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Theorem plyrem 26362
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16577). 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 26254 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simpl 482 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
31, 2sselid 3993 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4 plyrem.1 . . . . . . . . . 10 𝐺 = (Xpf − (ℂ × {𝐴}))
54plyremlem 26361 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))
65adantl 481 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))
76simp1d 1141 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
86simp2d 1142 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9 ax-1ne0 11222 . . . . . . . . . 10 1 ≠ 0
109a1i 11 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
118, 10eqnetrd 3006 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12 fveq2 6907 . . . . . . . . . 10 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13 dgr0 26317 . . . . . . . . . 10 (deg‘0𝑝) = 0
1412, 13eqtrdi 2791 . . . . . . . . 9 (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
1514necon3i 2971 . . . . . . . 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 26360 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
193, 7, 16, 18syl3anc 1370 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20 0lt1 11783 . . . . . . . 8 0 < 1
2120, 8breqtrrid 5186 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22 fveq2 6907 . . . . . . . . 9 (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
2322, 13eqtrdi 2791 . . . . . . . 8 (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
2423breq1d 5158 . . . . . . 7 (𝑅 = 0𝑝 → ((deg‘𝑅) < (deg‘𝐺) ↔ 0 < (deg‘𝐺)))
2521, 24syl5ibrcom 247 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)))
26 pm2.62 899 . . . . . 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 5174 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29 quotcl2 26359 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
303, 7, 16, 29syl3anc 1370 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31 plymulcl 26275 . . . . . . . . 9 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
327, 30, 31syl2anc 584 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33 plysubcl 26276 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺f · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐹f − (𝐺f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
343, 32, 33syl2anc 584 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹f − (𝐺f · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
3517, 34eqeltrid 2843 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36 dgrcl 26287 . . . . . 6 (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
3735, 36syl 17 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38 nn0lt10b 12678 . . . . 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 232 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41 0dgrb 26300 . . . 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 232 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
4443fveq1d 6909 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
4517fveq1i 6908 . . . . . . 7 (𝑅𝐴) = ((𝐹f − (𝐺f · (𝐹 quot 𝐺)))‘𝐴)
46 plyf 26252 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
4746adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
4847ffnd 6738 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
49 plyf 26252 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
507, 49syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
5150ffnd 6738 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
52 plyf 26252 . . . . . . . . . . . 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 6738 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
55 cnex 11234 . . . . . . . . . . 11 ℂ ∈ V
5655a1i 11 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
57 inidm 4235 . . . . . . . . . 10 (ℂ ∩ ℂ) = ℂ
5851, 54, 56, 56, 57offn 7710 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺f · (𝐹 quot 𝐺)) Fn ℂ)
59 eqidd 2736 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝐹𝐴))
606simp3d 1143 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 “ {0}) = {𝐴})
61 ssun1 4188 . . . . . . . . . . . . . . 15 (𝐺 “ {0}) ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))
6260, 61eqsstrrdi 4051 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
63 snssg 4788 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6463adantl 481 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6562, 64mpbird 257 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
66 ofmulrt 26338 . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺f · (𝐹 quot 𝐺)) “ {0}) = ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
6865, 67eleqtrrd 2842 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺f · (𝐹 quot 𝐺)) “ {0}))
69 fniniseg 7080 . . . . . . . . . . . . 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 232 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0))
7271simprd 495 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0)
7372adantr 480 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺f · (𝐹 quot 𝐺))‘𝐴) = 0)
7448, 58, 56, 56, 57, 59, 73ofval 7708 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹f − (𝐺f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7574anabss3 675 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹f − (𝐺f · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7645, 75eqtrid 2787 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((𝐹𝐴) − 0))
7746ffvelcdmda 7104 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) ∈ ℂ)
7877subid1d 11607 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹𝐴) − 0) = (𝐹𝐴))
7976, 78eqtrd 2775 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = (𝐹𝐴))
80 fvex 6920 . . . . . . 7 (𝑅‘0) ∈ V
8180fvconst2 7224 . . . . . 6 (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8281adantl 481 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8344, 79, 823eqtr3d 2783 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝑅‘0))
8483sneqd 4643 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹𝐴)} = {(𝑅‘0)})
8584xpeq2d 5719 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹𝐴)}) = (ℂ × {(𝑅‘0)}))
8643, 85eqtr4d 2778 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  cun 3961  wss 3963  {csn 4631   class class class wbr 5148   × cxp 5687  ccnv 5688  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  cc 11151  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293  cmin 11490  0cn0 12524  0𝑝c0p 25718  Polycply 26238  Xpcidp 26239  degcdgr 26241   quot cquot 26347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-0p 25719  df-ply 26242  df-idp 26243  df-coe 26244  df-dgr 26245  df-quot 26348
This theorem is referenced by:  facth  26363
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