Theorem nzerooringczr 42072

Description: There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)

Hypotheses

Ref	Expression
nzerooringczr.u	|- ( ph -> U e. V )
nzerooringczr.c	|- C = (RingCat ` U )
nzerooringczr.z	|- ( ph -> Z e. ( Ring \ NzRing ) )
nzerooringczr.e	|- ( ph -> Z e. U )
nzerooringczr.i	|- ( ph -> ℤring e. U )

Assertion

Ref	Expression
nzerooringczr	|- ( ph -> (ZeroO ` C ) = (/) )

Proof of Theorem nzerooringczr

Dummy variables f h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 |- ( (ZeroO ` C ) = (/) -> ( ph -> (ZeroO ` C ) = (/) ) )
2		neq0 3930	. . 3 |- ( -. (ZeroO ` C ) = (/) <-> E. h h e. (ZeroO ` C ) )
3		nzerooringczr.u	. . . . . . . 8 |- ( ph -> U e. V )
4		nzerooringczr.c	. . . . . . . . 9 |- C = (RingCat ` U )
5	4	ringccat 42024	. . . . . . . 8 |- ( U e. V -> C e. Cat )
6	3, 5	syl 17	. . . . . . 7 |- ( ph -> C e. Cat )
7		iszeroi 16659	. . . . . . 7 |- ( ( C e. Cat /\ h e. (ZeroO ` C ) ) -> ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) )
8	6, 7	sylan 488	. . . . . 6 |- ( ( ph /\ h e. (ZeroO ` C ) ) -> ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) )
9		nzerooringczr.z	. . . . . . . . 9 |- ( ph -> Z e. ( Ring \ NzRing ) )
10		nzerooringczr.e	. . . . . . . . 9 |- ( ph -> Z e. U )
11	3, 4, 9, 10	zrtermoringc 42070	. . . . . . . 8 |- ( ph -> Z e. (TermO ` C ) )
12		nzerooringczr.i	. . . . . . . . . 10 |- ( ph -> ℤring e. U )
13	3, 12, 4	irinitoringc 42069	. . . . . . . . 9 |- ( ph -> ℤring e. (InitO ` C ) )
14	6	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> C e. Cat )
15		simplr 792	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> h e. (InitO ` C ) )
16		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> ℤring e. (InitO ` C ) )
17	14, 15, 16	initoeu1w 16662	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> h ( ~=c𝑐 ` C )ℤring )
18	6	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> C e. Cat )
19		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> Z e. (TermO ` C ) )
20		simplr 792	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> h e. (TermO ` C ) )
21	18, 19, 20	termoeu1w 16669	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> Z ( ~=c𝑐 ` C ) h )
22		cictr 16465	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( C e. Cat /\ Z ( ~=c𝑐 ` C ) h /\ h ( ~=c𝑐 ` C )ℤring ) -> Z ( ~=c𝑐 ` C )ℤring )
23	6, 22	syl3an1 1359	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ph /\ Z ( ~=c𝑐 ` C ) h /\ h ( ~=c𝑐 ` C )ℤring ) -> Z ( ~=c𝑐 ` C )ℤring )
24		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( Iso ` C ) = ( Iso ` C )
25		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( Base ` C ) = ( Base ` C )
26	9	eldifad 3586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ph -> Z e. Ring )
27	10, 26	elind 3798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> Z e. ( U i^i Ring ) )
28	4, 25, 3	ringcbas 42011	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> ( Base ` C ) = ( U i^i Ring ) )
29	27, 28	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ph -> Z e. ( Base ` C ) )
30		zringring 19821	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ℤring e. Ring
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ph -> ℤring e. Ring )
32	12, 31	elind 3798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> ℤring e. ( U i^i Ring ) )
33	32, 28	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ph -> ℤring e. ( Base ` C ) )
34	24, 25, 6, 29, 33	cic 16459	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> ( Z ( ~=c𝑐 ` C )ℤring <-> E. f f e. ( Z ( Iso ` C )ℤring ) ) )
