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Theorem srg1zr 17048
Description: The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
Hypotheses
Ref Expression
srg1zr.b  |-  B  =  ( Base `  R
)
srg1zr.p  |-  .+  =  ( +g  `  R )
srg1zr.t  |-  .*  =  ( .r `  R )
Assertion
Ref Expression
srg1zr  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  (  .+  =  { <. <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) ) )

Proof of Theorem srg1zr
StepHypRef Expression
1 pm4.24 643 . 2  |-  ( B  =  { Z }  <->  ( B  =  { Z }  /\  B  =  { Z } ) )
2 srgmnd 17029 . . . . . . 7  |-  ( R  e. SRing  ->  R  e.  Mnd )
323ad2ant1 1016 . . . . . 6  |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  ->  R  e.  Mnd )
43adantr 465 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e.  Mnd )
5 mndmgm 15797 . . . . 5  |-  ( R  e.  Mnd  ->  R  e. Mgm )
64, 5syl 16 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e. Mgm )
7 simpr 461 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  Z  e.  B )
8 simpl2 999 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  .+  Fn  ( B  X.  B
) )
9 srg1zr.b . . . . 5  |-  B  =  ( Base `  R
)
10 srg1zr.p . . . . 5  |-  .+  =  ( +g  `  R )
119, 10mgmb1mgm1 15752 . . . 4  |-  ( ( R  e. Mgm  /\  Z  e.  B  /\  .+  Fn  ( B  X.  B
) )  ->  ( B  =  { Z } 
<-> 
.+  =  { <. <. Z ,  Z >. ,  Z >. } ) )
126, 7, 8, 11syl3anc 1227 . . 3  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<-> 
.+  =  { <. <. Z ,  Z >. ,  Z >. } ) )
13 simpl1 998 . . . . . 6  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e. SRing )
14 eqid 2441 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
1514srgmgp 17030 . . . . . 6  |-  ( R  e. SRing  ->  (mulGrp `  R )  e.  Mnd )
16 mndmgm 15797 . . . . . 6  |-  ( (mulGrp `  R )  e.  Mnd  ->  (mulGrp `  R )  e. Mgm )
1713, 15, 163syl 20 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (mulGrp `  R )  e. Mgm )
18 srg1zr.t . . . . . . . . . 10  |-  .*  =  ( .r `  R )
1914, 18mgpplusg 17013 . . . . . . . . 9  |-  .*  =  ( +g  `  (mulGrp `  R ) )
2019fneq1i 5661 . . . . . . . 8  |-  (  .*  Fn  ( B  X.  B )  <->  ( +g  `  (mulGrp `  R )
)  Fn  ( B  X.  B ) )
2120biimpi 194 . . . . . . 7  |-  (  .*  Fn  ( B  X.  B )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
22213ad2ant3 1018 . . . . . 6  |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
2322adantr 465 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
2414, 9mgpbas 17015 . . . . . 6  |-  B  =  ( Base `  (mulGrp `  R ) )
25 eqid 2441 . . . . . 6  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
2624, 25mgmb1mgm1 15752 . . . . 5  |-  ( ( (mulGrp `  R )  e. Mgm  /\  Z  e.  B  /\  ( +g  `  (mulGrp `  R ) )  Fn  ( B  X.  B
) )  ->  ( B  =  { Z } 
<->  ( +g  `  (mulGrp `  R ) )  =  { <. <. Z ,  Z >. ,  Z >. } ) )
2717, 7, 23, 26syl3anc 1227 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  ( +g  `  (mulGrp `  R ) )  =  { <. <. Z ,  Z >. ,  Z >. } ) )
2819eqcomi 2454 . . . . . 6  |-  ( +g  `  (mulGrp `  R )
)  =  .*
2928a1i 11 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( +g  `  (mulGrp `  R
) )  =  .*  )
3029eqeq1d 2443 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (
( +g  `  (mulGrp `  R ) )  =  { <. <. Z ,  Z >. ,  Z >. }  <->  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) )
3127, 30bitrd 253 . . 3  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) )
3212, 31anbi12d 710 . 2  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (
( B  =  { Z }  /\  B  =  { Z } )  <-> 
(  .+  =  { <. <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) ) )
331, 32syl5bb 257 1  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  (  .+  =  { <. <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   {csn 4010   <.cop 4016    X. cxp 4983    Fn wfn 5569   ` cfv 5574   Basecbs 14504   +g cplusg 14569   .rcmulr 14570  Mgmcmgm 15739   Mndcmnd 15788  mulGrpcmgp 17009  SRingcsrg 17025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-plusg 14582  df-plusf 15740  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-cmn 16669  df-mgp 17010  df-srg 17026
This theorem is referenced by:  srgen1zr  17049  ring1zr  17791
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