Theorem dgrsub2 37705

Description: Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)

Hypothesis

Ref	Expression
dgrsub2.a	|- N = (deg ` F )

Assertion

Ref	Expression
dgrsub2	|- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (deg ` ( F oF - G ) ) < N )

Proof of Theorem dgrsub2

Step	Hyp	Ref	Expression
1		simpr2 1068	. . 3 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> N e. NN )
2		dgr0 24018	. . . . 5 |- (deg ` 0p ) = 0
3		nngt0 11049	. . . . 5 |- ( N e. NN -> 0 < N )
4	2, 3	syl5eqbr 4688	. . . 4 |- ( N e. NN -> (deg ` 0p ) < N )
5		fveq2 6191	. . . . 5 |- ( ( F oF - G ) = 0p -> (deg ` ( F oF - G ) ) = (deg ` 0p ) )
6	5	breq1d 4663	. . . 4 |- ( ( F oF - G ) = 0p -> ( (deg ` ( F oF - G ) ) < N <-> (deg ` 0p ) < N ) )
7	4, 6	syl5ibrcom 237	. . 3 |- ( N e. NN -> ( ( F oF - G ) = 0p -> (deg ` ( F oF - G ) ) < N ) )
8	1, 7	syl 17	. 2 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( ( F oF - G ) = 0p -> (deg ` ( F oF - G ) ) < N ) )
9		plyssc 23956	. . . . . . . 8 |- (Poly ` S ) C_ (Poly ` CC )
10	9	sseli 3599	. . . . . . 7 |- ( F e. (Poly ` S ) -> F e. (Poly ` CC ) )
11		plyssc 23956	. . . . . . . 8 |- (Poly ` T ) C_ (Poly ` CC )
12	11	sseli 3599	. . . . . . 7 |- ( G e. (Poly ` T ) -> G e. (Poly ` CC ) )
13		eqid 2622	. . . . . . . 8 |- (deg ` F ) = (deg ` F )
14		eqid 2622	. . . . . . . 8 |- (deg ` G ) = (deg ` G )
15	13, 14	dgrsub 24028	. . . . . . 7 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` CC ) ) -> (deg ` ( F oF - G ) ) <_ if ( (deg ` F ) <_ (deg ` G ) , (deg ` G ) , (deg ` F ) ) )
16	10, 12, 15	syl2an 494	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) -> (deg ` ( F oF - G ) ) <_ if ( (deg ` F ) <_ (deg ` G ) , (deg ` G ) , (deg ` F ) ) )
17	16	adantr 481	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (deg ` ( F oF - G ) ) <_ if ( (deg ` F ) <_ (deg ` G ) , (deg ` G ) , (deg ` F ) ) )
18		simpr1 1067	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (deg ` G ) = N )
19		dgrsub2.a	. . . . . . . . 9 |- N = (deg ` F )
20	19	eqcomi 2631	. . . . . . . 8 |- (deg ` F ) = N
21	20	a1i 11	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (deg ` F ) = N )
22	18, 21	ifeq12d 4106	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> if ( (deg ` F ) <_ (deg ` G ) , (deg ` G ) , (deg ` F ) ) = if ( (deg ` F ) <_ (deg ` G ) , N , N ) )
23		ifid 4125	. . . . . 6 |- if ( (deg ` F ) <_ (deg ` G ) , N , N ) = N
24	22, 23	syl6eq 2672	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> if ( (deg ` F ) <_ (deg ` G ) , (deg ` G ) , (deg ` F ) ) = N )
25	17, 24	breqtrd 4679	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (deg ` ( F oF - G ) ) <_ N )
26		eqid 2622	. . . . . . . . 9 |- (coeff ` F ) = (coeff ` F )
27		eqid 2622	. . . . . . . . 9 |- (coeff ` G ) = (coeff ` G )
28	26, 27	coesub 24013	. . . . . . . 8 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` CC ) ) -> (coeff ` ( F oF - G ) ) = ( (coeff ` F ) oF - (coeff ` G ) ) )
29	10, 12, 28	syl2an 494	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) -> (coeff ` ( F oF - G ) ) = ( (coeff ` F ) oF - (coeff ` G ) ) )
30	29	adantr 481	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (coeff ` ( F oF - G ) ) = ( (coeff ` F ) oF - (coeff ` G ) ) )
31	30	fveq1d 6193	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( (coeff ` ( F oF - G ) ) ` N ) = ( ( (coeff ` F ) oF - (coeff ` G ) ) ` N ) )
32	1	nnnn0d 11351	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> N e. NN0 )
