -onto->wfo 5370   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   Mndcmnd 13706  CMndccmn 14064

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-cmn 14066

This theorem is referenced by:  ghmabl  14109