Theorem dgrlt 24022

Description: Two ways to say that the degree of F is strictly less than N. (Contributed by Mario Carneiro, 25-Jul-2014.)

Hypotheses

Ref	Expression
dgreq0.1	|- N = (deg ` F )
dgreq0.2	|- A = (coeff ` F )

Assertion

Ref	Expression
dgrlt	|- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( ( F = 0p \/ N < M ) <-> ( N <_ M /\ ( A ` M ) = 0 ) ) )

Proof of Theorem dgrlt

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> F = 0p )
2	1	fveq2d 6195	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> (deg ` F ) = (deg ` 0p ) )
3		dgreq0.1	. . . . . 6 |- N = (deg ` F )
4		dgr0 24018	. . . . . . 7 |- (deg ` 0p ) = 0
5	4	eqcomi 2631	. . . . . 6 |- 0 = (deg ` 0p )
6	2, 3, 5	3eqtr4g 2681	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> N = 0 )
7		nn0ge0 11318	. . . . . 6 |- ( M e. NN0 -> 0 <_ M )
8	7	ad2antlr 763	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> 0 <_ M )
9	6, 8	eqbrtrd 4675	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> N <_ M )
10	1	fveq2d 6195	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> (coeff ` F ) = (coeff ` 0p ) )
11		dgreq0.2	. . . . . . 7 |- A = (coeff ` F )
12		coe0 24012	. . . . . . . 8 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
13	12	eqcomi 2631	. . . . . . 7 |- ( NN0 X. { 0 } ) = (coeff ` 0p )
14	10, 11, 13	3eqtr4g 2681	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> A = ( NN0 X. { 0 } ) )
15	14	fveq1d 6193	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> ( A ` M ) = ( ( NN0 X. { 0 } ) ` M ) )
16		c0ex 10034	. . . . . . 7 |- 0 e. _V
17	16	fvconst2 6469	. . . . . 6 |- ( M e. NN0 -> ( ( NN0 X. { 0 } ) ` M ) = 0 )
18	17	ad2antlr 763	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> ( ( NN0 X. { 0 } ) ` M ) = 0 )
19	15, 18	eqtrd 2656	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> ( A ` M ) = 0 )
20	9, 19	jca 554	. . 3 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ F = 0p ) -> ( N <_ M /\ ( A ` M ) = 0 ) )
21		dgrcl 23989	. . . . . . . 8 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
22	3, 21	syl5eqel 2705	. . . . . . 7 |- ( F e. (Poly ` S ) -> N e. NN0 )
23	22	nn0red 11352	. . . . . 6 |- ( F e. (Poly ` S ) -> N e. RR )
24		nn0re 11301	. . . . . 6 |- ( M e. NN0 -> M e. RR )
25		ltle 10126	. . . . . 6 |- ( ( N e. RR /\ M e. RR ) -> ( N < M -> N <_ M ) )
26	23, 24, 25	syl2an 494	. . . . 5 |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( N < M -> N <_ M ) )
27	26	imp 445	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ N < M ) -> N <_ M )
28	11, 3	dgrub 23990	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M <_ N )
29	28	3expia 1267	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( ( A ` M ) =/= 0 -> M <_ N ) )
30		lenlt 10116	. . . . . . . 8 |- ( ( M e. RR /\ N e. RR ) -> ( M <_ N <-> -. N < M ) )
31	24, 23, 30	syl2anr 495	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( M <_ N <-> -. N < M ) )
32	29, 31	sylibd 229	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( ( A ` M ) =/= 0 -> -. N < M ) )
33	32	necon4ad 2813	. . . . 5 |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( N < M -> ( A ` M ) = 0 ) )
34	33	imp 445	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ N < M ) -> ( A ` M ) = 0 )
35	27, 34	jca 554	. . 3 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ N < M ) -> ( N <_ M /\ ( A ` M ) = 0 ) )
36	20, 35	jaodan 826	. 2 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( F = 0p \/ N < M ) ) -> ( N <_ M /\ ( A ` M ) = 0 ) )
37		leloe 10124	. . . . . . 7 |- ( ( N e. RR /\ M e. RR ) -> ( N <_ M <-> ( N < M \/ N = M ) ) )
38	23, 24, 37	syl2an 494	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( N <_ M <-> ( N < M \/ N = M ) ) )
39	38	biimpa 501	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ N <_ M ) -> ( N < M \/ N = M ) )
40	39	adantrr 753	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( N < M \/ N = M ) )
41		fveq2 6191	. . . . . 6 |- ( N = M -> ( A ` N ) = ( A ` M ) )
42	3, 11	dgreq0 24021	. . . . . . . 8 |- ( F e. (Poly ` S ) -> ( F = 0p <-> ( A ` N ) = 0 ) )
43	42	ad2antrr 762	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( F = 0p <-> ( A ` N ) = 0 ) )
44		simprr 796	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( A ` M ) = 0 )
45	44	eqeq2d 2632	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( ( A ` N ) = ( A ` M ) <-> ( A ` N ) = 0 ) )
46	43, 45	bitr4d 271	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( F = 0p <-> ( A ` N ) = ( A ` M ) ) )
47	41, 46	syl5ibr 236	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( N = M -> F = 0p ) )
48	47	orim2d 885	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( ( N < M \/ N = M ) -> ( N < M \/ F = 0p ) ) )
49	40, 48	mpd 15	. . 3 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( N < M \/ F = 0p ) )
50	49	orcomd 403	. 2 |- ( ( ( F e. (Poly ` S ) /\ M e. NN0 ) /\ ( N <_ M /\ ( A ` M ) = 0 ) ) -> ( F = 0p \/ N < M ) )
51	36, 50	impbida 877	1 |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( ( F = 0p \/ N < M ) <-> ( N <_ M /\ ( A ` M ) = 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {csn 4177   class class class wbr 4653   X. cxp 5112   ` cfv 5888   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   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:  dgrcolem2  24030  plydivlem4  24051  plydiveu  24053  dgrsub2  37705  elaa2lem  40450