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Theorem dgrlt 26492

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 490	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐹 = 0_𝑝)
2	1	fveq2d 6882	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (deg‘𝐹) = (deg‘0_𝑝))
3		dgreq0.1	. . . . . 6 ⊢ 𝑁 = (deg‘𝐹)
4		dgr0 26488	. . . . . . 7 ⊢ (deg‘0_𝑝) = 0
5	4	eqcomi 2769	. . . . . 6 ⊢ 0 = (deg‘0_𝑝)
6	2, 3, 5	3eqtr4g 2820	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 = 0)
7		nn0ge0 12553	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
8	7	ad2antlr 740	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 0 ≤ 𝑀)
9	6, 8	eqbrtrd 5127	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 ≤ 𝑀)
10	1	fveq2d 6882	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (coeff‘𝐹) = (coeff‘0_𝑝))
11		dgreq0.2	. . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹)
12		coe0 26482	. . . . . . . 8 ⊢ (coeff‘0_𝑝) = (ℕ₀ × {0})
13	12	eqcomi 2769	. . . . . . 7 ⊢ (ℕ₀ × {0}) = (coeff‘0_𝑝)
14	10, 11, 13	3eqtr4g 2820	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐴 = (ℕ₀ × {0}))
15	14	fveq1d 6880	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = ((ℕ₀ × {0})‘𝑀))
16		c0ex 11224	. . . . . . 7 ⊢ 0 ∈ V
17	16	fvconst2 7203	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → ((ℕ₀ × {0})‘𝑀) = 0)
18	17	ad2antlr 740	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → ((ℕ₀ × {0})‘𝑀) = 0)
19	15, 18	eqtrd 2795	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = 0)
20	9, 19	jca 521	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
21		dgrcl 26459	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
22	3, 21	eqeltrid 2864	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
23	22	nn0red 12590	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℝ)
24		nn0re 12537	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
25		ltle 11322	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀))
26	23, 24, 25	syl2an 608	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀))
27	26	imp 412	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → 𝑁 ≤ 𝑀)
28	11, 3	dgrub 26460	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀ ∧ (𝐴‘𝑀) ≠ 0) → 𝑀 ≤ 𝑁)
29	28	3expia 1139	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐴‘𝑀) ≠ 0 → 𝑀 ≤ 𝑁))
30		lenlt 11312	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
31	24, 23, 30	syl2anr 609	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
32	29, 31	sylibd 242	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐴‘𝑀) ≠ 0 → ¬ 𝑁 < 𝑀))
33	32	necon4ad 2974	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 < 𝑀 → (𝐴‘𝑀) = 0))
34	33	imp 412	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → (𝐴‘𝑀) = 0)
35	27, 34	jca 521	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 < 𝑀) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
36	20, 35	jaodan 972	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀)) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))
37		leloe 11320	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)))
38	23, 24, 37	syl2an 608	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)))
39	38	biimpa 482	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
40	39	adantrr 730	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))
41		fveq2 6878	. . . . . 6 ⊢ (𝑁 = 𝑀 → (𝐴‘𝑁) = (𝐴‘𝑀))
42	3, 11	dgreq0 26491	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
43	42	ad2antrr 739	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
44		simprr 785	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐴‘𝑀) = 0)
45	44	eqeq2d 2771	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → ((𝐴‘𝑁) = (𝐴‘𝑀) ↔ (𝐴‘𝑁) = 0))
46	43, 45	bitr4d 285	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = (𝐴‘𝑀)))
47	41, 46	imbitrrid 249	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 = 𝑀 → 𝐹 = 0_𝑝))
48	47	orim2d 982	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → ((𝑁 < 𝑀 ∨ 𝑁 = 𝑀) → (𝑁 < 𝑀 ∨ 𝐹 = 0_𝑝)))
49	40, 48	mpd 16	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝐹 = 0_𝑝))
50	49	orcomd 885	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀))
51	36, 50	impbida 813	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝐹 = 0_𝑝 ∨ 𝑁 < 𝑀) ↔ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 {csn 4584 class class class wbr 5103 × cxp 5653 ‘cfv 6533 ℝcr 11123 0cc0 11124 < clt 11267 ≤ cle 11268 ℕ₀cn0 12528 0_𝑝c0p 25897 Polycply 26409 coeffccoe 26411 degcdgr 26412

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-z 12616 df-uz 12888 df-rp 13043 df-fz 13562 df-fzo 13710 df-fl 13853 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-rlim 15576 df-sum 15774 df-0p 25898 df-ply 26413 df-coe 26415 df-dgr 26416

This theorem is used by: dgrcolem2 26500 plydivlem4 26526 plydiveu 26528 dgrsub2 43976 elaa2lem 47061

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