Theorem 0ringnnzr 19269

Description: A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)

Assertion

Ref	Expression
0ringnnzr	|- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 <-> -. R e. NzRing ) )

Proof of Theorem 0ringnnzr

Step	Hyp	Ref	Expression
1		1re 10039	. . . . . . . 8 |- 1 e. RR
2	1	ltnri 10146	. . . . . . 7 |- -. 1 < 1
3		breq2 4657	. . . . . . 7 |- ( ( # ` ( Base ` R ) ) = 1 -> ( 1 < ( # ` ( Base ` R ) ) <-> 1 < 1 ) )
4	2, 3	mtbiri 317	. . . . . 6 |- ( ( # ` ( Base ` R ) ) = 1 -> -. 1 < ( # ` ( Base ` R ) ) )
5	4	adantl 482	. . . . 5 |- ( ( R e. Ring /\ ( # ` ( Base ` R ) ) = 1 ) -> -. 1 < ( # ` ( Base ` R ) ) )
6	5	intnand 962	. . . 4 |- ( ( R e. Ring /\ ( # ` ( Base ` R ) ) = 1 ) -> -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) )
7	6	ex 450	. . 3 |- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 -> -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) ) )
8		ianor 509	. . . . 5 |- ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) <-> ( -. R e. Ring \/ -. 1 < ( # ` ( Base ` R ) ) ) )
9		pm2.21 120	. . . . . 6 |- ( -. R e. Ring -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) )
10		fvex 6201	. . . . . . . . . 10 |- ( Base ` R ) e. _V
11		hashxrcl 13148	. . . . . . . . . 10 |- ( ( Base ` R ) e. _V -> ( # ` ( Base ` R ) ) e. RR* )
12	10, 11	ax-mp 5	. . . . . . . . 9 |- ( # ` ( Base ` R ) ) e. RR*
13	1	rexri 10097	. . . . . . . . 9 |- 1 e. RR*
14		xrlenlt 10103	. . . . . . . . 9 |- ( ( ( # ` ( Base ` R ) ) e. RR* /\ 1 e. RR* ) -> ( ( # ` ( Base ` R ) ) <_ 1 <-> -. 1 < ( # ` ( Base ` R ) ) ) )
15	12, 13, 14	mp2an 708	. . . . . . . 8 |- ( ( # ` ( Base ` R ) ) <_ 1 <-> -. 1 < ( # ` ( Base ` R ) ) )
16	15	bicomi 214	. . . . . . 7 |- ( -. 1 < ( # ` ( Base ` R ) ) <-> ( # ` ( Base ` R ) ) <_ 1 )
17		simpr 477	. . . . . . . . . 10 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) <_ 1 )
18	10	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( Base ` R ) e. _V )
19		1nn0 11308	. . . . . . . . . . . . . 14 |- 1 e. NN0
20	19	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> 1 e. NN0 )
21		hashbnd 13123	. . . . . . . . . . . . 13 |- ( ( ( Base ` R ) e. _V /\ 1 e. NN0 /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( Base ` R ) e. Fin )
22	18, 20, 17, 21	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( Base ` R ) e. Fin )
23		hashcl 13147	. . . . . . . . . . . . 13 |- ( ( Base ` R ) e. Fin -> ( # ` ( Base ` R ) ) e. NN0 )
24		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) e. NN0 ) -> ( # ` ( Base ` R ) ) e. NN0 )
25	10	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` ( Base ` R ) ) e. NN0 -> ( Base ` R ) e. _V )
26		hasheq0 13154	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( Base ` R ) e. _V -> ( ( # ` ( Base ` R ) ) = 0 <-> ( Base ` R ) = (/) ) )
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` ( Base ` R ) ) e. NN0 -> ( ( # ` ( Base ` R ) ) = 0 <-> ( Base ` R ) = (/) ) )
28	27	biimpd 219	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` ( Base ` R ) ) e. NN0 -> ( ( # ` ( Base ` R ) ) = 0 -> ( Base ` R ) = (/) ) )
29	28	necon3d 2815	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` ( Base ` R ) ) e. NN0 -> ( ( Base ` R ) =/= (/) -> ( # ` ( Base ` R ) ) =/= 0 ) )
30	29	impcom 446	. . . . . . . . . . . . . . . . 17 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) e. NN0 ) -> ( # ` ( Base ` R ) ) =/= 0 )
