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Theorem srg1zr 17293
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 pm4.24 641 . 2  |-  ( B  =  { Z }  <->  ( B  =  { Z }  /\  B  =  { Z } ) )
2 srgmnd 17274 . . . . . . 7  |-  ( R  e. SRing  ->  R  e.  Mnd )
323ad2ant1 1015 . . . . . 6  |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  ->  R  e.  Mnd )
43adantr 463 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e.  Mnd )
5 mndmgm 16045 . . . . 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 459 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  Z  e.  B )
8 simpl2 998 . . . 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 16000 . . . 4  |-  ( ( R  e. Mgm  /\  Z  e.  B  /\  .+  Fn  ( B  X.  B
) )  ->  ( B  =  { Z } 
<-> 
.+  =  { <. <. Z ,  Z >. ,  Z >. } ) )
126, 7, 8, 11syl3anc 1226 . . 3  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<-> 
.+  =  { <. <. Z ,  Z >. ,  Z >. } ) )
13 simpl1 997 . . . . . 6  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e. SRing )
14 eqid 2382 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
1514srgmgp 17275 . . . . . 6  |-  ( R  e. SRing  ->  (mulGrp `  R )  e.  Mnd )
16 mndmgm 16045 . . . . . 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 17258 . . . . . . . . 9  |-  .*  =  ( +g  `  (mulGrp `  R ) )
2019fneq1i 5583 . . . . . . . 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 1017 . . . . . 6  |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
2322adantr 463 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
2414, 9mgpbas 17260 . . . . . 6  |-  B  =  ( Base `  (mulGrp `  R ) )
25 eqid 2382 . . . . . 6  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
2624, 25mgmb1mgm1 16000 . . . . 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 1226 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  ( +g  `  (mulGrp `  R ) )  =  { <. <. Z ,  Z >. ,  Z >. } ) )
2819eqcomi 2395 . . . . . 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 2384 . . . 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 708 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   {csn 3944   <.cop 3950    X. cxp 4911    Fn wfn 5491   ` cfv 5496   Basecbs 14634   +g cplusg 14702   .rcmulr 14703  Mgmcmgm 15987   Mndcmnd 16036  mulGrpcmgp 17254  SRingcsrg 17270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-plusg 14715  df-plusf 15988  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-cmn 16917  df-mgp 17255  df-srg 17271
This theorem is referenced by:  srgen1zr  17294  ring1zr  18036
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