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Theorem dgrlt 26219

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 6835	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (deg‘𝐹) = (deg‘0_𝑝))
3		dgreq0.1	. . . . . 6 ⊢ 𝑁 = (deg‘𝐹)
4		dgr0 26215	. . . . . . 7 ⊢ (deg‘0_𝑝) = 0
5	4	eqcomi 2742	. . . . . 6 ⊢ 0 = (deg‘0_𝑝)
6	2, 3, 5	3eqtr4g 2793	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 = 0)
7		nn0ge0 12417	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
8	7	ad2antlr 727	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 0 ≤ 𝑀)
9	6, 8	eqbrtrd 5117	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝑁 ≤ 𝑀)
10	1	fveq2d 6835	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (coeff‘𝐹) = (coeff‘0_𝑝))
11		dgreq0.2	. . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹)
12		coe0 26208	. . . . . . . 8 ⊢ (coeff‘0_𝑝) = (ℕ₀ × {0})
13	12	eqcomi 2742	. . . . . . 7 ⊢ (ℕ₀ × {0}) = (coeff‘0_𝑝)
14	10, 11, 13	3eqtr4g 2793	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → 𝐴 = (ℕ₀ × {0}))
15	14	fveq1d 6833	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ₀) ∧ 𝐹 = 0_𝑝) → (𝐴‘𝑀) = ((ℕ₀ × {0})‘𝑀))
16		c0ex 11117	. . . . . . 7 ⊢ 0 ∈ V
17	16	fvconst2 7147	. . . . . 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 2768	. . . 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 26185	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
22	3, 21	eqeltrid 2837	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
23	22	nn0red 12454	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℝ)
24		nn0re 12401	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
25		ltle 11212	. . . . . 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 26186	. . . . . . . 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 11202	. . . . . . . 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 2948	. . . . 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 11210	. . . . . . 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 6831	. . . . . 6 ⊢ (𝑁 = 𝑀 → (𝐴‘𝑁) = (𝐴‘𝑀))
42	3, 11	dgreq0 26218	. . . . . . . 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 2744	. . . . . . 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 1541 ∈ wcel 2113 ≠ wne 2929 {csn 4577 class class class wbr 5095 × cxp 5619 ‘cfv 6489 ℝcr 11016 0cc0 11017 < clt 11157 ≤ cle 11158 ℕ₀cn0 12392 0_𝑝c0p 25617 Polycply 26136 coeffccoe 26138 degcdgr 26139

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-pm 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-fz 13415 df-fzo 13562 df-fl 13703 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-clim 15402 df-rlim 15403 df-sum 15601 df-0p 25618 df-ply 26140 df-coe 26142 df-dgr 26143

This theorem is referenced by: dgrcolem2 26227 plydivlem4 26251 plydiveu 26253 dgrsub2 43292 elaa2lem 46393

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