Theorem mhmid 13080

Description: A surjective monoid morphism preserves identity element. (Contributed 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 )
mhmmnd.3	|- ( ph -> G e. Mnd )
mhmid.0	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
mhmid	|- ( ph -> ( F ` .0. ) = ( 0g ` H ) )

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   x, .0. , y

Proof of Theorem mhmid

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

Step	Hyp	Ref	Expression
1		ghmgrp.y	. 2 |- Y = ( Base ` H )
2		eqid 2189	. 2 |- ( 0g ` H ) = ( 0g ` H )
3		ghmgrp.q	. 2 |- .+^ = ( +g ` H )
4		ghmgrp.1	. . . 4 |- ( ph -> F : X -onto-> Y )
5		fof 5460	. . . 4 |- ( F : X -onto-> Y -> F : X --> Y )
6	4, 5	syl 14	. . 3 |- ( ph -> F : X --> Y )
7		mhmmnd.3	. . . 4 |- ( ph -> G e. Mnd )
8		ghmgrp.x	. . . . 5 |- X = ( Base ` G )
9		mhmid.0	. . . . 5 |- .0. = ( 0g ` G )
10	8, 9	mndidcl 12914	. . . 4 |- ( G e. Mnd -> .0. e. X )
11	7, 10	syl 14	. . 3 |- ( ph -> .0. e. X )
12	6, 11	ffvelcdmd 5676	. 2 |- ( ph -> ( F ` .0. ) e. Y )
13		simplll 533	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ph )
14		ghmgrp.f	. . . . . . 7 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
15	13, 14	syl3an1 1282	. . . . . 6 |- ( ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
16	7	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> G e. Mnd )
17	16, 10	syl 14	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> .0. e. X )
18		simplr 528	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> i e. X )
19	15, 17, 18	mhmlem 13079	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( .0. .+ i ) ) = ( ( F ` .0. ) .+^ ( F ` i ) ) )
20		ghmgrp.p	. . . . . . . 8 |- .+ = ( +g ` G )
21	8, 20, 9	mndlid 12919	. . . . . . 7 |- ( ( G e. Mnd /\ i e. X ) -> ( .0. .+ i ) = i )
22	16, 18, 21	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( .0. .+ i ) = i )
23	22	fveq2d 5541	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( .0. .+ i ) ) = ( F ` i ) )
24	19, 23	eqtr3d 2224	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ ( F ` i ) ) = ( F ` i ) )
25		simpr 110	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` i ) = a )
26	25	oveq2d 5916	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ ( F ` i ) ) = ( ( F ` .0. ) .+^ a ) )
27	24, 26, 25	3eqtr3d 2230	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ a ) = a )
28		foelcdmi 5592	. . . 4 |- ( ( F : X -onto-> Y /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
29	4, 28	sylan 283	. . 3 |- ( ( ph /\ a e. Y ) -> E. i e. X ( F ` i ) = a )
30	27, 29	r19.29a 2633	. 2 |- ( ( ph /\ a e. Y ) -> ( ( F ` .0. ) .+^ a ) = a )
31	15, 18, 17	mhmlem 13079	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( ( F ` i ) .+^ ( F ` .0. ) ) )
32	8, 20, 9	mndrid 12920	. . . . . . 7 |- ( ( G e. Mnd /\ i e. X ) -> ( i .+ .0. ) = i )
33	16, 18, 32	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( i .+ .0. ) = i )
34	33	fveq2d 5541	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( F ` i ) )
35	31, 34	eqtr3d 2224	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( F ` i ) )
36	25	oveq1d 5915	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( a .+^ ( F ` .0. ) ) )
37	35, 36, 25	3eqtr3d 2230	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( a .+^ ( F ` .0. ) ) = a )
38	37, 29	r19.29a 2633	. 2 |- ( ( ph /\ a e. Y ) -> ( a .+^ ( F ` .0. ) ) = a )
39	1, 2, 3, 12, 30, 38	ismgmid2 12867	1 |- ( ph -> ( F ` .0. ) = ( 0g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   E.wrex 2469   -->wf 5234   -onto->wfo 5236   ` cfv 5238  (class class class)co 5900   Basecbs 12523   +g cplusg 12600   0gc0g 12772   Mndcmnd 12900

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-cnex 7937  ax-resscn 7938  ax-1re 7940  ax-addrcl 7943

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fo 5244  df-fv 5246  df-riota 5855  df-ov 5903  df-inn 8955  df-2 9013  df-ndx 12526  df-slot 12527  df-base 12529  df-plusg 12613  df-0g 12774  df-mgm 12843  df-sgrp 12888  df-mnd 12901

This theorem is referenced by:  mhmfmhm  13082  ghmgrp  13083