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Theorem dgrlt 26179

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 6865	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (deg‘𝐹) = (deg‘0_𝑝))
3		dgreq0.1	. . . . . 6 ⊢ 𝑁 = (deg‘𝐹)
4		dgr0 26175	. . . . . . 7 ⊢ (deg‘0_𝑝) = 0
5	4	eqcomi 2739	. . . . . 6 ⊢ 0 = (deg‘0_𝑝)
6	2, 3, 5	3eqtr4g 2790	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 = 0)
7		nn0ge0 12474	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
8	7	ad2antlr 727	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 0 ≤ 𝑀)
9	6, 8	eqbrtrd 5132	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 ≤ 𝑀)
10	1	fveq2d 6865	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (coeff‘𝐹) = (coeff‘0_𝑝))
11		dgreq0.2	. . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹)
12		coe0 26168	. . . . . . . 8 ⊢ (coeff‘0_𝑝) = (ℕ₀ × {0})
13	12	eqcomi 2739	. . . . . . 7 ⊢ (ℕ₀ × {0}) = (coeff‘0_𝑝)
14	10, 11, 13	3eqtr4g 2790	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐴 = (ℕ₀ × {0}))
15	14	fveq1d 6863	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = ((ℕ₀ × {0})‘𝑀))
16		c0ex 11175	. . . . . . 7 ⊢ 0 ∈ V
17	16	fvconst2 7181	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → ((ℕ₀ × {0})‘𝑀) = 0)
18	17	ad2antlr 727	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → ((ℕ₀ × {0})‘𝑀) = 0)
19	15, 18	eqtrd 2765	. . . 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 26145	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
22	3, 21	eqeltrid 2833	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
23	22	nn0red 12511	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℝ)
24		nn0re 12458	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
25		ltle 11269	. . . . . 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 406	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → 𝑁 ≤ 𝑀)
28	11, 3	dgrub 26146	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀ ∧ (𝐴‘𝑀) ≠ 0) → 𝑀 ≤ 𝑁)
29	28	3expia 1121	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐴‘𝑀) ≠ 0 → 𝑀 ≤ 𝑁))
30		lenlt 11259	. . . . . . . 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 239	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐴‘𝑀) ≠ 0 → ¬ 𝑁 < 𝑀))
33	32	necon4ad 2945	. . . . 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 959	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀)) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
37		leloe 11267	. . . . . . 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 476	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
40	39	adantrr 717	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
41		fveq2 6861	. . . . . 6 ⊢ (𝑁 = 𝑀 → (𝐴‘𝑁) = (𝐴‘𝑀))
42	3, 11	dgreq0 26178	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
43	42	ad2antrr 726	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
44		simprr 772	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐴‘𝑀) = 0)
45	44	eqeq2d 2741	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → ((𝐴‘𝑁) = (𝐴‘𝑀) ↔ (𝐴‘𝑁) = 0))
46	43, 45	bitr4d 282	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = (𝐴‘𝑀)))
47	41, 46	imbitrrid 246	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 = 𝑀 → 𝐹 = 0_𝑝))
48	47	orim2d 968	. . . 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 871	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀))
51	36, 50	impbida 800	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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 {csn 4592 class class class wbr 5110 × cxp 5639 ‘cfv 6514 ℝcr 11074 0cc0 11075 < clt 11215 ≤ cle 11216 ℕ₀cn0 12449 0_𝑝c0p 25577 Polycply 26096 coeffccoe 26098 degcdgr 26099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-0p 25578 df-ply 26100 df-coe 26102 df-dgr 26103

This theorem is referenced by: dgrcolem2 26187 plydivlem4 26211 plydiveu 26213 dgrsub2 43131 elaa2lem 46238

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