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Theorem deg1mul2 19906
Description: Degree of multiplication of two nonzero polynomials when the first leads with a non-zero-divisor coefficient. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Hypotheses
Ref Expression
deg1mul2.d  |-  D  =  ( deg1  `  R )
deg1mul2.p  |-  P  =  (Poly1 `  R )
deg1mul2.e  |-  E  =  (RLReg `  R )
deg1mul2.b  |-  B  =  ( Base `  P
)
deg1mul2.t  |-  .x.  =  ( .r `  P )
deg1mul2.z  |-  .0.  =  ( 0g `  P )
deg1mul2.r  |-  ( ph  ->  R  e.  Ring )
deg1mul2.fb  |-  ( ph  ->  F  e.  B )
deg1mul2.fz  |-  ( ph  ->  F  =/=  .0.  )
deg1mul2.fc  |-  ( ph  ->  ( (coe1 `  F ) `  ( D `  F ) )  e.  E )
deg1mul2.gb  |-  ( ph  ->  G  e.  B )
deg1mul2.gz  |-  ( ph  ->  G  =/=  .0.  )
Assertion
Ref Expression
deg1mul2  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( ( D `
 F )  +  ( D `  G
) ) )

Proof of Theorem deg1mul2
StepHypRef Expression
1 deg1mul2.p . . 3  |-  P  =  (Poly1 `  R )
2 deg1mul2.d . . 3  |-  D  =  ( deg1  `  R )
3 deg1mul2.r . . 3  |-  ( ph  ->  R  e.  Ring )
4 deg1mul2.b . . 3  |-  B  =  ( Base `  P
)
5 deg1mul2.t . . 3  |-  .x.  =  ( .r `  P )
6 deg1mul2.fb . . 3  |-  ( ph  ->  F  e.  B )
7 deg1mul2.gb . . 3  |-  ( ph  ->  G  e.  B )
8 deg1mul2.fz . . . 4  |-  ( ph  ->  F  =/=  .0.  )
9 deg1mul2.z . . . . 5  |-  .0.  =  ( 0g `  P )
102, 1, 9, 4deg1nn0cl 19880 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
113, 6, 8, 10syl3anc 1184 . . 3  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
12 deg1mul2.gz . . . 4  |-  ( ph  ->  G  =/=  .0.  )
132, 1, 9, 4deg1nn0cl 19880 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  ( D `  G )  e.  NN0 )
143, 7, 12, 13syl3anc 1184 . . 3  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
1511nn0red 10209 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  RR )
1615leidd 9527 . . 3  |-  ( ph  ->  ( D `  F
)  <_  ( D `  F ) )
1714nn0red 10209 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  RR )
1817leidd 9527 . . 3  |-  ( ph  ->  ( D `  G
)  <_  ( D `  G ) )
191, 2, 3, 4, 5, 6, 7, 11, 14, 16, 18deg1mulle2 19901 . 2  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( ( D `
 F )  +  ( D `  G
) ) )
201ply1rng 16571 . . . . 5  |-  ( R  e.  Ring  ->  P  e. 
