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Theorem 0rngnnzr 30790
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 9397 . . . . . . . 8  |-  1  e.  RR
21ltnri 9495 . . . . . . 7  |-  -.  1  <  1
3 breq2 4308 . . . . . . 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 907 . . . 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 5713 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
11 hashxrcl 12139 . . . . . . . . . 10  |-  ( (
Base `  R )  e.  _V  ->  ( # `  ( Base `  R ) )  e.  RR* )
1210, 11ax-mp 5 . . . . . . . . 9  |-  ( # `  ( Base `  R
) )  e.  RR*
131rexri 9448 . . . . . . . . 9  |-  1  e.  RR*
14 xrlenlt 9454 . . . . . . . . 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 16662 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
18 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
1918grpbn0 15579 . . . . . . . . . 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 10607 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
2423a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  1  e.  NN0 )
25 hashbnd 12121 . . . . . . . . . . . . . 14  |-  ( ( ( Base `  R
)  e.  _V  /\  1  e.  NN0  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  Fin )
2622, 24, 21, 25syl3anc 1218 . . . . . . . . . . . . 13  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  <_  1
)  ->  ( Base `  R )  e.  Fin )
27 hashcl 12138 . . . . . . . . . . . . . 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 12143 . . . . . . . . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( Base `  R ) )  e. 
NN0  ->  ( ( Base `  R )  =/=  (/)  ->  ( # `
 ( Base `  R
) )  =/=  0
) )
3433impcom 430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( Base `  R
)  =/=  (/)  /\  ( # `
 ( Base `  R
) )  e.  NN0 )  ->  ( # `  ( Base `  R ) )  =/=  0 )
35 elnnne0 10605 . . . . . . . . . . . . . . . . . 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 10362 . . . . . . . . . . . 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 30786 . . . 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 1369    e. wcel 1756    =/= wne 2618   _Vcvv 2984   (/)c0 3649   class class class wbr 4304   ` cfv 5430   Fincfn 7322   0cc0 9294   1c1 9295   RR*cxr 9429    < clt 9430    <_ cle 9431   NNcn 10334   NN0cn0 10591   #chash 12115   Basecbs 14186   Grpcgrp 15422   Ringcrg 16657  NzRingcnzr 17351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-hash 12116  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-plusg 14263  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-mgp 16604  df-ur 16616  df-rng 16659  df-nzr 17352
This theorem is referenced by:  lmod0rng  30791  lindszr  31015  rng1nnzr  31038
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