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Theorem dgrlt 25525

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 485	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐹 = 0_𝑝)
2	1	fveq2d 6823	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (deg‘𝐹) = (deg‘0_𝑝))
3		dgreq0.1	. . . . . 6 ⊢ 𝑁 = (deg‘𝐹)
4		dgr0 25521	. . . . . . 7 ⊢ (deg‘0_𝑝) = 0
5	4	eqcomi 2745	. . . . . 6 ⊢ 0 = (deg‘0_𝑝)
6	2, 3, 5	3eqtr4g 2801	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 = 0)
7		nn0ge0 12351	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
8	7	ad2antlr 724	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 0 ≤ 𝑀)
9	6, 8	eqbrtrd 5111	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 ≤ 𝑀)
10	1	fveq2d 6823	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (coeff‘𝐹) = (coeff‘0_𝑝))
11		dgreq0.2	. . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹)
12		coe0 25515	. . . . . . . 8 ⊢ (coeff‘0_𝑝) = (ℕ₀ × {0})
13	12	eqcomi 2745	. . . . . . 7 ⊢ (ℕ₀ × {0}) = (coeff‘0_𝑝)
14	10, 11, 13	3eqtr4g 2801	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐴 = (ℕ₀ × {0}))
15	14	fveq1d 6821	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = ((ℕ₀ × {0})‘𝑀))
16		c0ex 11062	. . . . . . 7 ⊢ 0 ∈ V
17	16	fvconst2 7129	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → ((ℕ₀ × {0})‘𝑀) = 0)
18	17	ad2antlr 724	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → ((ℕ₀ × {0})‘𝑀) = 0)
19	15, 18	eqtrd 2776	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = 0)
20	9, 19	jca 512	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
21		dgrcl 25492	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
22	3, 21	eqeltrid 2841	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
23	22	nn0red 12387	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℝ)
24		nn0re 12335	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
25		ltle 11156	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀))
26	23, 24, 25	syl2an 596	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀))
27	26	imp 407	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → 𝑁 ≤ 𝑀)
28	11, 3	dgrub 25493	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀ ∧ (𝐴‘𝑀) ≠ 0) → 𝑀 ≤ 𝑁)
29	28	3expia 1120	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐴‘𝑀) ≠ 0 → 𝑀 ≤ 𝑁))
30		lenlt 11146	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
31	24, 23, 30	syl2anr 597	. . . . . . 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 2959	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 < 𝑀 → (𝐴‘𝑀) = 0))
34	33	imp 407	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → (𝐴‘𝑀) = 0)
35	27, 34	jca 512	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
36	20, 35	jaodan 955	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀)) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
37		leloe 11154	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)))
38	23, 24, 37	syl2an 596	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)))
39	38	biimpa 477	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
40	39	adantrr 714	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
41		fveq2 6819	. . . . . 6 ⊢ (𝑁 = 𝑀 → (𝐴‘𝑁) = (𝐴‘𝑀))
42	3, 11	dgreq0 25524	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
43	42	ad2antrr 723	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
44		simprr 770	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐴‘𝑀) = 0)
45	44	eqeq2d 2747	. . . . . . 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 964	. . . 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 868	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀))
51	36, 50	impbida 798	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 396 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 {csn 4572 class class class wbr 5089 × cxp 5612 ‘cfv 6473 ℝcr 10963 0cc0 10964 < clt 11102 ≤ cle 11103 ℕ₀cn0 12326 0_𝑝c0p 24931 Polycply 25443 coeffccoe 25445 degcdgr 25446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-pm 8681 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-sup 9291 df-inf 9292 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-n0 12327 df-z 12413 df-uz 12676 df-rp 12824 df-fz 13333 df-fzo 13476 df-fl 13605 df-seq 13815 df-exp 13876 df-hash 14138 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-clim 15288 df-rlim 15289 df-sum 15489 df-0p 24932 df-ply 25447 df-coe 25449 df-dgr 25450

This theorem is referenced by: dgrcolem2 25533 plydivlem4 25554 plydiveu 25556 dgrsub2 41211 elaa2lem 44099

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