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Theorem plyrem 26347
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16580). 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 26239 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simpl 482 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
31, 2sselid 3981 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4 plyrem.1 . . . . . . . . . 10 𝐺 = (Xpf − (ℂ × {𝐴}))
54plyremlem 26346 . . . . . . . . 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 1143 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
86simp2d 1144 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9 ax-1ne0 11224 . . . . . . . . . 10 1 ≠ 0
109a1i 11 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
118, 10eqnetrd 3008 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12 fveq2 6906 . . . . . . . . . 10 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13 dgr0 26302 . . . . . . . . . 10 (deg‘0𝑝) = 0
1412, 13eqtrdi 2793 . . . . . . . . 9 (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
1514necon3i 2973 . . . . . . . 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 26345 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
193, 7, 16, 18syl3anc 1373 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20 0lt1 11785 . . . . . . . 8 0 < 1
2120, 8breqtrrid 5181 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22 fveq2 6906 . . . . . . . . 9 (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
2322, 13eqtrdi 2793 . . . . . . . 8 (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
2423breq1d 5153 . . . . . . 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 pm2.62 900 . . . . . 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 5169 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29 quotcl2 26344 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
303, 7, 16, 29syl3anc 1373 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31 plymulcl 26260 . . . . . . . . 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 26261 . . . . . . . 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 2845 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36 dgrcl 26272 . . . . . 6 (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
3735, 36syl 17 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38 nn0lt10b 12680 . . . . 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 26285 . . . 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 6908 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
4517fveq1i 6907 . . . . . . 7 (𝑅𝐴) = ((𝐹f − (𝐺f · (𝐹 quot 𝐺)))‘𝐴)
46 plyf 26237 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
4746adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
4847ffnd 6737 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
49 plyf 26237 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
507, 49syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
5150ffnd 6737 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
52 plyf 26237 . . . . . . . . . . . 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 6737 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
55 cnex 11236 . . . . . . . . . . 11 ℂ ∈ V
5655a1i 11 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
57 inidm 4227 . . . . . . . . . 10 (ℂ ∩ ℂ) = ℂ
5851, 54, 56, 56, 57offn 7710 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺f · (𝐹 quot 𝐺)) Fn ℂ)
59 eqidd 2738 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝐹𝐴))
606simp3d 1145 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 “ {0}) = {𝐴})
61 ssun1 4178 . . . . . . . . . . . . . . 15 (𝐺 “ {0}) ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))
6260, 61eqsstrrdi 4029 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
63 snssg 4783 . . . . . . . . . . . . . . 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 26323 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺f · (𝐹 quot 𝐺)) “ {0}) = ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
6865, 67eleqtrrd 2844 . . . . . . . . . . . 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 2789 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((𝐹𝐴) − 0))
7746ffvelcdmda 7104 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) ∈ ℂ)
7877subid1d 11609 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹𝐴) − 0) = (𝐹𝐴))
7976, 78eqtrd 2777 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = (𝐹𝐴))
80 fvex 6919 . . . . . . 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 2785 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝑅‘0))
8483sneqd 4638 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹𝐴)} = {(𝑅‘0)})
8584xpeq2d 5715 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹𝐴)}) = (ℂ × {(𝑅‘0)}))
8643, 85eqtr4d 2780 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cun 3949  wss 3951  {csn 4626   class class class wbr 5143   × cxp 5683  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  cc 11153  0cc0 11155  1c1 11156   · cmul 11160   < clt 11295  cmin 11492  0cn0 12526  0𝑝c0p 25704  Polycply 26223  Xpcidp 26224  degcdgr 26226   quot cquot 26332
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227  df-idp 26228  df-coe 26229  df-dgr 26230  df-quot 26333
This theorem is referenced by:  facth  26348
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