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Theorem 0rngnnzr 17711
Description: A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)
Assertion
Ref Expression
0rngnnzr  |-  ( R  e.  Ring  ->  ( (
# `  ( Base `  R ) )  =  1  <->  -.  R  e. NzRing ) )

Proof of Theorem 0rngnnzr
StepHypRef Expression
1 1re 9594 . . . . . . . 8  |-  1  e.  RR
21ltnri 9692 . . . . . . 7  |-  -.  1  <  1
3 breq2 4451 . . . . . . 7  |-  ( (
# `  ( Base `  R ) )  =  1  ->  ( 1  <  ( # `  ( Base `  R ) )  <->  1  <  1 ) )
42, 3mtbiri 303 . . . . . 6  |-  ( (
# `  ( Base `  R ) )  =  1  ->  -.  1  <  ( # `  ( Base `  R ) ) )
54adantl 466 . . . . 5  |-  ( ( R  e.  Ring  /\  ( # `
 ( Base `  R
) )  =  1 )  ->  -.  1  <  ( # `  ( Base `  R ) ) )
65intnand 914 . . . 4  |-  ( ( R  e.  Ring  /\  ( # `
 ( Base `  R
) )  =  1 )  ->  -.  ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R ) ) ) )
76ex 434 . . 3  |-  ( R  e.  Ring  ->  ( (
# `  ( Base `  R ) )  =  1  ->  -.  ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R ) ) ) ) )
8 ianor 488 . . . . 5  |-  ( -.  ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R
) ) )  <->  ( -.  R  e.  Ring  \/  -.  1  <  ( # `  ( Base `  R ) ) ) )
9 pm2.21 108 . . . . . 6  |-  ( -.  R  e.  Ring  ->  ( R  e.  Ring  ->  (
# `  ( Base `  R ) )  =  1 ) )
10 fvex 5875 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
11 hashxrcl 12396 . . . . . . . . . 10  |-  ( (
Base `  R )  e.  _V  ->  ( # `  ( Base `  R ) )  e.  RR* )
1210, 11ax-mp 5 . . . . . . . . 9  |-  ( # `  ( Base `  R
) )  e.  RR*
131rexri 9645 . . . . . . . . 9  |-  1  e.  RR*
14 xrlenlt 9651 . . . . . . . . 9  |-  ( ( ( # `  ( Base `  R ) )  e.  RR*  /\  1  e.  RR* )  ->  (
( # `  ( Base `  R ) )  <_ 
1  <->  -.  1  <  (
# `  ( Base `  R ) ) ) )
1512, 13, 14mp2an 672 . . . . . . . 8  |-  ( (
# `  ( Base `  R ) )  <_ 
1  <->  -.  1  <  (
# `  ( Base `  R ) ) )
1615bicomi 202 . . . . . . 7  |-  ( -.  1  <  ( # `  ( Base `  R
) )  <->  ( # `  ( Base `  R ) )  <_  1 )
17 rnggrp 17000 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
18 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
1918grpbn0 15886 . . . . . . . . . 10  |-  ( R  e.  Grp  ->  ( Base `  R )  =/=  (/) )
2017, 19syl 16 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( Base `  R )  =/=  (/) )
21 simpr 461 . . . . . . . . . . 11  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( # `  ( Base `  R ) )  <_  1 )
2210a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  _V )
23 1nn0 10810 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
2423a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  1  e.  NN0 )
25 hashbnd 12378 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  e.  _V  /\  1  e.  NN0  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  Fin )
2622, 24, 21, 25syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  Fin )
27 hashcl 12395 . . . . . . . . . . . . . 14  |-  ( (
Base `  R )  e.  Fin  ->  ( # `  ( Base `  R ) )  e.  NN0 )
28 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  e.  NN0 )  ->  ( # `  ( Base `  R ) )  e.  NN0 )
2910a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( Base `  R
)  e.  _V )
30 hasheq0 12400 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
Base `  R )  e.  _V  ->  ( ( # `
 ( Base `  R
) )  =  0  <-> 
( Base `  R )  =  (/) ) )
3129, 30syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( ( # `  ( Base `  R
) )  =  0  <-> 
( Base `  R )  =  (/) ) )
3231biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( ( # `  ( Base `  R
) )  =  0  ->  ( Base `  R
)  =  (/) ) )
3332necon3d 2691 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( ( Base `  R )  =/=  (/)  ->  ( # `
 ( Base `  R
) )  =/=  0
) )
3433impcom 430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  e.  NN0 )  ->  ( # `  ( Base `  R ) )  =/=  0 )
