Theorem idmhm 13753

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 5674	. . . 4 |- ( _I |` B ) : B -1-1-onto-> B
3		f1of 5634	. . . 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 2238	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
7	5, 6	mndcl 13713	. . . . . . 7 |- ( ( M e. Mnd /\ a e. B /\ b e. B ) -> ( a ( +g ` M ) b ) e. B )
8	7	3expb 1235	. . . . . 6 |- ( ( M e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
9		fvresi 5899	. . . . . 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 5899	. . . . . . 7 |- ( a e. B -> ( ( _I |` B ) ` a ) = a )
12		fvresi 5899	. . . . . . 7 |- ( b e. B -> ( ( _I |` B ) ` b ) = b )
13	11, 12	oveqan12d 6094	. . . . . 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 2274	. . . 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 2632	. . 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 2238	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
18	5, 17	mndidcl 13720	. . . 4 |- ( M e. Mnd -> ( 0g ` M ) e. B )
19		fvresi 5899	. . . 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 1208	. 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 13745	. 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 1213	1 |- ( M e. Mnd -> ( _I |` B ) e. ( M MndHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _I cid 4428   |` cres 4771   -->wf 5368   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706   MndHom cmhm 13741

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 623  ax-in2 624  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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743

This theorem is referenced by: (None)