Theorem mhmid 13832

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 2232	. 2 |- ( 0g ` H ) = ( 0g ` H )
3		ghmgrp.q	. 2 |- .+^ = ( +g ` H )
4		ghmgrp.1	. . . 4 |- ( ph -> F : X -onto-> Y )
5		fof 5590	. . . 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 13643	. . . 4 |- ( G e. Mnd -> .0. e. X )
11	7, 10	syl 14	. . 3 |- ( ph -> .0. e. X )
12	6, 11	ffvelcdmd 5813	. 2 |- ( ph -> ( F ` .0. ) e. Y )
13		simplll 535	. . . . . . 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 1307	. . . . . 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 529	. . . . . 6 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> i e. X )
19	15, 17, 18	mhmlem 13831	. . . . 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 13648	. . . . . . 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 5674	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( .0. .+ i ) ) = ( F ` i ) )
24	19, 23	eqtr3d 2267	. . . 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 6066	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ ( F ` i ) ) = ( ( F ` .0. ) .+^ a ) )
27	24, 26, 25	3eqtr3d 2273	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ a ) = a )
28		foelcdmi 5729	. . . 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 2686	. 2 |- ( ( ph /\ a e. Y ) -> ( ( F ` .0. ) .+^ a ) = a )
31	15, 18, 17	mhmlem 13831	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( ( F ` i ) .+^ ( F ` .0. ) ) )
32	8, 20, 9	mndrid 13649	. . . . . . 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 5674	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( F ` i ) )
35	31, 34	eqtr3d 2267	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( F ` i ) )
36	25	oveq1d 6065	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( a .+^ ( F ` .0. ) ) )
37	35, 36, 25	3eqtr3d 2273	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( a .+^ ( F ` .0. ) ) = a )
38	37, 29	r19.29a 2686	. 2 |- ( ( ph /\ a e. Y ) -> ( a .+^ ( F ` .0. ) ) = a )
39	1, 2, 3, 12, 30, 38	ismgmid2 13593	1 |- ( ph -> ( F ` .0. ) = ( 0g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   -->wf 5348   -onto->wfo 5350   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   Mndcmnd 13629

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630

This theorem is referenced by:  mhmfmhm  13834  ghmgrp  13835