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Theorem ply1domn 22259
Description: Corollary of deg1mul2 22250: 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 17715 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 ply1domn.p . . . 4  |-  P  =  (Poly1 `  R )
32ply1nz 22257 . . 3  |-  ( R  e. NzRing  ->  P  e. NzRing )
41, 3syl 16 . 2  |-  ( R  e. Domn  ->  P  e. NzRing )
5 neanior 2792 . . . . 5  |-  ( ( x  =/=  ( 0g
`  P )  /\  y  =/=  ( 0g `  P ) )  <->  -.  (
x  =  ( 0g
`  P )  \/  y  =  ( 0g
`  P ) ) )
6 eqid 2467 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
7 eqid 2467 . . . . . . . . 9  |-  (RLReg `  R )  =  (RLReg `  R )
8 eqid 2467 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
9 eqid 2467 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
10 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  P )  =  ( 0g `  P
)
11 domnrng 17716 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
1211ad2antrr 725 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  R  e.  Ring )
13 simplrl 759 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  x  e.  ( Base `  P
) )
14 simprl 755 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  x  =/=  ( 0g `  P
) )
15 simpll 753 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  R  e. Domn )
16 eqid 2467 . . . . . . . . . . 11  |-  (coe1 `  x
)  =  (coe1 `  x
)
176, 2, 10, 8, 7, 16deg1ldgdomn 22229 . . . . . . . . . 10  |-  ( ( R  e. Domn  /\  x  e.  ( Base `  P
)  /\  x  =/=  ( 0g `  P ) )  ->  ( (coe1 `  x ) `  (
( deg1  `
 R ) `  x ) )  e.  (RLReg `  R )
)
1815, 13, 14, 17syl3anc 1228 . . . . . . . . 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 760 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  y  e.  ( Base `  P
) )
20 simprr 756 . . . . . . . . 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 22250 . . . . . . . 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 22223 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  P
)  /\  x  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  x
)  e.  NN0 )
2312, 13, 14, 22syl3anc 1228 . . . . . . . . 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 22223 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  P
)  /\  y  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  y
)  e.  NN0 )
2512, 19, 20, 24syl3anc 1228 . . . . . . . . 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 10852 . . . . . . . 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 2555 . . . . . . 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 18060 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  P  e. 
Ring )
2911, 28syl 16 . . . . . . . . . 10  |-  ( R  e. Domn  ->  P  e.  Ring )
3029ad2antrr 725 . . . . . . . . 9  |-  ( ( ( R  e. Domn  /\  ( x  e.  ( Base `  P )  /\  y  e.  ( Base `  P ) ) )  /\  ( x  =/=  ( 0g `  P
)  /\  y  =/=  ( 0g `  P ) ) )  ->  P  e.  Ring )
318, 9rngcl 16999 . . . . . . . . 9  |-  ( ( P  e.  Ring  /\  x  e.  ( Base `  P
)  /\  y  e.  ( Base `  P )
)  ->  ( x
( .r `  P
) y )  e.  ( Base `  P
) )
3230, 13, 19, 31syl3anc 1228 . . . . . . . 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 22225 . . . . . . . 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 232 . . . . . 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 218 . . . 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 2689 . . 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 2885 . 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 17714 . 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 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   ` cfv 5586  (class class class)co 6282    + caddc 9491   NN0cn0 10791   Basecbs 14486   .rcmulr 14552   0gc0g 14691   Ringcrg 16986  NzRingcnzr 17687  RLRegcrlreg 17698  Domncdomn 17699  Poly1cpl1 17987  coe1cco1 17988   deg1 cdg1 22187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-ofr 6523  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-0g 14693  df-gsum 14694  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-cring 16989  df-subrg 17210  df-lmod 17297  df-lss 17362  df-nzr 17688  df-rlreg 17702  df-domn 17703  df-ascl 17734  df-psr 17776  df-mvr 17777  df-mpl 17778  df-opsr 17780  df-psr1 17990  df-vr1 17991  df-ply1 17992  df-coe1 17993  df-cnfld 18192  df-mdeg 22188  df-deg1 22189
This theorem is referenced by:  ply1idom  22260  deg1mhm  30772
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