31		elnnne0 11306	. . . . . . . . . . . . . . . . 17 |- ( ( # ` ( Base ` R ) ) e. NN <-> ( ( # ` ( Base ` R ) ) e. NN0 /\ ( # ` ( Base ` R ) ) =/= 0 ) )
32	24, 30, 31	sylanbrc 698	. . . . . . . . . . . . . . . 16 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) e. NN0 ) -> ( # ` ( Base ` R ) ) e. NN )
33	32	ex 450	. . . . . . . . . . . . . . 15 |- ( ( Base ` R ) =/= (/) -> ( ( # ` ( Base ` R ) ) e. NN0 -> ( # ` ( Base ` R ) ) e. NN ) )
34	33	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( ( # ` ( Base ` R ) ) e. NN0 -> ( # ` ( Base ` R ) ) e. NN ) )
35	34	com12 32	. . . . . . . . . . . . 13 |- ( ( # ` ( Base ` R ) ) e. NN0 -> ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) e. NN ) )
36	23, 35	syl 17	. . . . . . . . . . . 12 |- ( ( Base ` R ) e. Fin -> ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) e. NN ) )
37	22, 36	mpcom 38	. . . . . . . . . . 11 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) e. NN )
38		nnle1eq1 11048	. . . . . . . . . . 11 |- ( ( # ` ( Base ` R ) ) e. NN -> ( ( # ` ( Base ` R ) ) <_ 1 <-> ( # ` ( Base ` R ) ) = 1 ) )
39	37, 38	syl 17	. . . . . . . . . 10 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( ( # ` ( Base ` R ) ) <_ 1 <-> ( # ` ( Base ` R ) ) = 1 ) )
40	17, 39	mpbid 222	. . . . . . . . 9 |- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) = 1 )
41	40	ex 450	. . . . . . . 8 |- ( ( Base ` R ) =/= (/) -> ( ( # ` ( Base ` R ) ) <_ 1 -> ( # ` ( Base ` R ) ) = 1 ) )
42		ringgrp 18552	. . . . . . . . 9 |- ( R e. Ring -> R e. Grp )
43		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
44	43	grpbn0 17451	. . . . . . . . 9 |- ( R e. Grp -> ( Base ` R ) =/= (/) )
45	42, 44	syl 17	. . . . . . . 8 |- ( R e. Ring -> ( Base ` R ) =/= (/) )
46	41, 45	syl11 33	. . . . . . 7 |- ( ( # ` ( Base ` R ) ) <_ 1 -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) )
47	16, 46	sylbi 207	. . . . . 6 |- ( -. 1 < ( # ` ( Base ` R ) ) -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) )
48	9, 47	jaoi 394	. . . . 5 |- ( ( -. R e. Ring \/ -. 1 < ( # ` ( Base ` R ) ) ) -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) )
49	8, 48	sylbi 207	. . . 4 |- ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) )
50	49	com12 32	. . 3 |- ( R e. Ring -> ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) -> ( # ` ( Base ` R ) ) = 1 ) )
51	7, 50	impbid 202	. 2 |- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 <-> -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) ) )
52	43	isnzr2hash 19264	. . . 4 |- ( R e. NzRing <-> ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) )
53	52	bicomi 214	. . 3 |- ( ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) <-> R e. NzRing )
54	53	notbii 310	. 2 |- ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) <-> -. R e. NzRing )
55	51, 54	syl6bb 276	1 |- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 <-> -. R e. NzRing ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   ` cfv 5888   Fincfn 7955   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   #chash 13117   Basecbs 15857   Grpcgrp 17422   Ringcrg 18547  NzRingcnzr 19257

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-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-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-nzr 19258

This theorem is referenced by:  rng1nnzr  19274  lmod0rng  41868  0ringdif  41870  0ring1eq0  41872  lindszr  42258