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Theorem ply1domn 20038
Description: Corollary of deg1mul2 20029: 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 16347 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 ply1domn.p . . . 4  |-  P  =  (Poly1 `  R )
32ply1nz 20036 . . 3  |-  ( R  e. NzRing  ->  P  e. NzRing )
41, 3syl 16 . 2  |-  ( R  e. Domn  ->  P  e. NzRing )
5 neanior 2683 . . . . 5  |-  ( ( x  =/=  ( 0g
`  P )  /\  y  =/=  ( 0g `  P ) )  <->  -.  (
x  =  ( 0g
`  P )  \/  y  =  ( 0g
`  P ) ) )
6 eqid 2435 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
7 eqid 2435 . . . . . . . . 9  |-  (RLReg `  R )  =  (RLReg `  R )
8 eqid 2435 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
9 eqid 2435 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
10 eqid 2435 . . . . . . . . 9  |-  ( 0g
`  P )  =  ( 0g `  P
)
11 domnrng 16348 . . . . . . . . . 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 2435 . . . . . . . . . . 11  |-  (coe1 `  x
)  =  (coe1 `  x
)
176, 2, 10, 8, 7, 16deg1ldgdomn 20009 . . . . . . . . . 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 20029 . . . . . . . 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 20003 . . . . . . . . . 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 20003 . . . . . . . . . 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 10270 . . . . . . . 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 2509 . . . . . . 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 16634 . . . . . . . . . . 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 15669 . . . . . . . . 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 20005 . . . . . . . 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 2660 . . 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 2790 . 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 16346 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   ` cfv 5446  (class class class)co 6073    + caddc 8985   NN0cn0 10213   Basecbs 13461   .rcmulr 13522   0gc0g 13715   Ringcrg 15652  NzRingcnzr 16320  RLRegcrlreg 16331  Domncdomn 16332  Poly1cpl1 16563  coe1cco1 16566   deg1 cdg1 19969
This theorem is referenced by:  ply1idom  20039  deg1mhm  27494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-subrg 15858  df-lmod 15944  df-lss 16001  df-nzr 16321  df-rlreg 16335  df-domn 16336  df-ascl 16366  df-psr 16409  df-mvr 16410  df-mpl 16411  df-opsr 16417  df-psr1 16568  df-vr1 16569  df-ply1 16570  df-coe1 16573  df-cnfld 16696  df-mdeg 19970  df-deg1 19971
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