33	26	coef3 23988	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
34	33	ad2antrr 762	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (coeff ` F ) : NN0 --> CC )
35		ffn 6045	. . . . . . . 8 |- ( (coeff ` F ) : NN0 --> CC -> (coeff ` F ) Fn NN0 )
36	34, 35	syl 17	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (coeff ` F ) Fn NN0 )
37	27	coef3 23988	. . . . . . . . 9 |- ( G e. (Poly ` T ) -> (coeff ` G ) : NN0 --> CC )
38	37	ad2antlr 763	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (coeff ` G ) : NN0 --> CC )
39		ffn 6045	. . . . . . . 8 |- ( (coeff ` G ) : NN0 --> CC -> (coeff ` G ) Fn NN0 )
40	38, 39	syl 17	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (coeff ` G ) Fn NN0 )
41		nn0ex 11298	. . . . . . . 8 |- NN0 e. _V
42	41	a1i 11	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> NN0 e. _V )
43		inidm 3822	. . . . . . 7 |- ( NN0 i^i NN0 ) = NN0
44		simplr3 1105	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) /\ N e. NN0 ) -> ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) )
45		eqidd 2623	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) /\ N e. NN0 ) -> ( (coeff ` G ) ` N ) = ( (coeff ` G ) ` N ) )
46	36, 40, 42, 42, 43, 44, 45	ofval 6906	. . . . . 6 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) /\ N e. NN0 ) -> ( ( (coeff ` F ) oF - (coeff ` G ) ) ` N ) = ( ( (coeff ` G ) ` N ) - ( (coeff ` G ) ` N ) ) )
47	32, 46	mpdan 702	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( ( (coeff ` F ) oF - (coeff ` G ) ) ` N ) = ( ( (coeff ` G ) ` N ) - ( (coeff ` G ) ` N ) ) )
48	38, 32	ffvelrnd 6360	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( (coeff ` G ) ` N ) e. CC )
49	48	subidd 10380	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( ( (coeff ` G ) ` N ) - ( (coeff ` G ) ` N ) ) = 0 )
50	31, 47, 49	3eqtrd 2660	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( (coeff ` ( F oF - G ) ) ` N ) = 0 )
51		plysubcl 23978	. . . . . . 7 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` CC ) ) -> ( F oF - G ) e. (Poly ` CC ) )
52	10, 12, 51	syl2an 494	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) -> ( F oF - G ) e. (Poly ` CC ) )
53	52	adantr 481	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( F oF - G ) e. (Poly ` CC ) )
54		eqid 2622	. . . . . 6 |- (deg ` ( F oF - G ) ) = (deg ` ( F oF - G ) )
55		eqid 2622	. . . . . 6 |- (coeff ` ( F oF - G ) ) = (coeff ` ( F oF - G ) )
56	54, 55	dgrlt 24022	. . . . 5 |- ( ( ( F oF - G ) e. (Poly ` CC ) /\ N e. NN0 ) -> ( ( ( F oF - G ) = 0p \/ (deg ` ( F oF - G ) ) < N ) <-> ( (deg ` ( F oF - G ) ) <_ N /\ ( (coeff ` ( F oF - G ) ) ` N ) = 0 ) ) )
57	53, 32, 56	syl2anc 693	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( ( ( F oF - G ) = 0p \/ (deg ` ( F oF - G ) ) < N ) <-> ( (deg ` ( F oF - G ) ) <_ N /\ ( (coeff ` ( F oF - G ) ) ` N ) = 0 ) ) )
58	25, 50, 57	mpbir2and 957	. . 3 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( ( F oF - G ) = 0p \/ (deg ` ( F oF - G ) ) < N ) )
59	58	ord 392	. 2 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> ( -. ( F oF - G ) = 0p -> (deg ` ( F oF - G ) ) < N ) )
60	8, 59	pm2.61d 170	1 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` T ) ) /\ ( (deg ` G ) = N /\ N e. NN /\ ( (coeff ` F ) ` N ) = ( (coeff ` G ) ` N ) ) ) -> (deg ` ( F oF - G ) ) < N )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   class class class wbr 4653   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  mpaaeu  37720