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Theorem deg1mul2 19516
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 19490 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
113, 6, 8, 10syl3anc 1182 . . 3  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
12 deg1mul2.gz . . . 4  |-  ( ph  ->  G  =/=  .0.  )
132, 1, 9, 4deg1nn0cl 19490 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  ( D `  G )  e.  NN0 )
143, 7, 12, 13syl3anc 1182 . . 3  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
1511nn0red 10035 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  RR )
1615leidd 9355 . . 3  |-  ( ph  ->  ( D `  F
)  <_  ( D `  F ) )
1714nn0red 10035 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  RR )
1817leidd 9355 . . 3  |-  ( ph  ->  ( D `  G
)  <_  ( D `  G ) )
191, 2, 3, 4, 5, 6, 7, 11, 14, 16, 18deg1mulle2 19511 . 2  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( ( D `
 F )  +  ( D `  G
) ) )
201ply1rng 16342 . . . . 5  |-  ( R  e.  Ring  ->  P  e. 
Ring )
213, 20syl 15 . . . 4  |-  ( ph  ->  P  e.  Ring )
224, 5rngcl 15370 . . . 4  |-  ( ( P  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .x.  G )  e.  B )
2321, 6, 7, 22syl3anc 1182 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
2411, 14nn0addcld 10038 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  NN0 )
25 eqid 2296 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
261, 5, 25, 4, 2, 9, 3, 6, 8, 7, 12coe1mul4 19502 . . . 4  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =  ( ( (coe1 `  F ) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) ) )
27 eqid 2296 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
28 eqid 2296 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
292, 1, 9, 4, 27, 28deg1ldg 19494 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  (
(coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
303, 7, 12, 29syl3anc 1182 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
31 deg1mul2.fc . . . . . . 7  |-  ( ph  ->  ( (coe1 `  F ) `  ( D `  F ) )  e.  E )
32 eqid 2296 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
3328, 4, 1, 32coe1f 16308 . . . . . . . . 9  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
347, 33syl 15 . . . . . . . 8  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
35 ffvelrn 5679 . . . . . . . 8  |-  ( ( (coe1 `  G ) : NN0 --> ( Base `  R
)  /\  ( D `  G )  e.  NN0 )  ->  ( (coe1 `  G
) `  ( D `  G ) )  e.  ( Base `  R
) )
3634, 14, 35syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  ( Base `  R ) )
37 deg1mul2.e . . . . . . . 8  |-  E  =  (RLReg `  R )
3837, 32, 25, 27rrgeq0i 16046 . . . . . . 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
) ) )
3931, 36, 38syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( ( (coe1 `  F ) `  ( D `  F )
) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) )  =  ( 0g `  R )  ->  ( (coe1 `  G
) `  ( D `  G ) )  =  ( 0g `  R
) ) )
4039necon3d 2497 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  G
) `  ( D `  G ) )  =/=  ( 0g `  R
)  ->  ( (
(coe1 `  F ) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) )  =/=  ( 0g `  R ) ) )
4130, 40mpd 14 . . . 4  |-  ( ph  ->  ( ( (coe1 `  F
) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G ) ) )  =/=  ( 0g `  R ) )
4226, 41eqnetrd 2477 . . 3  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =/=  ( 0g `  R ) )
43 eqid 2296 . . . 4  |-  (coe1 `  ( F  .x.  G ) )  =  (coe1 `  ( F  .x.  G ) )
442, 1, 4, 27, 43deg1ge 19500 . . 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 ) ) )
4523, 24, 42, 44syl3anc 1182 . 2  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  <_  ( D `  ( F  .x.  G
) ) )
462, 1, 4deg1xrcl 19484 . . . 4  |-  ( ( F  .x.  G )  e.  B  ->  ( D `  ( F  .x.  G ) )  e. 
RR* )
4723, 46syl 15 . . 3  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  e.  RR* )
4824nn0red 10035 . . . 4  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR )
4948rexrd 8897 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR* )
50 xrletri3 10502 . . 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 ) ) ) ) )
5147, 49, 50syl2anc 642 . 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 ) ) ) ) )
5219, 45, 51mpbir2and 888 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( ( D `
 F )  +  ( D `  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    + caddc 8756   RR*cxr 8882    <_ cle 8884   NN0cn0 9981   Basecbs 13164   .rcmulr 13225   0gc0g 13416   Ringcrg 15353  RLRegcrlreg 16036  Poly1cpl1 16268  coe1cco1 16271   deg1 cdg1 19456
This theorem is referenced by:  ply1domn  19525  ply1divmo  19537  fta1glem1  19567  mon1psubm  27628  deg1mhm  27629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-subrg 15559  df-rlreg 16040  df-psr 16114  df-mpl 16116  df-opsr 16122  df-psr1 16273  df-ply1 16275  df-coe1 16278  df-cnfld 16394  df-mdeg 19457  df-deg1 19458
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