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Theorem ply1domn 22818
Description: Corollary of deg1mul2 22809: 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 18266 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 ply1domn.p . . . 4  |-  P  =  (Poly1 `  R )
32ply1nz 22816 . . 3  |-  ( R  e. NzRing  ->  P  e. NzRing )
41, 3syl 17 . 2  |-  ( R  e. Domn  ->  P  e. NzRing )
5 neanior 2730 . . . . 5  |-  ( ( x  =/=  ( 0g
`  P )  /\  y  =/=  ( 0g `  P ) )  <->  -.  (
x  =  ( 0g
`  P )  \/  y  =  ( 0g
`  P ) ) )
6 eqid 2404 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
7 eqid 2404 . . . . . . . . 9  |-  (RLReg `  R )  =  (RLReg `  R )
8 eqid 2404 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
9 eqid 2404 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
10 eqid 2404 . . . . . . . . 9  |-  ( 0g
`  P )  =  ( 0g `  P
)
11 domnring 18267 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
1211ad2antrr 726 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  R  e.  Ring )
13 simplrl 764 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  x  e.  ( Base `  P
) )
14 simprl 758 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  x  =/=  ( 0g `  P
) )
15 simpll 754 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  R  e. Domn )
16 eqid 2404 . . . . . . . . . . 11  |-  (coe1 `  x
)  =  (coe1 `  x
)
176, 2, 10, 8, 7, 16deg1ldgdomn 22788 . . . . . . . . . 10  |-  ( ( R  e. Domn  /\  x  e.  ( Base `  P
)  /\  x  =/=  ( 0g `  P ) )  ->  ( (coe1 `  x ) `  (
( deg1  `
 R ) `  x ) )  e.  (RLReg `  R )
)
1815, 13, 14, 17syl3anc 1232 . . . . . . . . 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 765 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  y  e.  ( Base `  P
) )
20 simprr 760 . . . . . . . . 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 22809 . . . . . . . 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 22782 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  P
)  /\  x  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  x
)  e.  NN0 )
2312, 13, 14, 22syl3anc 1232 . . . . . . . . 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 22782 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  P
)  /\  y  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  y
)  e.  NN0 )
2512, 19, 20, 24syl3anc 1232 . . . . . . . . 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 10899 . . . . . . . 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 2492 . . . . . . 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 )
282ply1ring 18611 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  P  e. 
Ring )
2911, 28syl 17 . . . . . . . . . 10  |-  ( R  e. Domn  ->  P  e.  Ring )
3029ad2antrr 726 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  P  e.  Ring )
318, 9ringcl 17534 . . . . . . . . 9  |-  ( ( P  e.  Ring  /\  x  e.  ( Base `  P
)  /\  y  e.  ( Base `  P )
)  ->  ( x
( .r `  P
) y )  e.  ( Base `  P
) )
3230, 13, 19, 31syl3anc 1232 . . . . . . . 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 22784 . . . . . . . 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 661 . . . . . . 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 234 . . . . . 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 434 . . . . 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 220 . . . 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 2627 . . 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 2827 . 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 18265 . 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 664 1  |-  ( R  e. Domn  ->  P  e. Domn )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    \/ wo 368    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   A.wral 2756   ` cfv 5571  (class class class)co 6280    + caddc 9527   NN0cn0 10838   Basecbs 14843   .rcmulr 14912   0gc0g 15056   Ringcrg 17520  NzRingcnzr 18227  RLRegcrlreg 18249  Domncdomn 18250  Poly1cpl1 18538  coe1cco1 18539   deg1 cdg1 22746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-ofr 6524  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-sup 7937  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-fz 11729  df-fzo 11857  df-seq 12154  df-hash 12455  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-0g 15058  df-gsum 15059  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mhm 16292  df-submnd 16293  df-grp 16383  df-minusg 16384  df-sbg 16385  df-mulg 16386  df-subg 16524  df-ghm 16591  df-cntz 16681  df-cmn 17126  df-abl 17127  df-mgp 17464  df-ur 17476  df-ring 17522  df-cring 17523  df-subrg 17749  df-lmod 17836  df-lss 17901  df-nzr 18228  df-rlreg 18253  df-domn 18254  df-ascl 18285  df-psr 18327  df-mvr 18328  df-mpl 18329  df-opsr 18331  df-psr1 18541  df-vr1 18542  df-ply1 18543  df-coe1 18544  df-cnfld 18743  df-mdeg 22747  df-deg1 22748
This theorem is referenced by:  ply1idom  22819  deg1mhm  35544
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