35		n0 3931	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Z ( Iso ` C )ℤring ) =/= (/) <-> E. f f e. ( Z ( Iso ` C )ℤring ) )
36		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( Hom ` C ) = ( Hom ` C )
37	25, 36, 24, 6, 29, 33	isohom 16436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> ( Z ( Iso ` C )ℤring ) C_ ( Z ( Hom ` C )ℤring ) )
38		ssn0 3976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( ( Z ( Iso ` C )ℤring ) C_ ( Z ( Hom ` C )ℤring ) /\ ( Z ( Iso ` C )ℤring ) =/= (/) ) -> ( Z ( Hom ` C )ℤring ) =/= (/) )
39	4, 25, 3, 36, 29, 33	ringchom 42013	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ph -> ( Z ( Hom ` C )ℤring ) = ( Z RingHom ℤring ) )
40	39	neeq1d 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ph -> ( ( Z ( Hom ` C )ℤring ) =/= (/) <-> ( Z RingHom ℤring ) =/= (/) ) )
41		zringnzr 19830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ℤring e. NzRing
42		nrhmzr 41873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( Z e. ( Ring \ NzRing ) /\ ℤring e. NzRing ) -> ( Z RingHom ℤring ) = (/) )
43	9, 41, 42	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ph -> ( Z RingHom ℤring ) = (/) )
44		eqneqall 2805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( Z RingHom ℤring ) = (/) -> ( ( Z RingHom ℤring ) =/= (/) -> (ZeroO ` C ) = (/) ) )
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ph -> ( ( Z RingHom ℤring ) =/= (/) -> (ZeroO ` C ) = (/) ) )
46	40, 45	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ph -> ( ( Z ( Hom ` C )ℤring ) =/= (/) -> (ZeroO ` C ) = (/) ) )
47	38, 46	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( ( Z ( Iso ` C )ℤring ) C_ ( Z ( Hom ` C )ℤring ) /\ ( Z ( Iso ` C )ℤring ) =/= (/) ) -> ( ph -> (ZeroO ` C ) = (/) ) )
48	47	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( Z ( Iso ` C )ℤring ) =/= (/) -> ( ( Z ( Iso ` C )ℤring ) C_ ( Z ( Hom ` C )ℤring ) -> ( ph -> (ZeroO ` C ) = (/) ) ) )
49	48	com13 88	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> ( ( Z ( Iso ` C )ℤring ) C_ ( Z ( Hom ` C )ℤring ) -> ( ( Z ( Iso ` C )ℤring ) =/= (/) -> (ZeroO ` C ) = (/) ) ) )
50	37, 49	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ph -> ( ( Z ( Iso ` C )ℤring ) =/= (/) -> (ZeroO ` C ) = (/) ) )
51	35, 50	syl5bir 233	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> ( E. f f e. ( Z ( Iso ` C )ℤring ) -> (ZeroO ` C ) = (/) ) )
52	34, 51	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ph -> ( Z ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) )
53	52	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ph /\ Z ( ~=c𝑐 ` C ) h /\ h ( ~=c𝑐 ` C )ℤring ) -> ( Z ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) )
54	23, 53	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ph /\ Z ( ~=c𝑐 ` C ) h /\ h ( ~=c𝑐 ` C )ℤring ) -> (ZeroO ` C ) = (/) )
55	54	3exp 1264	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> ( Z ( ~=c𝑐 ` C ) h -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) )
56	55	a1dd 50	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( Z ( ~=c𝑐 ` C ) h -> ( h e. ( Base ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) ) )
57	56	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> ( Z ( ~=c𝑐 ` C ) h -> ( h e. ( Base ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) ) )
58	21, 57	mpd 15	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> ( h e. ( Base ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) )