35 elnnne0 10808 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  ( Base `  R ) )  e.  NN  <->  ( ( # `  ( Base `  R
) )  e.  NN0  /\  ( # `  ( Base `  R ) )  =/=  0 ) )
3628, 34, 35sylanbrc 664 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  e.  NN0 )  ->  ( # `  ( Base `  R ) )  e.  NN )
3736ex 434 . . . . . . . . . . . . . . . 16  |-  ( (
Base `  R )  =/=  (/)  ->  ( ( # `
 ( Base `  R
) )  e.  NN0  ->  ( # `  ( Base `  R ) )  e.  NN ) )
3837adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( ( # `
 ( Base `  R
) )  e.  NN0  ->  ( # `  ( Base `  R ) )  e.  NN ) )
3938com12 31 . . . . . . . . . . . . . 14  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( ( (
Base `  R )  =/=  (/)  /\  ( # `  ( Base `  R
) )  <_  1
)  ->  ( # `  ( Base `  R ) )  e.  NN ) )
4027, 39syl 16 . . . . . . . . . . . . 13  |-  ( (
Base `  R )  e.  Fin  ->  ( (
( Base `  R )  =/=  (/)  /\  ( # `  ( Base `  R
) )  <_  1
)  ->  ( # `  ( Base `  R ) )  e.  NN ) )
4126, 40mpcom 36 . . . . . . . . . . . 12  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( # `  ( Base `  R ) )  e.  NN )
42 nnle1eq1 10563 . . . . . . . . . . . 12  |-  ( (
# `  ( Base `  R ) )  e.  NN  ->  ( ( # `
 ( Base `  R
) )  <_  1  <->  (
# `  ( Base `  R ) )  =  1 ) )
4341, 42syl 16 . . . . . . . . . . 11  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( ( # `
 ( Base `  R
) )  <_  1  <->  (
# `  ( Base `  R ) )  =  1 ) )
4421, 43mpbid 210 . . . . . . . . . 10  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( # `  ( Base `  R ) )  =  1 )
4544ex 434 . . . . . . . . 9  |-  ( (
Base `  R )  =/=  (/)  ->  ( ( # `
 ( Base `  R
) )  <_  1  ->  ( # `  ( Base `  R ) )  =  1 ) )
4620, 45syl 16 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( (
# `  ( Base `  R ) )  <_ 
1  ->  ( # `  ( Base `  R ) )  =  1 ) )
4746com12 31 . . . . . . 7  |-  ( (
# `  ( Base `  R ) )  <_ 
1  ->  ( R  e.  Ring  ->  ( # `  ( Base `  R ) )  =  1 ) )
4816, 47sylbi 195 . . . . . 6  |-  ( -.  1  <  ( # `  ( Base `  R
) )  ->  ( R  e.  Ring  ->  ( # `
 ( Base `  R
) )  =  1 ) )
499, 48jaoi 379 . . . . 5  |-  ( ( -.  R  e.  Ring  \/ 
-.  1  <  ( # `
 ( Base `  R
) ) )  -> 
( R  e.  Ring  -> 
( # `  ( Base `  R ) )  =  1 ) )
508, 49sylbi 195 . . . 4  |-  ( -.  ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R
) ) )  -> 
( R  e.  Ring  -> 
( # `  ( Base `  R ) )  =  1 ) )
5150com12 31 . . 3  |-  ( R  e.  Ring  ->  ( -.  ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R
) ) )  -> 
( # `  ( Base `  R ) )  =  1 ) )
527, 51impbid 191 . 2  |-  ( R  e.  Ring  ->  ( (
# `  ( Base `  R ) )  =  1  <->  -.  ( R  e.  Ring  /\  1  <  (
# `  ( Base `  R ) ) ) ) )
5318isnzr2hash 17706 . . . 4  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R
) ) ) )
5453bicomi 202 . . 3  |-  ( ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R ) ) )  <->  R  e. NzRing )
5554notbii 296 . 2  |-  ( -.  ( R  e.  Ring  /\  1  <  ( # `  ( Base `  R
) ) )  <->  -.  R  e. NzRing )
5652, 55syl6bb 261 1  |-  ( R  e.  Ring  ->  ( (
# `  ( Base `  R ) )  =  1  <->  -.  R  e. NzRing ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   class class class wbr 4447   ` cfv 5587   Fincfn 7516   0cc0 9491   1c1 9492   RR*cxr 9626    < clt 9627    <_ cle 9628   NNcn 10535   NN0cn0 10794   #chash 12372   Basecbs 14489   Grpcgrp 15726   Ringcrg 16995  NzRingcnzr 17699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-hash 12373  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-plusg 14567  df-0g 14696  df-mnd 15731  df-grp 15864  df-minusg 15865  df-mgp 16941  df-ur 16953  df-rng 16997  df-nzr 17700
This theorem is referenced by:  rng1nnzr  17712  lmod0rng  32049  lindszr  32160
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