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Theorem deg1mul2 21712
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 21685 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
113, 6, 8, 10syl3anc 1219 . . 3  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
12 deg1mul2.gz . . . 4  |-  ( ph  ->  G  =/=  .0.  )
132, 1, 9, 4deg1nn0cl 21685 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  ( D `  G )  e.  NN0 )
143, 7, 12, 13syl3anc 1219 . . 3  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
1511nn0red 10741 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  RR )
1615leidd 10010 . . 3  |-  ( ph  ->  ( D `  F
)  <_  ( D `  F ) )
1714nn0red 10741 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  RR )
1817leidd 10010 . . 3  |-  ( ph  ->  ( D `  G
)  <_  ( D `  G ) )
191, 2, 3, 4, 5, 6, 7, 11, 14, 16, 18deg1mulle2 21707 . 2  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( ( D `
 F )  +  ( D `  G
) ) )
201ply1rng 17819 . . . . 5  |-  ( R  e.  Ring  ->  P  e. 
Ring )
213, 20syl 16 . . . 4  |-  ( ph  ->  P  e.  Ring )
224, 5rngcl 16773 . . . 4  |-  ( ( P  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .x.  G )  e.  B )
2321, 6, 7, 22syl3anc 1219 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
2411, 14nn0addcld 10744 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  NN0 )
25 eqid 2451 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
261, 5, 25, 4, 2, 9, 3, 6, 8, 7, 12coe1mul4 21698 . . . 4  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =  ( ( (coe1 `  F ) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) ) )
27 eqid 2451 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
28 eqid 2451 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
292, 1, 9, 4, 27, 28deg1ldg 21689 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  (
(coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
303, 7, 12, 29syl3anc 1219 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
31 deg1mul2.fc . . . . . . 7  |-  ( ph  ->  ( (coe1 `  F ) `  ( D `  F ) )  e.  E )
32 eqid 2451 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
3328, 4, 1, 32coe1f 17783 . . . . . . . . 9  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
347, 33syl 16 . . . . . . . 8  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
3534, 14ffvelrnd 5946 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  ( Base `  R ) )
36 deg1mul2.e . . . . . . . 8  |-  E  =  (RLReg `  R )
3736, 32, 25, 27rrgeq0i 17475 . . . . . . 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 661 . . . . . 6  |-  ( ph  ->  ( ( ( (coe1 `  F ) `  ( D `  F )
) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) )  =  ( 0g `  R )  ->  ( (coe1 `  G
) `  ( D `  G ) )  =  ( 0g `  R
) ) )
3938necon3d 2672 . . . . 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 2741 . . 3  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =/=  ( 0g `  R ) )
42 eqid 2451 . . . 4  |-  (coe1 `  ( F  .x.  G ) )  =  (coe1 `  ( F  .x.  G ) )
432, 1, 4, 27, 42deg1ge 21696 . . 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 1219 . 2  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  <_  ( D `  ( F  .x.  G
) ) )
452, 1, 4deg1xrcl 21679 . . . 4  |-  ( ( F  .x.  G )  e.  B  ->  ( D `  ( F  .x.  G ) )  e. 
RR* )
4623, 45syl 16 . . 3  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  e.  RR* )
4724nn0red 10741 . . . 4  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR )
4847rexrd 9537 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR* )
49 xrletri3 11233 . . 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 661 . 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 913 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( ( D `
 F )  +  ( D `  G
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   -->wf 5515   ` cfv 5519  (class class class)co 6193    + caddc 9389   RR*cxr 9521    <_ cle 9523   NN0cn0 10683   Basecbs 14285   .rcmulr 14350   0gc0g 14489   Ringcrg 16760  RLRegcrlreg 17465  Poly1cpl1 17749  coe1cco1 17750   deg1 cdg1 21649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-ofr 6424  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-0g 14491  df-gsum 14492  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-mulg 15659  df-subg 15789  df-ghm 15856  df-cntz 15946  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-cring 16763  df-subrg 16978  df-rlreg 17469  df-psr 17538  df-mpl 17540  df-opsr 17542  df-psr1 17752  df-ply1 17754  df-coe1 17755  df-cnfld 17937  df-mdeg 21650  df-deg1 21651
This theorem is referenced by:  ply1domn  21721  ply1divmo  21733  fta1glem1  21763  mon1psubm  29715  deg1mhm  29716
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