59	58	exp31 630	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( h e. (TermO ` C ) -> ( Z e. (TermO ` C ) -> ( h e. ( Base ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) ) ) )
60	59	com34 91	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( h e. (TermO ` C ) -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) ) ) )
61	60	com25 99	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( h ( ~=c𝑐 ` C )ℤring -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> ( h e. (TermO ` C ) -> (ZeroO ` C ) = (/) ) ) ) ) )
62	61	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> ( h ( ~=c𝑐 ` C )ℤring -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> ( h e. (TermO ` C ) -> (ZeroO ` C ) = (/) ) ) ) ) )
63	17, 62	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> ( h e. (TermO ` C ) -> (ZeroO ` C ) = (/) ) ) ) )
64	63	ex 450	. . . . . . . . . . . . . 14 |- ( ( ph /\ h e. (InitO ` C ) ) -> (ℤring e. (InitO ` C ) -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> ( h e. (TermO ` C ) -> (ZeroO ` C ) = (/) ) ) ) ) )
65	64	com25 99	. . . . . . . . . . . . 13 |- ( ( ph /\ h e. (InitO ` C ) ) -> ( h e. (TermO ` C ) -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> (ℤring e. (InitO ` C ) -> (ZeroO ` C ) = (/) ) ) ) ) )
66	65	expimpd 629	. . . . . . . . . . . 12 |- ( ph -> ( ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> (ℤring e. (InitO ` C ) -> (ZeroO ` C ) = (/) ) ) ) ) )
67	66	com23 86	. . . . . . . . . . 11 |- ( ph -> ( h e. ( Base ` C ) -> ( ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) -> ( Z e. (TermO ` C ) -> (ℤring e. (InitO ` C ) -> (ZeroO ` C ) = (/) ) ) ) ) )
68	67	impd 447	. . . . . . . . . 10 |- ( ph -> ( ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) -> ( Z e. (TermO ` C ) -> (ℤring e. (InitO ` C ) -> (ZeroO ` C ) = (/) ) ) ) )
69	68	com24 95	. . . . . . . . 9 |- ( ph -> (ℤring e. (InitO ` C ) -> ( Z e. (TermO ` C ) -> ( ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) -> (ZeroO ` C ) = (/) ) ) ) )
70	13, 69	mpd 15	. . . . . . . 8 |- ( ph -> ( Z e. (TermO ` C ) -> ( ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) -> (ZeroO ` C ) = (/) ) ) )
71	11, 70	mpd 15	. . . . . . 7 |- ( ph -> ( ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) -> (ZeroO ` C ) = (/) ) )
72	71	adantr 481	. . . . . 6 |- ( ( ph /\ h e. (ZeroO ` C ) ) -> ( ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) -> (ZeroO ` C ) = (/) ) )
73	8, 72	mpd 15	. . . . 5 |- ( ( ph /\ h e. (ZeroO ` C ) ) -> (ZeroO ` C ) = (/) )
74	73	expcom 451	. . . 4 |- ( h e. (ZeroO ` C ) -> ( ph -> (ZeroO ` C ) = (/) ) )
75	74	exlimiv 1858	. . 3 |- ( E. h h e. (ZeroO ` C ) -> ( ph -> (ZeroO ` C ) = (/) ) )
76	2, 75	sylbi 207	. 2 |- ( -. (ZeroO ` C ) = (/) -> ( ph -> (ZeroO ` C ) = (/) ) )
77	1, 76	pm2.61i 176	1 |- ( ph -> (ZeroO ` C ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Catccat 16325   Iso ciso 16406   ~=c_𝑐 ccic 16455  InitOcinito 16638  TermOctermo 16639  ZeroOczeroo 16640   Ringcrg 18547   RingHom crh 18712  NzRingcnzr 19257  ℤ_ringzring 19818  RingCatcringc 42003

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-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-cat 16329  df-cid 16330  df-homf 16331  df-sect 16407  df-inv 16408  df-iso 16409  df-cic 16456  df-ssc 16470  df-resc 16471  df-subc 16472  df-inito 16641  df-termo 16642  df-zeroo 16643  df-estrc 16763  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-nzr 19258  df-cnfld 19747  df-zring 19819  df-ringc 42005

This theorem is referenced by: (None)