Theorem nzerooringczr 40178

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 3744	. . 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 40130	. . . . . . . 8 |- ( U e. V -> C e. Cat )
6	3, 5	syl 17	. . . . . . 7 |- ( ph -> C e. Cat )
7		iszeroi 15916	. . . . . . 7 |- ( ( C e. Cat /\ h e. (ZeroO ` C ) ) -> ( h e. ( Base ` C ) /\ ( h e. (InitO ` C ) /\ h e. (TermO ` C ) ) ) )
8	6, 7	sylan 474	. . . . . 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 40176	. . . . . . . 8 |- ( ph -> Z e. (TermO ` C ) )
12		nzerooringczr.i	. . . . . . . . . 10 |- ( ph -> ℤring e. U )
13	3, 12, 4	irinitoringc 40175	. . . . . . . . 9 |- ( ph -> ℤring e. (InitO ` C ) )
14	6	ad2antrr 733	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> C e. Cat )
15		simplr 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> h e. (InitO ` C ) )
16		simpr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> ℤring e. (InitO ` C ) )
17	14, 15, 16	initoeu1w 15919	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ h e. (InitO ` C ) ) /\ ℤring e. (InitO ` C ) ) -> h ( ~=c𝑐 ` C )ℤring )
18	6	ad2antrr 733	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> C e. Cat )
19		simpr 463	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> Z e. (TermO ` C ) )
20		simplr 763	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> h e. (TermO ` C ) )
21	18, 19, 20	termoeu1w 15926	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ h e. (TermO ` C ) ) /\ Z e. (TermO ` C ) ) -> Z ( ~=c𝑐 ` C ) h )
22		cictr 15722	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( C e. Cat /\ Z ( ~=c𝑐 ` C ) h /\ h ( ~=c𝑐 ` C )ℤring ) -> Z ( ~=c𝑐 ` C )ℤring )
23	6, 22	syl3an1 1302	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ph /\ Z ( ~=c𝑐 ` C ) h /\ h ( ~=c𝑐 ` C )ℤring ) -> Z ( ~=c𝑐 ` C )ℤring )
24		eqid 2453	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( Iso ` C ) = ( Iso ` C )
25		eqid 2453	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( Base ` C ) = ( Base ` C )
26	9	eldifad 3418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ph -> Z e. Ring )
27	10, 26	elind 3620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> Z e. ( U i^i Ring ) )
28	4, 25, 3	ringcbas 40117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> ( Base ` C ) = ( U i^i Ring ) )
29	27, 28	eleqtrrd 2534	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ph -> Z e. ( Base ` C ) )
30		zringring 19054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ℤring e. Ring
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ph -> ℤring e. Ring )
32	12, 31	elind 3620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> ℤring e. ( U i^i Ring ) )
33	32, 28	eleqtrrd 2534	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ph -> ℤring e. ( Base ` C ) )
34	24, 25, 6, 29, 33	cic 15716	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> ( Z ( ~=c𝑐 ` C )ℤring <-> E. f f e. ( Z ( Iso ` C )ℤring ) ) )
35		n0 3743	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Z ( Iso ` C )ℤring ) =/= (/) <-> E. f f e. ( Z ( Iso ` C )ℤring ) )
36		eqid 2453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( Hom ` C ) = ( Hom ` C )
37	25, 36, 24, 6, 29, 33	isohom 15693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ph -> ( Z ( Iso ` C )ℤring ) C_ ( Z ( Hom ` C )ℤring ) )
38		ssn0 3769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 40119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ph -> ( Z ( Hom ` C )ℤring ) = ( Z RingHom ℤring ) )
40	39	neeq1d 2685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ph -> ( ( Z ( Hom ` C )ℤring ) =/= (/) <-> ( Z RingHom ℤring ) =/= (/) ) )
41		zringnzr 19063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ℤring e. NzRing
