Theorem idmhm 12696

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 5483	. . . 4 |- ( _I |` B ) : B -1-1-onto-> B
3		f1of 5445	. . . 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 2171	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
7	5, 6	mndcl 12663	. . . . . . 7 |- ( ( M e. Mnd /\ a e. B /\ b e. B ) -> ( a ( +g ` M ) b ) e. B )
8	7	3expb 1200	. . . . . 6 |- ( ( M e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
9		fvresi 5693	. . . . . 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 5693	. . . . . . 7 |- ( a e. B -> ( ( _I |` B ) ` a ) = a )
12		fvresi 5693	. . . . . . 7 |- ( b e. B -> ( ( _I |` B ) ` b ) = b )
13	11, 12	oveqan12d 5876	. . . . . 6 |- ( ( a e. B /\ b e. B ) -> ( ( ( _I |` B ) ` a ) ( +g ` M ) ( ( _I |` B ) ` b ) ) = ( a ( +g ` M ) b ) )
14	13	adantl 275	. . . . 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 2207	. . . 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 2553	. . 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 2171	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
18	5, 17	mndidcl 12670	. . . 4 |- ( M e. Mnd -> ( 0g ` M ) e. B )
19		fvresi 5693	. . . 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 1173	. 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 12689	. 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 1178	1 |- ( M e. Mnd -> ( _I |` B ) e. ( M MndHom M ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 974   = wceq 1349   e. wcel 2142   A.wral 2449   _I cid 4274   |` cres 4614   -->wf 5196   -1-1-onto->wf1o 5199   ` cfv 5200  (class class class)co 5857   Basecbs 12420   +g cplusg 12484   0gc0g 12600   Mndcmnd 12656   MndHom cmhm 12685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-cnex 7869  ax-resscn 7870  ax-1re 7872  ax-addrcl 7875

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-map 6632  df-inn 8883  df-2 8941  df-ndx 12423  df-slot 12424  df-base 12426  df-plusg 12497  df-0g 12602  df-mgm 12614  df-sgrp 12647  df-mnd 12657  df-mhm 12687

This theorem is referenced by: (None)