Theorem zrrnghm 41917

Description: The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
zrrnghm	|- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> H e. ( T RngHomo S ) )

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

Allowed substitution hint:   H( x)

Proof of Theorem zrrnghm

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

Step	Hyp	Ref	Expression
1		eldifi 3732	. . . . 5 |- ( T e. ( Ring \ NzRing ) -> T e. Ring )
2		ringrng 41879	. . . . 5 |- ( T e. Ring -> T e. Rng )
3	1, 2	syl 17	. . . 4 |- ( T e. ( Ring \ NzRing ) -> T e. Rng )
4	3	anim1i 592	. . 3 |- ( ( T e. ( Ring \ NzRing ) /\ S e. Rng ) -> ( T e. Rng /\ S e. Rng ) )
5	4	ancoms 469	. 2 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> ( T e. Rng /\ S e. Rng ) )
6		rngabl 41877	. . . . . 6 |- ( S e. Rng -> S e. Abel )
7		ablgrp 18198	. . . . . 6 |- ( S e. Abel -> S e. Grp )
8	6, 7	syl 17	. . . . 5 |- ( S e. Rng -> S e. Grp )
9	8	adantr 481	. . . 4 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> S e. Grp )
10		ringgrp 18552	. . . . . 6 |- ( T e. Ring -> T e. Grp )
11	1, 10	syl 17	. . . . 5 |- ( T e. ( Ring \ NzRing ) -> T e. Grp )
12	11	adantl 482	. . . 4 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> T e. Grp )
13		zrrhm.b	. . . . . 6 |- B = ( Base ` T )
14		eqid 2622	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
15	13, 14	0ringbas 41871	. . . . 5 |- ( T e. ( Ring \ NzRing ) -> B = { ( 0g ` T ) } )
16	15	adantl 482	. . . 4 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> B = { ( 0g ` T ) } )
17		zrrhm.0	. . . . 5 |- .0. = ( 0g ` S )
18		zrrhm.h	. . . . 5 |- H = ( x e. B |-> .0. )
19	13, 17, 18, 14	c0snghm 41916	. . . 4 |- ( ( S e. Grp /\ T e. Grp /\ B = { ( 0g ` T ) } ) -> H e. ( T GrpHom S ) )
20	9, 12, 16, 19	syl3anc 1326	. . 3 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> H e. ( T GrpHom S ) )
21	18	a1i 11	. . . . . . . 8 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> H = ( x e. B |-> .0. ) )
22		eqidd 2623	. . . . . . . 8 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ x = ( 0g ` T ) ) -> .0. = .0. )
23	13, 14	ring0cl 18569	. . . . . . . . . 10 |- ( T e. Ring -> ( 0g ` T ) e. B )
24	1, 23	syl 17	. . . . . . . . 9 |- ( T e. ( Ring \ NzRing ) -> ( 0g ` T ) e. B )
25	24	ad2antlr 763	. . . . . . . 8 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( 0g ` T ) e. B )
26		fvex 6201	. . . . . . . . . 10 |- ( 0g ` S ) e. _V
27	17, 26	eqeltri 2697	. . . . . . . . 9 |- .0. e. _V
28	27	a1i 11	. . . . . . . 8 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> .0. e. _V )
29	21, 22, 25, 28	fvmptd 6288	. . . . . . 7 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( H ` ( 0g ` T ) ) = .0. )
30		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` S ) = ( Base ` S )
31	30, 17	grpidcl 17450	. . . . . . . . . . . . 13 |- ( S e. Grp -> .0. e. ( Base ` S ) )
32	8, 31	syl 17	. . . . . . . . . . . 12 |- ( S e. Rng -> .0. e. ( Base ` S ) )
33		eqid 2622	. . . . . . . . . . . . 13 |- ( .r ` S ) = ( .r ` S )
34	30, 33, 17	rnglz 41884	. . . . . . . . . . . 12 |- ( ( S e. Rng /\ .0. e. ( Base ` S ) ) -> ( .0. ( .r ` S ) .0. ) = .0. )
35	32, 34	mpdan 702	. . . . . . . . . . 11 |- ( S e. Rng -> ( .0. ( .r ` S ) .0. ) = .0. )
36	35	adantr 481	. . . . . . . . . 10 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> ( .0. ( .r ` S ) .0. ) = .0. )
