Theorem mhmid 13701

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 2231	. 2 |- ( 0g ` H ) = ( 0g ` H )
3		ghmgrp.q	. 2 |- .+^ = ( +g ` H )
4		ghmgrp.1	. . . 4 |- ( ph -> F : X -onto-> Y )
5		fof 5559	. . . 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 13512	. . . 4 |- ( G e. Mnd -> .0. e. X )
11	7, 10	syl 14	. . 3 |- ( ph -> .0. e. X )
12	6, 11	ffvelcdmd 5783	. 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 1306	. . . . . 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 13700	. . . . 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 13517	. . . . . . 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 5643	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( .0. .+ i ) ) = ( F ` i ) )
24	19, 23	eqtr3d 2266	. . . 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 6033	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ ( F ` i ) ) = ( ( F ` .0. ) .+^ a ) )
27	24, 26, 25	3eqtr3d 2272	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` .0. ) .+^ a ) = a )
28		foelcdmi 5698	. . . 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 2676	. 2 |- ( ( ph /\ a e. Y ) -> ( ( F ` .0. ) .+^ a ) = a )
31	15, 18, 17	mhmlem 13700	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( ( F ` i ) .+^ ( F ` .0. ) ) )
32	8, 20, 9	mndrid 13518	. . . . . . 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 5643	. . . . 5 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( F ` ( i .+ .0. ) ) = ( F ` i ) )
35	31, 34	eqtr3d 2266	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( F ` i ) )
36	25	oveq1d 6032	. . . 4 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( ( F ` i ) .+^ ( F ` .0. ) ) = ( a .+^ ( F ` .0. ) ) )
37	35, 36, 25	3eqtr3d 2272	. . 3 |- ( ( ( ( ph /\ a e. Y ) /\ i e. X ) /\ ( F ` i ) = a ) -> ( a .+^ ( F ` .0. ) ) = a )
38	37, 29	r19.29a 2676	. 2 |- ( ( ph /\ a e. Y ) -> ( a .+^ ( F ` .0. ) ) = a )
39	1, 2, 3, 12, 30, 38	ismgmid2 13462	1 |- ( ph -> ( F ` .0. ) = ( 0g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   -->wf 5322   -onto->wfo 5324   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Mndcmnd 13498

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499

This theorem is referenced by:  mhmfmhm  13703  ghmgrp  13704