42		nrhmzr 39977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( Z e. ( Ring \ NzRing ) /\ ℤring e. NzRing ) -> ( Z RingHom ℤring ) = (/) )
43	9, 41, 42	sylancl 669	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ph -> ( Z RingHom ℤring ) = (/) )
44		eqneqall 2636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( Z ( Iso ` C )ℤring ) =/= (/) -> ( ( Z ( Iso ` C )ℤring ) C_ ( Z ( Hom ` C )ℤring ) -> ( ph -> (ZeroO ` C ) = (/) ) ) )
49	48	com13 83	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> ( E. f f e. ( Z ( Iso ` C )ℤring ) -> (ZeroO ` C ) = (/) ) )
52	34, 51	sylbid 219	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ph -> ( Z ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) )
53	52	3ad2ant1 1030	. . . . . . . . . . . . . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> ( Z ( ~=c𝑐 ` C ) h -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) )
56	55	a1dd 47	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( Z ( ~=c𝑐 ` C ) h -> ( h e. ( Base ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) ) )
57	56	ad2antrr 733	. . . . . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( h e. (TermO ` C ) -> ( Z e. (TermO ` C ) -> ( h e. ( Base ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) ) ) )
60	59	com34 86	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( h e. (TermO ` C ) -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> ( h ( ~=c𝑐 ` C )ℤring -> (ZeroO ` C ) = (/) ) ) ) ) )
61	60	com25 94	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( h ( ~=c𝑐 ` C )ℤring -> ( h e. ( Base ` C ) -> ( Z e. (TermO ` C ) -> ( h e. (TermO ` C ) -> (ZeroO ` C ) = (/) ) ) ) ) )
62	61	ad2antrr 733	. . . . . . . . . . . . . . . 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 436	. . . . . . . . . . . . . 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 94	. . . . . . . . . . . . 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 608	. . . . . . . . . . . 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 81	. . . . . . . . . . 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 433	. . . . . . . . . 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 90	. . . . . . . . 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 467	. . . . . 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 437	. . . 4 |- ( h e. (ZeroO ` C ) -> ( ph -> (ZeroO ` C ) = (/) ) )
75	74	exlimiv 1778	. . 3 |- ( E. h h e. (ZeroO ` C ) -> ( ph -> (ZeroO ` C ) = (/) ) )
76	2, 75	sylbi 199	. 2 |- ( -. (ZeroO ` C ) = (/) -> ( ph -> (ZeroO ` C ) = (/) ) )
77	1, 76	pm2.61i 168	1 |- ( ph -> (ZeroO ` C ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 986   = wceq 1446   E.wex 1665   e. wcel 1889   =/= wne 2624   \ cdif 3403   i^i cin 3405   C_ wss 3406   (/)c0 3733   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   Basecbs 15133   Hom chom 15213   Catccat 15582   Iso ciso 15663   ~=c_𝑐 ccic 15712  InitOcinito 15895  TermOctermo 15896  ZeroOczeroo 15897   Ringcrg 17792   RingHom crh 17952  NzRingcnzr 18493  ℤ_ringzring 19051  RingCatcringc 40109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-addf 9623  ax-mulf 9624

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-seq 12221  df-hash 12523  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-0g 15352  df-cat 15586  df-cid 15587  df-homf 15588  df-sect 15664  df-inv 15665  df-iso 15666  df-cic 15713  df-ssc 15727  df-resc 15728  df-subc 15729  df-inito 15898  df-termo 15899  df-zeroo 15900  df-estrc 16020  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mhm 16594  df-grp 16685  df-minusg 16686  df-mulg 16688  df-subg 16826  df-ghm 16893  df-cmn 17444  df-mgp 17736  df-ur 17748  df-ring 17794  df-cring 17795  df-rnghom 17955  df-subrg 18018  df-nzr 18494  df-cnfld 18983  df-zring 19052  df-ringc 40111

This theorem is referenced by: (None)