Ring )
213, 20syl 16 . . . 4  |-  ( ph  ->  P  e.  Ring )
224, 5rngcl 15606 . . . 4  |-  ( ( P  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .x.  G )  e.  B )
2321, 6, 7, 22syl3anc 1184 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
2411, 14nn0addcld 10212 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  NN0 )
25 eqid 2389 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
261, 5, 25, 4, 2, 9, 3, 6, 8, 7, 12coe1mul4 19892 . . . 4  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =  ( ( (coe1 `  F ) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) ) )
27 eqid 2389 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
28 eqid 2389 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
292, 1, 9, 4, 27, 28deg1ldg 19884 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  (
(coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
303, 7, 12, 29syl3anc 1184 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
31 deg1mul2.fc . . . . . . 7  |-  ( ph  ->  ( (coe1 `  F ) `  ( D `  F ) )  e.  E )
32 eqid 2389 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
3328, 4, 1, 32coe1f 16538 . . . . . . . . 9  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
347, 33syl 16 . . . . . . . 8  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
3534, 14ffvelrnd 5812 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  ( Base `  R ) )
36 deg1mul2.e . . . . . . . 8  |-  E  =  (RLReg `  R )
3736, 32, 25, 27rrgeq0i 16278 . . . . . . 7  |-  ( ( ( (coe1 `  F ) `  ( D `  F ) )  e.  E  /\  ( (coe1 `  G ) `  ( D `  G ) )  e.  ( Base `  R ) )  -> 
( ( ( (coe1 `  F ) `  ( D `  F )
) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) )  =  ( 0g `  R )  ->  ( (coe1 `  G
) `  ( D `  G ) )  =  ( 0g `  R
) ) )
3831, 35, 37syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( ( (coe1 `  F ) `  ( D `  F )
) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) )  =  ( 0g `  R )  ->  ( (coe1 `  G
) `  ( D `  G ) )  =  ( 0g `  R
) ) )
3938necon3d 2590 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  G
) `  ( D `  G ) )  =/=  ( 0g `  R
)  ->  ( (
(coe1 `  F ) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) )  =/=  ( 0g `  R ) ) )
4030, 39mpd 15 . . . 4  |-  ( ph  ->  ( ( (coe1 `  F
) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G ) ) )  =/=  ( 0g `  R ) )
4126, 40eqnetrd 2570 . . 3  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =/=  ( 0g `  R ) )
42 eqid 2389 . . . 4  |-  (coe1 `  ( F  .x.  G ) )  =  (coe1 `  ( F  .x.  G ) )
432, 1, 4, 27, 42deg1ge 19890 . . 3  |-  ( ( ( F  .x.  G
)  e.  B  /\  ( ( D `  F )  +  ( D `  G ) )  e.  NN0  /\  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =/=  ( 0g `  R ) )  ->  ( ( D `
 F )  +  ( D `  G
) )  <_  ( D `  ( F  .x.  G ) ) )
4423, 24, 41, 43syl3anc 1184 . 2  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  <_  ( D `  ( F  .x.  G
) ) )
452, 1, 4deg1xrcl 19874 . . . 4  |-  ( ( F  .x.  G )  e.  B  ->  ( D `  ( F  .x.  G ) )  e. 
RR* )
4623, 45syl 16 . . 3  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  e.  RR* )
4724nn0red 10209 . . . 4  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR )
4847rexrd 9069 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR* )
49 xrletri3 10679 . . 3  |-  ( ( ( D `  ( F  .x.  G ) )  e.  RR*  /\  (
( D `  F
)  +  ( D `
 G ) )  e.  RR* )  ->  (
( D `  ( F  .x.  G ) )  =  ( ( D `
 F )  +  ( D `  G
) )  <->  ( ( D `  ( F  .x.  G ) )  <_ 
( ( D `  F )  +  ( D `  G ) )  /\  ( ( D `  F )  +  ( D `  G ) )  <_ 
( D `  ( F  .x.  G ) ) ) ) )
5046, 48, 49syl2anc 643 . 2  |-  ( ph  ->  ( ( D `  ( F  .x.  G ) )  =  ( ( D `  F )  +  ( D `  G ) )  <->  ( ( D `  ( F  .x.  G ) )  <_ 
( ( D `  F )  +  ( D `  G ) )  /\  ( ( D `  F )  +  ( D `  G ) )  <_ 
( D `  ( F  .x.  G ) ) ) ) )
5119, 44, 50mpbir2and 889 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( ( D `
 F )  +  ( D `  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   -->wf 5392   ` cfv 5396  (class class class)co 6022    + caddc 8928   RR*cxr 9054    <_ cle 9056   NN0cn0 10155   Basecbs 13398   .rcmulr 13459   0gc0g 13652   Ringcrg 15589  RLRegcrlreg 16268  Poly1cpl1 16500  coe1cco1 16503   deg1 cdg1 19846
This theorem is referenced by:  ply1domn  19915  ply1divmo  19927  fta1glem1  19957  mon1psubm  27196  deg1mhm  27197
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-ofr 6247  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-0g 13656  df-gsum 13657  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-mhm 14667  df-submnd 14668  df-grp 14741  df-minusg 14742  df-mulg 14744  df-subg 14870  df-ghm 14933  df-cntz 15045  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-cring 15593  df-ur 15594  df-subrg 15795  df-rlreg 16272  df-psr 16346  df-mpl 16348  df-opsr 16354  df-psr1 16505  df-ply1 16507  df-coe1 16510  df-cnfld 16629  df-mdeg 19847  df-deg1 19848
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