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Theorem ply1domn 19913
Description: Corollary of deg1mul2 19904: the univariate polynomials over a domain are a domain. This is true for multivariate but with a much more complicated proof. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypothesis
Ref Expression
ply1domn.p  |-  P  =  (Poly1 `  R )
Assertion
Ref Expression
ply1domn  |-  ( R  e. Domn  ->  P  e. Domn )

Proof of Theorem ply1domn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 domnnzr 16282 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 ply1domn.p . . . 4  |-  P  =  (Poly1 `  R )
32ply1nz 19911 . . 3  |-  ( R  e. NzRing  ->  P  e. NzRing )
41, 3syl 16 . 2  |-  ( R  e. Domn  ->  P  e. NzRing )
5 neanior 2635 . . . . 5  |-  ( ( x  =/=  ( 0g
`  P )  /\  y  =/=  ( 0g `  P ) )  <->  -.  (
x  =  ( 0g
`  P )  \/  y  =  ( 0g
`  P ) ) )
6 eqid 2387 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
7 eqid 2387 . . . . . . . . 9  |-  (RLReg `  R )  =  (RLReg `  R )
8 eqid 2387 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
9 eqid 2387 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
10 eqid 2387 . . . . . . . . 9  |-  ( 0g
`  P )  =  ( 0g `  P
)
11 domnrng 16283 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
1211ad2antrr 707 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  R  e.  Ring )
13 simplrl 737 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  x  e.  ( Base `  P
) )
14 simprl 733 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  x  =/=  ( 0g `  P
) )
15 simpll 731 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  R  e. Domn )
16 eqid 2387 . . . . . . . . . . 11  |-  (coe1 `  x
)  =  (coe1 `  x
)
176, 2, 10, 8, 7, 16deg1ldgdomn 19884 . . . . . . . . . 10  |-  ( ( R  e. Domn  /\  x  e.  ( Base `  P
)  /\  x  =/=  ( 0g `  P ) )  ->  ( (coe1 `  x ) `  (
( deg1  `
 R ) `  x ) )  e.  (RLReg `  R )
)
1815, 13, 14, 17syl3anc 1184 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
(coe1 `  x ) `  ( ( deg1  `  R ) `  x ) )  e.  (RLReg `  R )
)
19 simplrr 738 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  y  e.  ( Base `  P
) )
20 simprr 734 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  y  =/=  ( 0g `  P
) )
216, 2, 7, 8, 9, 10, 12, 13, 14, 18, 19, 20deg1mul2 19904 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
( deg1  `
 R ) `  ( x ( .r
`  P ) y ) )  =  ( ( ( deg1  `  R ) `  x )  +  ( ( deg1  `  R ) `  y ) ) )
226, 2, 10, 8deg1nn0cl 19878 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  P
)  /\  x  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  x
)  e.  NN0 )
2312, 13, 14, 22syl3anc 1184 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
( deg1  `
 R ) `  x )  e.  NN0 )
246, 2, 10, 8deg1nn0cl 19878 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  P
)  /\  y  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  y
)  e.  NN0 )
2512, 19, 20, 24syl3anc 1184 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
( deg1  `
 R ) `  y )  e.  NN0 )
2623, 25nn0addcld 10210 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
( ( deg1  `  R ) `  x )  +  ( ( deg1  `  R ) `  y ) )  e. 
NN0 )
2721, 26eqeltrd 2461 . . . . . . 7  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
( deg1  `
 R ) `  ( x ( .r
`  P ) y ) )  e.  NN0 )
282ply1rng 16569 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  P  e. 
Ring )
2911, 28syl 16 . . . . . . . . . 10  |-  ( R  e. Domn  ->  P  e.  Ring )
3029ad2antrr 707 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  P  e.  Ring )
318, 9rngcl 15604 . . . . . . . . 9  |-  ( ( P  e.  Ring  /\  x  e.  ( Base `  P
)  /\  y  e.  ( Base `  P )
)  ->  ( x
( .r `  P
) y )  e.  ( Base `  P
) )
3230, 13, 19, 31syl3anc 1184 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
x ( .r `  P ) y )  e.  ( Base `  P
) )
336, 2, 10, 8deg1nn0clb 19880 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x ( .r `  P ) y )  e.  ( Base `  P
) )  ->  (
( x ( .r
`  P ) y )  =/=  ( 0g
`  P )  <->  ( ( deg1  `  R ) `  (
x ( .r `  P ) y ) )  e.  NN0 )
)
3412, 32, 33syl2anc 643 . . . . . . 7  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
( x ( .r
`  P ) y )  =/=  ( 0g
`  P )  <->  ( ( deg1  `  R ) `  (
x ( .r `  P ) y ) )  e.  NN0 )
)
3527, 34mpbird 224 . . . . . 6  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  (
x ( .r `  P ) y )  =/=  ( 0g `  P ) )
3635ex 424 . . . . 5  |-  ( ( R  e. Domn  /\  (
x  e.  ( Base `  P )  /\  y  e.  ( Base `  P
) ) )  -> 
( ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) )  ->  ( x
( .r `  P
) y )  =/=  ( 0g `  P
) ) )
375, 36syl5bir 210 . . . 4  |-  ( ( R  e. Domn  /\  (
x  e.  ( Base `  P )  /\  y  e.  ( Base `  P
) ) )  -> 
( -.  ( x  =  ( 0g `  P )  \/  y  =  ( 0g `  P ) )  -> 
( x ( .r
`  P ) y )  =/=  ( 0g
`  P ) ) )
3837necon4bd 2612 . . 3  |-  ( ( R  e. Domn  /\  (
x  e.  ( Base `  P )  /\  y  e.  ( Base `  P
) ) )  -> 
( ( x ( .r `  P ) y )  =  ( 0g `  P )  ->  ( x  =  ( 0g `  P
)  \/  y  =  ( 0g `  P
) ) ) )
3938ralrimivva 2741 . 2  |-  ( R  e. Domn  ->  A. x  e.  (
Base `  P ) A. y  e.  ( Base `  P ) ( ( x ( .r
`  P ) y )  =  ( 0g
`  P )  -> 
( x  =  ( 0g `  P )  \/  y  =  ( 0g `  P ) ) ) )
408, 9, 10isdomn 16281 . 2  |-  ( P  e. Domn 
<->  ( P  e. NzRing  /\  A. x  e.  ( Base `  P ) A. y  e.  ( Base `  P
) ( ( x ( .r `  P
) y )  =  ( 0g `  P
)  ->  ( x  =  ( 0g `  P )  \/  y  =  ( 0g `  P ) ) ) ) )
414, 39, 40sylanbrc 646 1  |-  ( R  e. Domn  ->  P  e. Domn )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   ` cfv 5394  (class class class)co 6020    + caddc 8926   NN0cn0 10153   Basecbs 13396   .rcmulr 13457   0gc0g 13650   Ringcrg 15587  NzRingcnzr 16255  RLRegcrlreg 16266  Domncdomn 16267  Poly1cpl1 16498  coe1cco1 16501   deg1 cdg1 19844
This theorem is referenced by:  ply1idom  19914  deg1mhm  27195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-ofr 6245  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-cntz 15043  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-subrg 15793  df-lmod 15879  df-lss 15936  df-nzr 16256  df-rlreg 16270  df-domn 16271  df-ascl 16301  df-psr 16344  df-mvr 16345  df-mpl 16346  df-opsr 16352  df-psr1 16503  df-vr1 16504  df-ply1 16505  df-coe1 16508  df-cnfld 16627  df-mdeg 19845  df-deg1 19846
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