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Theorem plyrem 23864
 Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15098). If a polynomial 𝐹 is divided by the linear factor 𝑥 − 𝐴, 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 23760 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simpl 472 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3566 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4 plyrem.1 . . . . . . . . . 10 𝐺 = (Xp𝑓 − (ℂ × {𝐴}))
54plyremlem 23863 . . . . . . . . 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 1066 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
86simp2d 1067 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9 ax-1ne0 9884 . . . . . . . . . 10 1 ≠ 0
109a1i 11 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
118, 10eqnetrd 2849 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12 fveq2 6103 . . . . . . . . . 10 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13 dgr0 23822 . . . . . . . . . 10 (deg‘0𝑝) = 0
1412, 13syl6eq 2660 . . . . . . . . 9 (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
1514necon3i 2814 . . . . . . . 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 23862 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
193, 7, 16, 18syl3anc 1318 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20 0lt1 10429 . . . . . . . 8 0 < 1
2120, 8syl5breqr 4621 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22 fveq2 6103 . . . . . . . . 9 (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
2322, 13syl6eq 2660 . . . . . . . 8 (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
2423breq1d 4593 . . . . . . 7 (𝑅 = 0𝑝 → ((deg‘𝑅) < (deg‘𝐺) ↔ 0 < (deg‘𝐺)))
2521, 24syl5ibrcom 236 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)))
26 pm2.62 424 . . . . . 6 ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)) → ((𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)) → (deg‘𝑅) < (deg‘𝐺)))
2719, 25, 26sylc 63 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < (deg‘𝐺))
2827, 8breqtrd 4609 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29 quotcl2 23861 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
303, 7, 16, 29syl3anc 1318 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31 plymulcl 23781 . . . . . . . . 9 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
327, 30, 31syl2anc 691 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33 plysubcl 23782 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
343, 32, 33syl2anc 691 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
3517, 34syl5eqel 2692 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36 dgrcl 23793 . . . . . 6 (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
3735, 36syl 17 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38 nn0lt10b 11316 . . . . 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 221 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41 0dgrb 23806 . . . 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 221 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
4443fveq1d 6105 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
4517fveq1i 6104 . . . . . . 7 (𝑅𝐴) = ((𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))‘𝐴)
46 plyf 23758 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
4746adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
48 ffn 5958 . . . . . . . . . 10 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
4947, 48syl 17 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
50 plyf 23758 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
517, 50syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
52 ffn 5958 . . . . . . . . . . 11 (𝐺:ℂ⟶ℂ → 𝐺 Fn ℂ)
5351, 52syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
54 plyf 23758 . . . . . . . . . . . 12 ((𝐹 quot 𝐺) ∈ (Poly‘ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
5530, 54syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺):ℂ⟶ℂ)
56 ffn 5958 . . . . . . . . . . 11 ((𝐹 quot 𝐺):ℂ⟶ℂ → (𝐹 quot 𝐺) Fn ℂ)
5755, 56syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) Fn ℂ)
58 cnex 9896 . . . . . . . . . . 11 ℂ ∈ V
5958a1i 11 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
60 inidm 3784 . . . . . . . . . 10 (ℂ ∩ ℂ) = ℂ
6153, 57, 59, 59, 60offn 6806 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺𝑓 · (𝐹 quot 𝐺)) Fn ℂ)
62 eqidd 2611 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝐹𝐴))
636simp3d 1068 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 “ {0}) = {𝐴})
64 ssun1 3738 . . . . . . . . . . . . . . 15 (𝐺 “ {0}) ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))
6563, 64syl6eqssr 3619 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
66 snssg 4268 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6766adantl 481 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6865, 67mpbird 246 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
69 ofmulrt 23841 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ 𝐺:ℂ⟶ℂ ∧ (𝐹 quot 𝐺):ℂ⟶ℂ) → ((𝐺𝑓 · (𝐹 quot 𝐺)) “ {0}) = ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
7059, 51, 55, 69syl3anc 1318 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺𝑓 · (𝐹 quot 𝐺)) “ {0}) = ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
7168, 70eleqtrrd 2691 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺𝑓 · (𝐹 quot 𝐺)) “ {0}))
72 fniniseg 6246 . . . . . . . . . . . . 13 ((𝐺𝑓 · (𝐹 quot 𝐺)) Fn ℂ → (𝐴 ∈ ((𝐺𝑓 · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)))
7361, 72syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((𝐺𝑓 · (𝐹 quot 𝐺)) “ {0}) ↔ (𝐴 ∈ ℂ ∧ ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)))
7471, 73mpbid 221 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0))
7574simprd 478 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)
7675adantr 480 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)
7749, 61, 59, 59, 60, 62, 76ofval 6804 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7877anabss3 860 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7945, 78syl5eq 2656 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((𝐹𝐴) − 0))
8046ffvelrnda 6267 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) ∈ ℂ)
8180subid1d 10260 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹𝐴) − 0) = (𝐹𝐴))
8279, 81eqtrd 2644 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = (𝐹𝐴))
83 fvex 6113 . . . . . . 7 (𝑅‘0) ∈ V
8483fvconst2 6374 . . . . . 6 (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8584adantl 481 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8644, 82, 853eqtr3d 2652 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝑅‘0))
8786sneqd 4137 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹𝐴)} = {(𝑅‘0)})
8887xpeq2d 5063 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹𝐴)}) = (ℂ × {(𝑅‘0)}))
8943, 88eqtr4d 2647 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  {csn 4125   class class class wbr 4583   × cxp 5036  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  ℂcc 9813  0cc0 9815  1c1 9816   · cmul 9820   < clt 9953   − cmin 10145  ℕ0cn0 11169  0𝑝c0p 23242  Polycply 23744  Xpcidp 23745  degcdgr 23747   quot cquot 23849 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-idp 23749  df-coe 23750  df-dgr 23751  df-quot 23850 This theorem is referenced by:  facth  23865
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