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Theorem deg1mul2 22681
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 22654 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
113, 6, 8, 10syl3anc 1226 . . 3  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
12 deg1mul2.gz . . . 4  |-  ( ph  ->  G  =/=  .0.  )
132, 1, 9, 4deg1nn0cl 22654 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  ( D `  G )  e.  NN0 )
143, 7, 12, 13syl3anc 1226 . . 3  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
1511nn0red 10849 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  RR )
1615leidd 10115 . . 3  |-  ( ph  ->  ( D `  F
)  <_  ( D `  F ) )
1714nn0red 10849 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  RR )
1817leidd 10115 . . 3  |-  ( ph  ->  ( D `  G
)  <_  ( D `  G ) )
191, 2, 3, 4, 5, 6, 7, 11, 14, 16, 18deg1mulle2 22676 . 2  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( ( D `
 F )  +  ( D `  G
) ) )
201ply1ring 18484 . . . . 5  |-  ( R  e.  Ring  ->  P  e. 
Ring )
213, 20syl 16 . . . 4  |-  ( ph  ->  P  e.  Ring )
224, 5ringcl 17407 . . . 4  |-  ( ( P  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .x.  G )  e.  B )
2321, 6, 7, 22syl3anc 1226 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
2411, 14nn0addcld 10852 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  NN0 )
25 eqid 2454 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
261, 5, 25, 4, 2, 9, 3, 6, 8, 7, 12coe1mul4 22667 . . . 4  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =  ( ( (coe1 `  F ) `  ( D `  F ) ) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) ) )
27 eqid 2454 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
28 eqid 2454 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
292, 1, 9, 4, 27, 28deg1ldg 22658 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  G  =/= 
.0.  )  ->  (
(coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
303, 7, 12, 29syl3anc 1226 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =/=  ( 0g
`  R ) )
31 deg1mul2.fc . . . . . . 7  |-  ( ph  ->  ( (coe1 `  F ) `  ( D `  F ) )  e.  E )
32 eqid 2454 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
3328, 4, 1, 32coe1f 18445 . . . . . . . . 9  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
347, 33syl 16 . . . . . . . 8  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
3534, 14ffvelrnd 6008 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  ( Base `  R ) )
36 deg1mul2.e . . . . . . . 8  |-  E  =  (RLReg `  R )
3736, 32, 25, 27rrgeq0i 18132 . . . . . . 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 659 . . . . . 6  |-  ( ph  ->  ( ( ( (coe1 `  F ) `  ( D `  F )
) ( .r `  R ) ( (coe1 `  G ) `  ( D `  G )
) )  =  ( 0g `  R )  ->  ( (coe1 `  G
) `  ( D `  G ) )  =  ( 0g `  R
) ) )
3938necon3d 2678 . . . . 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 2747 . . 3  |-  ( ph  ->  ( (coe1 `  ( F  .x.  G ) ) `  ( ( D `  F )  +  ( D `  G ) ) )  =/=  ( 0g `  R ) )
42 eqid 2454 . . . 4  |-  (coe1 `  ( F  .x.  G ) )  =  (coe1 `  ( F  .x.  G ) )
432, 1, 4, 27, 42deg1ge 22665 . . 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 1226 . 2  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  <_  ( D `  ( F  .x.  G
) ) )
452, 1, 4deg1xrcl 22648 . . . 4  |-  ( ( F  .x.  G )  e.  B  ->  ( D `  ( F  .x.  G ) )  e. 
RR* )
4623, 45syl 16 . . 3  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  e.  RR* )
4724nn0red 10849 . . . 4  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR )
4847rexrd 9632 . . 3  |-  ( ph  ->  ( ( D `  F )  +  ( D `  G ) )  e.  RR* )
49 xrletri3 11361 . . 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 659 . 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 920 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( ( D `
 F )  +  ( D `  G
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   -->wf 5566   ` cfv 5570  (class class class)co 6270    + caddc 9484   RR*cxr 9616    <_ cle 9618   NN0cn0 10791   Basecbs 14716   .rcmulr 14785   0gc0g 14929   Ringcrg 17393  RLRegcrlreg 18122  Poly1cpl1 18411  coe1cco1 18412   deg1 cdg1 22618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-ofr 6514  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  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 10977  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-mulg 16259  df-subg 16397  df-ghm 16464  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-subrg 17622  df-rlreg 18126  df-psr 18200  df-mpl 18202  df-opsr 18204  df-psr1 18414  df-ply1 18416  df-coe1 18417  df-cnfld 18616  df-mdeg 22619  df-deg1 22620
This theorem is referenced by:  ply1domn  22690  ply1divmo  22702  fta1glem1  22732  mon1psubm  31407  deg1mhm  31408
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