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Theorem dgrlt 25332

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 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀) ↔ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)))

**Proof of Theorem dgrlt**
Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐹 = 0_𝑝)
2	1	fveq2d 6760	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (deg‘𝐹) = (deg‘0_𝑝))
3		dgreq0.1	. . . . . 6 ⊢ 𝑁 = (deg‘𝐹)
4		dgr0 25328	. . . . . . 7 ⊢ (deg‘0_𝑝) = 0
5	4	eqcomi 2747	. . . . . 6 ⊢ 0 = (deg‘0_𝑝)
6	2, 3, 5	3eqtr4g 2804	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 = 0)
7		nn0ge0 12188	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
8	7	ad2antlr 723	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 0 ≤ 𝑀)
9	6, 8	eqbrtrd 5092	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 ≤ 𝑀)
10	1	fveq2d 6760	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (coeff‘𝐹) = (coeff‘0_𝑝))
11		dgreq0.2	. . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹)
12		coe0 25322	. . . . . . . 8 ⊢ (coeff‘0_𝑝) = (ℕ₀ × {0})
13	12	eqcomi 2747	. . . . . . 7 ⊢ (ℕ₀ × {0}) = (coeff‘0_𝑝)
14	10, 11, 13	3eqtr4g 2804	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐴 = (ℕ₀ × {0}))
15	14	fveq1d 6758	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = ((ℕ₀ × {0})‘𝑀))
16		c0ex 10900	. . . . . . 7 ⊢ 0 ∈ V
17	16	fvconst2 7061	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → ((ℕ₀ × {0})‘𝑀) = 0)
18	17	ad2antlr 723	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → ((ℕ₀ × {0})‘𝑀) = 0)
19	15, 18	eqtrd 2778	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = 0)
20	9, 19	jca 511	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
21		dgrcl 25299	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
22	3, 21	eqeltrid 2843	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
23	22	nn0red 12224	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℝ)
24		nn0re 12172	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
25		ltle 10994	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀))
26	23, 24, 25	syl2an 595	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀))
27	26	imp 406	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → 𝑁 ≤ 𝑀)
28	11, 3	dgrub 25300	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀ ∧ (𝐴‘𝑀) ≠ 0) → 𝑀 ≤ 𝑁)
29	28	3expia 1119	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐴‘𝑀) ≠ 0 → 𝑀 ≤ 𝑁))
30		lenlt 10984	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
31	24, 23, 30	syl2anr 596	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
32	29, 31	sylibd 238	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐴‘𝑀) ≠ 0 → ¬ 𝑁 < 𝑀))
33	32	necon4ad 2961	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 < 𝑀 → (𝐴‘𝑀) = 0))
34	33	imp 406	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → (𝐴‘𝑀) = 0)
35	27, 34	jca 511	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
36	20, 35	jaodan 954	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀)) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
37		leloe 10992	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)))
38	23, 24, 37	syl2an 595	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)))
39	38	biimpa 476	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
40	39	adantrr 713	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
41		fveq2 6756	. . . . . 6 ⊢ (𝑁 = 𝑀 → (𝐴‘𝑁) = (𝐴‘𝑀))
42	3, 11	dgreq0 25331	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
43	42	ad2antrr 722	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
44		simprr 769	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐴‘𝑀) = 0)
45	44	eqeq2d 2749	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → ((𝐴‘𝑁) = (𝐴‘𝑀) ↔ (𝐴‘𝑁) = 0))
46	43, 45	bitr4d 281	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = (𝐴‘𝑀)))
47	41, 46	syl5ibr 245	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 = 𝑀 → 𝐹 = 0_𝑝))
48	47	orim2d 963	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → ((𝑁 < 𝑀 ∨ 𝑁 = 𝑀) → (𝑁 < 𝑀 ∨ 𝐹 = 0_𝑝)))
49	40, 48	mpd 15	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝐹 = 0_𝑝))
50	49	orcomd 867	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀))
51	36, 50	impbida 797	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀) ↔ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 {csn 4558 class class class wbr 5070 × cxp 5578 ‘cfv 6418 ℝcr 10801 0cc0 10802 < clt 10940 ≤ cle 10941 ℕ₀cn0 12163 0_𝑝c0p 24738 Polycply 25250 coeffccoe 25252 degcdgr 25253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-0p 24739 df-ply 25254 df-coe 25256 df-dgr 25257

This theorem is referenced by: dgrcolem2 25340 plydivlem4 25361 plydiveu 25363 dgrsub2 40876 elaa2lem 43664

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