37	36	adantr 481	. . . . . . . . 9 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( .0. ( .r ` S ) .0. ) = .0. )
38	37	adantr 481	. . . . . . . 8 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ ( H ` ( 0g ` T ) ) = .0. ) -> ( .0. ( .r ` S ) .0. ) = .0. )
39		simpr 477	. . . . . . . . 9 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ ( H ` ( 0g ` T ) ) = .0. ) -> ( H ` ( 0g ` T ) ) = .0. )
40	39, 39	oveq12d 6668	. . . . . . . 8 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ ( H ` ( 0g ` T ) ) = .0. ) -> ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` ( 0g ` T ) ) ) = ( .0. ( .r ` S ) .0. ) )
41		eqid 2622	. . . . . . . . . . . . . . 15 |- ( .r ` T ) = ( .r ` T )
42	13, 41, 14	ringlz 18587	. . . . . . . . . . . . . 14 |- ( ( T e. Ring /\ ( 0g ` T ) e. B ) -> ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) = ( 0g ` T ) )
43	23, 42	mpdan 702	. . . . . . . . . . . . 13 |- ( T e. Ring -> ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) = ( 0g ` T ) )
44	1, 43	syl 17	. . . . . . . . . . . 12 |- ( T e. ( Ring \ NzRing ) -> ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) = ( 0g ` T ) )
45	44	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) = ( 0g ` T ) )
46	45	adantr 481	. . . . . . . . . 10 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ ( H ` ( 0g ` T ) ) = .0. ) -> ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) = ( 0g ` T ) )
47	46	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ ( H ` ( 0g ` T ) ) = .0. ) -> ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) = ( H ` ( 0g ` T ) ) )
48	47, 39	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ ( H ` ( 0g ` T ) ) = .0. ) -> ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) = .0. )
49	38, 40, 48	3eqtr4rd 2667	. . . . . . 7 |- ( ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) /\ ( H ` ( 0g ` T ) ) = .0. ) -> ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` ( 0g ` T ) ) ) )
50	29, 49	mpdan 702	. . . . . 6 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` ( 0g ` T ) ) ) )
51	23, 23	jca 554	. . . . . . . . 9 |- ( T e. Ring -> ( ( 0g ` T ) e. B /\ ( 0g ` T ) e. B ) )
52	1, 51	syl 17	. . . . . . . 8 |- ( T e. ( Ring \ NzRing ) -> ( ( 0g ` T ) e. B /\ ( 0g ` T ) e. B ) )
53	52	ad2antlr 763	. . . . . . 7 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( ( 0g ` T ) e. B /\ ( 0g ` T ) e. B ) )
54		oveq1 6657	. . . . . . . . . 10 |- ( a = ( 0g ` T ) -> ( a ( .r ` T ) c ) = ( ( 0g ` T ) ( .r ` T ) c ) )
55	54	fveq2d 6195	. . . . . . . . 9 |- ( a = ( 0g ` T ) -> ( H ` ( a ( .r ` T ) c ) ) = ( H ` ( ( 0g ` T ) ( .r ` T ) c ) ) )
56		fveq2 6191	. . . . . . . . . 10 |- ( a = ( 0g ` T ) -> ( H ` a ) = ( H ` ( 0g ` T ) ) )
57	56	oveq1d 6665	. . . . . . . . 9 |- ( a = ( 0g ` T ) -> ( ( H ` a ) ( .r ` S ) ( H ` c ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` c ) ) )
58	55, 57	eqeq12d 2637	. . . . . . . 8 |- ( a = ( 0g ` T ) -> ( ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) <-> ( H ` ( ( 0g ` T ) ( .r ` T ) c ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` c ) ) ) )
59		oveq2 6658	. . . . . . . . . 10 |- ( c = ( 0g ` T ) -> ( ( 0g ` T ) ( .r ` T ) c ) = ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) )
60	59	fveq2d 6195	. . . . . . . . 9 |- ( c = ( 0g ` T ) -> ( H ` ( ( 0g ` T ) ( .r ` T ) c ) ) = ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) )
