Theorem c0mhm 41910

Description: The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)

Hypotheses

Ref	Expression
c0mhm.b	|- B = ( Base ` S )
c0mhm.0	|- .0. = ( 0g ` T )
c0mhm.h	|- H = ( x e. B |-> .0. )

Assertion

Ref	Expression
c0mhm	|- ( ( S e. Mnd /\ T e. Mnd ) -> H e. ( S MndHom T ) )

Distinct variable groups:   x, B   x, S   x, T   x, .0.

Allowed substitution hint:   H( x)

Proof of Theorem c0mhm

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . 8 |- ( Base ` T ) = ( Base ` T )
2		c0mhm.0	. . . . . . . 8 |- .0. = ( 0g ` T )
3	1, 2	mndidcl 17308	. . . . . . 7 |- ( T e. Mnd -> .0. e. ( Base ` T ) )
4	3	adantl 482	. . . . . 6 |- ( ( S e. Mnd /\ T e. Mnd ) -> .0. e. ( Base ` T ) )
5	4	adantr 481	. . . . 5 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ x e. B ) -> .0. e. ( Base ` T ) )
6		c0mhm.h	. . . . 5 |- H = ( x e. B |-> .0. )
7	5, 6	fmptd 6385	. . . 4 |- ( ( S e. Mnd /\ T e. Mnd ) -> H : B --> ( Base ` T ) )
8	3	ancli 574	. . . . . . . . 9 |- ( T e. Mnd -> ( T e. Mnd /\ .0. e. ( Base ` T ) ) )
9	8	adantl 482	. . . . . . . 8 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( T e. Mnd /\ .0. e. ( Base ` T ) ) )
10		eqid 2622	. . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
11	1, 10, 2	mndlid 17311	. . . . . . . 8 |- ( ( T e. Mnd /\ .0. e. ( Base ` T ) ) -> ( .0. ( +g ` T ) .0. ) = .0. )
12	9, 11	syl 17	. . . . . . 7 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( .0. ( +g ` T ) .0. ) = .0. )
13	12	adantr 481	. . . . . 6 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( .0. ( +g ` T ) .0. ) = .0. )
14	6	a1i 11	. . . . . . . 8 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> H = ( x e. B |-> .0. ) )
15		eqidd 2623	. . . . . . . 8 |- ( ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) /\ x = a ) -> .0. = .0. )
16		simprl 794	. . . . . . . 8 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> a e. B )
17	4	adantr 481	. . . . . . . 8 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> .0. e. ( Base ` T ) )
18	14, 15, 16, 17	fvmptd 6288	. . . . . . 7 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` a ) = .0. )
19		eqidd 2623	. . . . . . . 8 |- ( ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) /\ x = b ) -> .0. = .0. )
20		simprr 796	. . . . . . . 8 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> b e. B )
21	14, 19, 20, 17	fvmptd 6288	. . . . . . 7 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` b ) = .0. )
22	18, 21	oveq12d 6668	. . . . . 6 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( ( H ` a ) ( +g ` T ) ( H ` b ) ) = ( .0. ( +g ` T ) .0. ) )
23		eqidd 2623	. . . . . . 7 |- ( ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) /\ x = ( a ( +g ` S ) b ) ) -> .0. = .0. )
24		c0mhm.b	. . . . . . . . . 10 |- B = ( Base ` S )
25		eqid 2622	. . . . . . . . . 10 |- ( +g ` S ) = ( +g ` S )
26	24, 25	mndcl 17301	. . . . . . . . 9 |- ( ( S e. Mnd /\ a e. B /\ b e. B ) -> ( a ( +g ` S ) b ) e. B )
27	26	3expb 1266	. . . . . . . 8 |- ( ( S e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` S ) b ) e. B )
28	27	adantlr 751	. . . . . . 7 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` S ) b ) e. B )
29	14, 23, 28, 17	fvmptd 6288	. . . . . 6 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` ( a ( +g ` S ) b ) ) = .0. )
30	13, 22, 29	3eqtr4rd 2667	. . . . 5 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) )
31	30	ralrimivva 2971	. . . 4 |- ( ( S e. Mnd /\ T e. Mnd ) -> A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) )
32	6	a1i 11	. . . . 5 |- ( ( S e. Mnd /\ T e. Mnd ) -> H = ( x e. B |-> .0. ) )
33		eqidd 2623	. . . . 5 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ x = ( 0g ` S ) ) -> .0. = .0. )
34		eqid 2622	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
35	24, 34	mndidcl 17308	. . . . . 6 |- ( S e. Mnd -> ( 0g ` S ) e. B )
36	35	adantr 481	. . . . 5 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( 0g ` S ) e. B )
37	32, 33, 36, 4	fvmptd 6288	. . . 4 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( H ` ( 0g ` S ) ) = .0. )
38	7, 31, 37	3jca 1242	. . 3 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( H : B --> ( Base ` T ) /\ A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) /\ ( H ` ( 0g ` S ) ) = .0. ) )
39	38	ancli 574	. 2 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( ( S e. Mnd /\ T e. Mnd ) /\ ( H : B --> ( Base ` T ) /\ A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) /\ ( H ` ( 0g ` S ) ) = .0. ) ) )
40	24, 1, 25, 10, 34, 2	ismhm 17337	. 2 |- ( H e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( H : B --> ( Base ` T ) /\ A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) /\ ( H ` ( 0g ` S ) ) = .0. ) ) )
41	39, 40	sylibr 224	1 |- ( ( S e. Mnd /\ T e. Mnd ) -> H e. ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294   MndHom cmhm 17333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335

This theorem is referenced by:  c0ghm  41911  c0rhm  41912