Theorem idmhm 13724

Description: The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)

Hypothesis

Ref	Expression
idmhm.b	|- B = ( Base ` M )

Assertion

Ref	Expression
idmhm	|- ( M e. Mnd -> ( _I |` B ) e. ( M MndHom M ) )

Proof of Theorem idmhm

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

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( M e. Mnd -> M e. Mnd )
2		f1oi 5659	. . . 4 |- ( _I |` B ) : B -1-1-onto-> B
3		f1of 5619	. . . 4 |- ( ( _I |` B ) : B -1-1-onto-> B -> ( _I |` B ) : B --> B )
4	2, 3	mp1i 10	. . 3 |- ( M e. Mnd -> ( _I |` B ) : B --> B )
5		idmhm.b	. . . . . . . 8 |- B = ( Base ` M )
6		eqid 2234	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
7	5, 6	mndcl 13684	. . . . . . 7 |- ( ( M e. Mnd /\ a e. B /\ b e. B ) -> ( a ( +g ` M ) b ) e. B )
8	7	3expb 1231	. . . . . 6 |- ( ( M e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
9		fvresi 5882	. . . . . 6 |- ( ( a ( +g ` M ) b ) e. B -> ( ( _I |` B ) ` ( a ( +g ` M ) b ) ) = ( a ( +g ` M ) b ) )
10	8, 9	syl 14	. . . . 5 |- ( ( M e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( ( _I |` B ) ` ( a ( +g ` M ) b ) ) = ( a ( +g ` M ) b ) )
11		fvresi 5882	. . . . . . 7 |- ( a e. B -> ( ( _I |` B ) ` a ) = a )
12		fvresi 5882	. . . . . . 7 |- ( b e. B -> ( ( _I |` B ) ` b ) = b )
13	11, 12	oveqan12d 6077	. . . . . 6 |- ( ( a e. B /\ b e. B ) -> ( ( ( _I |` B ) ` a ) ( +g ` M ) ( ( _I |` B ) ` b ) ) = ( a ( +g ` M ) b ) )
14	13	adantl 277	. . . . 5 |- ( ( M e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( ( ( _I |` B ) ` a ) ( +g ` M ) ( ( _I |` B ) ` b ) ) = ( a ( +g ` M ) b ) )
15	10, 14	eqtr4d 2270	. . . 4 |- ( ( M e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( ( _I |` B ) ` ( a ( +g ` M ) b ) ) = ( ( ( _I |` B ) ` a ) ( +g ` M ) ( ( _I |` B ) ` b ) ) )
16	15	ralrimivva 2626	. . 3 |- ( M e. Mnd -> A. a e. B A. b e. B ( ( _I |` B ) ` ( a ( +g ` M ) b ) ) = ( ( ( _I |` B ) ` a ) ( +g ` M ) ( ( _I |` B ) ` b ) ) )
17		eqid 2234	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
18	5, 17	mndidcl 13691	. . . 4 |- ( M e. Mnd -> ( 0g ` M ) e. B )
19		fvresi 5882	. . . 4 |- ( ( 0g ` M ) e. B -> ( ( _I |` B ) ` ( 0g ` M ) ) = ( 0g ` M ) )
20	18, 19	syl 14	. . 3 |- ( M e. Mnd -> ( ( _I |` B ) ` ( 0g ` M ) ) = ( 0g ` M ) )
21	4, 16, 20	3jca 1204	. 2 |- ( M e. Mnd -> ( ( _I |` B ) : B --> B /\ A. a e. B A. b e. B ( ( _I |` B ) ` ( a ( +g ` M ) b ) ) = ( ( ( _I |` B ) ` a ) ( +g ` M ) ( ( _I |` B ) ` b ) ) /\ ( ( _I |` B ) ` ( 0g ` M ) ) = ( 0g ` M ) ) )
22	5, 5, 6, 6, 17, 17	ismhm 13716	. 2 |- ( ( _I |` B ) e. ( M MndHom M ) <-> ( ( M e. Mnd /\ M e. Mnd ) /\ ( ( _I |` B ) : B --> B /\ A. a e. B A. b e. B ( ( _I |` B ) ` ( a ( +g ` M ) b ) ) = ( ( ( _I |` B ) ` a ) ( +g ` M ) ( ( _I |` B ) ` b ) ) /\ ( ( _I |` B ) ` ( 0g ` M ) ) = ( 0g ` M ) ) ) )
23	1, 1, 21, 22	syl21anbrc 1209	1 |- ( M e. Mnd -> ( _I |` B ) e. ( M MndHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   _I cid 4414   |` cres 4756   -->wf 5353   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Mndcmnd 13677   MndHom cmhm 13712

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-in1 619  ax-in2 620  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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mhm 13714

This theorem is referenced by: (None)