Theorem mhmid 13918

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 2238	. 2 |- ( 0g ` H ) = ( 0g ` H )
3		ghmgrp.q	. 2 |- .+^ = ( +g ` H )
4		ghmgrp.1	. . . 4 |- ( ph -> F : X -onto-> Y )
5		fof 5615	. . . 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 13743	. . . 4 |- ( G e. Mnd -> .0. e. X )
11	7, 10	syl 14	. . 3 |- ( ph -> .0. e. X )
12	6, 11	ffvelcdmd 5844	. 2 |- ( ph -> ( F ` .0. ) e. Y )
13		simplll 539	. . . . . . 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 1311	. . . . . 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 496	. . . . . . 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 533	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> i e. X )
19	15, 17, 18	mhmlem 13917	. . . . 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 13748	. . . . . . 7 |- ( ( G e. Mnd /\ i e. X ) -> ( .0. .+ i ) = i )
22	16, 18, 21	syl2anc 415	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( .0. .+ i ) = i )
23	22	fveq2d 5699	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( .0. .+ i ) ) = ( F ` i ) )
24	19, 23	eqtr3d 2273	. . . 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 6101	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ ( F ` i ) ) = ( ( F ` .0. ) .+^ a ) )
27	24, 26, 25	3eqtr3d 2279	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ a ) = a )
28		foelcdmi 5755	. . . 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 2694	. 2 |- ( ( ph /\ a e. Y ) -> ( ( F ` .0. ) .+^ a ) = a )
31	15, 18, 17	mhmlem 13917	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( ( F ` i ) .+^ ( F ` .0. ) ) )
32	8, 20, 9	mndrid 13749	. . . . . . 7 |- ( ( G e. Mnd /\ i e. X ) -> ( i .+ .0. ) = i )
33	16, 18, 32	syl2anc 415	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( i .+ .0. ) = i )
34	33	fveq2d 5699	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( F ` i ) )
35	31, 34	eqtr3d 2273	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( F ` i ) )
36	25	oveq1d 6100	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( a .+^ ( F ` .0. ) ) )
37	35, 36, 25	3eqtr3d 2279	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( a .+^ ( F ` .0. ) ) = a )
38	37, 29	r19.29a 2694	. 2 |- ( ( ph /\ a e. Y ) -> ( a .+^ ( F ` .0. ) ) = a )
39	1, 2, 3, 12, 30, 38	ismgmid2 13700	1 |- ( ph -> ( F ` .0. ) = ( 0g ` H ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   -->wf 5373   -onto->wfo 5375   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   Mndcmnd 13729

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730

This theorem is used by:  mhmfmhm  13920  ghmgrp  13921