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Theorem plyrem 24880
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15869). If a polynomial 𝐹 is divided by the linear factor 𝑥𝐴, the remainder is equal to 𝐹(𝐴), the evaluation of the polynomial at 𝐴 (interpreted as a constant polynomial). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
plyrem.1 𝐺 = (Xpf − (ℂ × {𝐴}))
plyrem.2 𝑅 = (𝐹f − (𝐺f · (𝐹 quot 𝐺)))
Assertion
Ref Expression
plyrem ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))

Proof of Theorem plyrem
StepHypRef Expression
1 plyssc 24776 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simpl 485 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3944 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4 plyrem.1 . . . . . . . . . 10 𝐺 = (Xpf − (ℂ × {𝐴}))
54plyremlem 24879 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))
65adantl 484 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))
76simp1d 1138 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
86simp2d 1139 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9 ax-1ne0 10584 . . . . . . . . . 10 1 ≠ 0
109a1i 11 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
118, 10eqnetrd 3073 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12 fveq2 6646 . . . . . . . . . 10 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13 dgr0 24838 . . . . . . . . . 10 (deg‘0𝑝) = 0
1412, 13syl6eq 2871 . . . . . . . . 9 (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
1514necon3i 3038 . . . . . . . 8 ((deg‘𝐺) ≠ 0 → 𝐺 ≠ 0𝑝)
1611, 15syl 17 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ≠ 0𝑝)
17 plyrem.2 . . . . . . . 8 𝑅 = (𝐹f − (𝐺f · (𝐹 quot 𝐺)))
1817quotdgr 24878 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
193, 7, 16, 18syl3anc 1367 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20 0lt1 11140 . . . . . . . 8 0 < 1
2120, 8breqtrrid 5080 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22 fveq2 6646 . . . . . . . . 9 (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
2322, 13syl6eq 2871 . . . . . . . 8 (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
2423breq1d 5052 . . . . . . 7 (𝑅 = 0𝑝 → ((deg‘𝑅) < (deg‘𝐺) ↔ 0 < (deg‘𝐺)))
2521, 24syl5ibrcom 249 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)))
26 pm2.62 896 . . . . . 6 ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)) → ((𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)) → (deg‘𝑅) < (deg‘𝐺)))
2719, 25, 26sylc 65 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < (deg‘𝐺))
2827, 8breqtrd 5068 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29 quotcl2 24877 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
303, 7, 16, 29syl3anc 1367 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31 plymulcl 24797 . . . . . . . . 9 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
327, 30, 31syl2anc 586 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33 plysubcl 24798 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐹f − (𝐺f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
343, 32, 33syl2anc 586 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹f − (𝐺f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
3517, 34eqeltrid 2915 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36 dgrcl 24809 . . . . . 6 (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
3735, 36syl 17 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38 nn0lt10b 12023 . . . . 5 ((deg‘𝑅) ∈ ℕ0 → ((deg‘𝑅) < 1 ↔ (deg‘𝑅) = 0))
3937, 38syl 17 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((deg‘𝑅) < 1 ↔ (deg‘𝑅) = 0))
4028, 39mpbid 234 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41 0dgrb 24822 . . . 4 (𝑅 ∈ (Poly‘ℂ) → ((deg‘𝑅) = 0 ↔ 𝑅 = (ℂ × {(𝑅‘0)})))
4235, 41syl 17 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((deg‘𝑅) = 0 ↔ 𝑅 = (ℂ × {(𝑅‘0)})))
4340, 42mpbid 234 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
4443fveq1d 6648 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
4517fveq1i 6647 . . . . . . 7 (𝑅𝐴) = ((𝐹f − (𝐺f · (𝐹 quot 𝐺)))‘𝐴)
46 plyf 24774 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
4746adantr 483 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
4847ffnd 6491 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
49 plyf 24774 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
507, 49syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
5150ffnd 6491 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
52 plyf 24774 . . . . . . . . . . . 12 ((𝐹 quot 𝐺) ∈ (Poly‘ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
5330, 52syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
5453ffnd 6491 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
55 cnex 10596 . . . . . . . . . . 11 ℂ ∈ V
5655a1i 11 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
57 inidm 4173 . . . . . . . . . 10 (ℂ ∩ ℂ) = ℂ
5851, 54, 56, 56, 57offn 7398 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺f · (𝐹 quot 𝐺)) Fn ℂ)
59 eqidd 2821 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝐹𝐴))
606simp3d 1140 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 “ {0}) = {𝐴})
61 ssun1 4127 . . . . . . . . . . . . . . 15 (𝐺 “ {0}) ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))
6260, 61eqsstrrdi 4001 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
63 snssg 4693 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6463adantl 484 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6562, 64mpbird 259 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
66 ofmulrt 24857 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ 𝐺:ℂ⟶ℂ ∧ (𝐹 quot 𝐺):ℂ⟶ℂ) → ((𝐺f · (𝐹 quot 𝐺)) “ {0}) = ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
6756, 50, 53, 66syl3anc 1367 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺f · (𝐹 quot 𝐺)) “ {0}) = ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
6865, 67eleqtrrd 2914 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺f · (𝐹 quot 𝐺)) “ {0}))
69 fniniseg 6806 . . . . . . . . . . . . 13 ((𝐺f · (𝐹 quot 𝐺)) Fn ℂ → (𝐴 ∈ ((𝐺f · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0)))
7058, 69syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((𝐺f · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0)))
7168, 70mpbid 234 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0))
7271simprd 498 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0)
7372adantr 483 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0)
7448, 58, 56, 56, 57, 59, 73ofval 7396 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹f − (𝐺f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7574anabss3 673 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹f − (𝐺f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7645, 75syl5eq 2867 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((𝐹𝐴) − 0))
7746ffvelrnda 6827 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) ∈ ℂ)
7877subid1d 10964 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹𝐴) − 0) = (𝐹𝐴))
7976, 78eqtrd 2855 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = (𝐹𝐴))
80 fvex 6659 . . . . . . 7 (𝑅‘0) ∈ V
8180fvconst2 6942 . . . . . 6 (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8281adantl 484 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8344, 79, 823eqtr3d 2863 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝑅‘0))
8483sneqd 4555 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹𝐴)} = {(𝑅‘0)})
8584xpeq2d 5561 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹𝐴)}) = (ℂ × {(𝑅‘0)}))
8643, 85eqtr4d 2858 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843  w3a 1083   = wceq 1537  wcel 2114  wne 3006  Vcvv 3473  cun 3911  wss 3913  {csn 4543   class class class wbr 5042   × cxp 5529  ccnv 5530  cima 5534   Fn wfn 6326  wf 6327  cfv 6331  (class class class)co 7133  f cof 7385  cc 10513  0cc0 10515  1c1 10516   · cmul 10520   < clt 10653  cmin 10848  0cn0 11876  0𝑝c0p 24252  Polycply 24760  Xpcidp 24761  degcdgr 24763   quot cquot 24865
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593  ax-addf 10594
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-map 8386  df-pm 8387  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-sup 8884  df-inf 8885  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-n0 11877  df-z 11961  df-uz 12223  df-rp 12369  df-fz 12877  df-fzo 13018  df-fl 13146  df-seq 13354  df-exp 13415  df-hash 13676  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-clim 14825  df-rlim 14826  df-sum 15023  df-0p 24253  df-ply 24764  df-idp 24765  df-coe 24766  df-dgr 24767  df-quot 24866
This theorem is referenced by:  facth  24881
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