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Theorem 0rngnnzr 30678
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 9381 . . . . . . . 8  |-  1  e.  RR
21ltnri 9479 . . . . . . 7  |-  -.  1  <  1
3 breq2 4293 . . . . . . 7  |-  ( (
# `  ( Base `  R ) )  =  1  ->  ( 1  <  ( # `  ( Base `  R ) )  <->  1  <  1 ) )
42, 3mtbiri 303 . . . . . 6  |-  ( (
# `  ( Base `  R ) )  =  1  ->  -.  1  <  ( # `  ( Base `  R ) ) )
54adantl 463 . . . . 5  |-  ( ( R  e.  Ring  /\  ( # `
 ( Base `  R
) )  =  1 )  ->  -.  1  <  ( # `  ( Base `  R ) ) )
65intnand 902 . . . 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 485 . . . . 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 5698 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
11 hashxrcl 12123 . . . . . . . . . 10  |-  ( (
Base `  R )  e.  _V  ->  ( # `  ( Base `  R ) )  e.  RR* )
1210, 11ax-mp 5 . . . . . . . . 9  |-  ( # `  ( Base `  R
) )  e.  RR*
131rexri 9432 . . . . . . . . 9  |-  1  e.  RR*
14 xrlenlt 9438 . . . . . . . . 9  |-  ( ( ( # `  ( Base `  R ) )  e.  RR*  /\  1  e.  RR* )  ->  (
( # `  ( Base `  R ) )  <_ 
1  <->  -.  1  <  (
# `  ( Base `  R ) ) ) )
1512, 13, 14mp2an 667 . . . . . . . 8  |-  ( (
# `  ( Base `  R ) )  <_ 
1  <->  -.  1  <  (
# `  ( Base `  R ) ) )
1615bicomi 202 . . . . . . 7  |-  ( -.  1  <  ( # `  ( Base `  R
) )  <->  ( # `  ( Base `  R ) )  <_  1 )
17 rnggrp 16640 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
18 eqid 2441 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
1918grpbn0 15560 . . . . . . . . . 10  |-  ( R  e.  Grp  ->  ( Base `  R )  =/=  (/) )
2017, 19syl 16 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( Base `  R )  =/=  (/) )
21 simpr 458 . . . . . . . . . . 11  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( # `  ( Base `  R ) )  <_  1 )
2210a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  _V )
23 1nn0 10591 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
2423a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  1  e.  NN0 )
25 hashbnd 12105 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  e.  _V  /\  1  e.  NN0  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  Fin )
2622, 24, 21, 25syl3anc 1213 . . . . . . . . . . . . 13  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  Fin )
27 hashcl 12122 . . . . . . . . . . . . . 14  |-  ( (
Base `  R )  e.  Fin  ->  ( # `  ( Base `  R ) )  e.  NN0 )
28 simpr 458 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  e.  NN0 )  ->  ( # `  ( Base `  R ) )  e.  NN0 )
2910a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( Base `  R
)  e.  _V )
30 hasheq0 12127 . . . . . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( ( Base `  R )  =/=  (/)  ->  ( # `
 ( Base `  R
) )  =/=  0
) )
3433impcom 430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  e.  NN0 )  ->  ( # `  ( Base `  R ) )  =/=  0 )
35 elnnne0 10589 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  ( Base `  R ) )  e.  NN  <->  ( ( # `  ( Base `  R
) )  e.  NN0  /\  ( # `  ( Base `  R ) )  =/=  0 ) )
3628, 34, 35sylanbrc 659 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 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 10346 . . . . . . . . . . . 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 30674 . . . 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 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970   (/)c0 3634   class class class wbr 4289   ` cfv 5415   Fincfn 7306   0cc0 9278   1c1 9279   RR*cxr 9413    < clt 9414    <_ cle 9415   NNcn 10318   NN0cn0 10575   #chash 12099   Basecbs 14170   Grpcgrp 15406   Ringcrg 16635  NzRingcnzr 17317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-mgp 16582  df-ur 16594  df-rng 16637  df-nzr 17318
This theorem is referenced by:  lmod0rng  30679  lindszr  30827  rng1nnzr  30850
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