61		fveq2 6191	. . . . . . . . . 10 |- ( c = ( 0g ` T ) -> ( H ` c ) = ( H ` ( 0g ` T ) ) )
62	61	oveq2d 6666	. . . . . . . . 9 |- ( c = ( 0g ` T ) -> ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` c ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` ( 0g ` T ) ) ) )
63	60, 62	eqeq12d 2637	. . . . . . . 8 |- ( c = ( 0g ` T ) -> ( ( H ` ( ( 0g ` T ) ( .r ` T ) c ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` c ) ) <-> ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` ( 0g ` T ) ) ) ) )
64	58, 63	2ralsng 4220	. . . . . . 7 |- ( ( ( 0g ` T ) e. B /\ ( 0g ` T ) e. B ) -> ( A. a e. { ( 0g ` T ) } A. c e. { ( 0g ` T ) } ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) <-> ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` ( 0g ` T ) ) ) ) )
65	53, 64	syl 17	. . . . . 6 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( A. a e. { ( 0g ` T ) } A. c e. { ( 0g ` T ) } ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) <-> ( H ` ( ( 0g ` T ) ( .r ` T ) ( 0g ` T ) ) ) = ( ( H ` ( 0g ` T ) ) ( .r ` S ) ( H ` ( 0g ` T ) ) ) ) )
66	50, 65	mpbird 247	. . . . 5 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> A. a e. { ( 0g ` T ) } A. c e. { ( 0g ` T ) } ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) )
67		raleq 3138	. . . . . . 7 |- ( B = { ( 0g ` T ) } -> ( A. c e. B ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) <-> A. c e. { ( 0g ` T ) } ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) ) )
68	67	raleqbi1dv 3146	. . . . . 6 |- ( B = { ( 0g ` T ) } -> ( A. a e. B A. c e. B ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) <-> A. a e. { ( 0g ` T ) } A. c e. { ( 0g ` T ) } ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) ) )
69	68	adantl 482	. . . . 5 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> ( A. a e. B A. c e. B ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) <-> A. a e. { ( 0g ` T ) } A. c e. { ( 0g ` T ) } ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) ) )
70	66, 69	mpbird 247	. . . 4 |- ( ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) /\ B = { ( 0g ` T ) } ) -> A. a e. B A. c e. B ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) )
71	16, 70	mpdan 702	. . 3 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> A. a e. B A. c e. B ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) )
72	20, 71	jca 554	. 2 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> ( H e. ( T GrpHom S ) /\ A. a e. B A. c e. B ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) ) )
73	13, 41, 33	isrnghm 41892	. 2 |- ( H e. ( T RngHomo S ) <-> ( ( T e. Rng /\ S e. Rng ) /\ ( H e. ( T GrpHom S ) /\ A. a e. B A. c e. B ( H ` ( a ( .r ` T ) c ) ) = ( ( H ` a ) ( .r ` S ) ( H ` c ) ) ) ) )
74	5, 72, 73	sylanbrc 698	1 |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> H e. ( T RngHomo S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   {csn 4177   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   GrpHom cghm 17657   Abelcabl 18194   Ringcrg 18547  NzRingcnzr 19257  Rngcrng 41874   RngHomo crngh 41885

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-ghm 17658  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-nzr 19258  df-mgmhm 41779  df-rng0 41875  df-rnghomo 41887

This theorem is referenced by:  zrinitorngc  42000