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Theorem plyrem 24280
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15468). 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 24176 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simpl 468 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3750 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℂ))
4 plyrem.1 . . . . . . . . . 10 𝐺 = (Xp𝑓 − (ℂ × {𝐴}))
54plyremlem 24279 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))
65adantl 467 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))
76simp1d 1136 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 ∈ (Poly‘ℂ))
86simp2d 1137 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) = 1)
9 ax-1ne0 10207 . . . . . . . . . 10 1 ≠ 0
109a1i 11 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 1 ≠ 0)
118, 10eqnetrd 3010 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝐺) ≠ 0)
12 fveq2 6332 . . . . . . . . . 10 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
13 dgr0 24238 . . . . . . . . . 10 (deg‘0𝑝) = 0
1412, 13syl6eq 2821 . . . . . . . . 9 (𝐺 = 0𝑝 → (deg‘𝐺) = 0)
1514necon3i 2975 . . . . . . . 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 24278 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
193, 7, 16, 18syl3anc 1476 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
20 0lt1 10752 . . . . . . . 8 0 < 1
2120, 8syl5breqr 4824 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 0 < (deg‘𝐺))
22 fveq2 6332 . . . . . . . . 9 (𝑅 = 0𝑝 → (deg‘𝑅) = (deg‘0𝑝))
2322, 13syl6eq 2821 . . . . . . . 8 (𝑅 = 0𝑝 → (deg‘𝑅) = 0)
2423breq1d 4796 . . . . . . 7 (𝑅 = 0𝑝 → ((deg‘𝑅) < (deg‘𝐺) ↔ 0 < (deg‘𝐺)))
2521, 24syl5ibrcom 237 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅 = 0𝑝 → (deg‘𝑅) < (deg‘𝐺)))
26 pm2.62 885 . . . . . 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 4812 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) < 1)
29 quotcl2 24277 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
303, 7, 16, 29syl3anc 1476 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
31 plymulcl 24197 . . . . . . . . 9 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
327, 30, 31syl2anc 573 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ))
33 plysubcl 24198 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
343, 32, 33syl2anc 573 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺))) ∈ (Poly‘ℂ))
3517, 34syl5eqel 2854 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 ∈ (Poly‘ℂ))
36 dgrcl 24209 . . . . . 6 (𝑅 ∈ (Poly‘ℂ) → (deg‘𝑅) ∈ ℕ0)
3735, 36syl 17 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) ∈ ℕ0)
38 nn0lt10b 11641 . . . . 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 222 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (deg‘𝑅) = 0)
41 0dgrb 24222 . . . 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 222 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝑅‘0)}))
4443fveq1d 6334 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((ℂ × {(𝑅‘0)})‘𝐴))
4517fveq1i 6333 . . . . . . 7 (𝑅𝐴) = ((𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))‘𝐴)
46 plyf 24174 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
4746adantr 466 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹:ℂ⟶ℂ)
48 ffn 6185 . . . . . . . . . 10 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
4947, 48syl 17 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℂ)
50 plyf 24174 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘ℂ) → 𝐺:ℂ⟶ℂ)
517, 50syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺:ℂ⟶ℂ)
52 ffn 6185 . . . . . . . . . . 11 (𝐺:ℂ⟶ℂ → 𝐺 Fn ℂ)
5351, 52syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℂ)
54 plyf 24174 . . . . . . . . . . . 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 6185 . . . . . . . . . . 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 10219 . . . . . . . . . . 11 ℂ ∈ V
5958a1i 11 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ℂ ∈ V)
60 inidm 3971 . . . . . . . . . 10 (ℂ ∩ ℂ) = ℂ
6153, 57, 59, 59, 60offn 7055 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺𝑓 · (𝐹 quot 𝐺)) Fn ℂ)
62 eqidd 2772 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝐹𝐴))
636simp3d 1138 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐺 “ {0}) = {𝐴})
64 ssun1 3927 . . . . . . . . . . . . . . 15 (𝐺 “ {0}) ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))
6563, 64syl6eqssr 3805 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
66 snssg 4450 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6766adantl 467 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})) ↔ {𝐴} ⊆ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0}))))
6865, 67mpbird 247 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
69 ofmulrt 24257 . . . . . . . . . . . . . 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 1476 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺𝑓 · (𝐹 quot 𝐺)) “ {0}) = ((𝐺 “ {0}) ∪ ((𝐹 quot 𝐺) “ {0})))
7168, 70eleqtrrd 2853 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ((𝐺𝑓 · (𝐹 quot 𝐺)) “ {0}))
72 fniniseg 6481 . . . . . . . . . . . . 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 222 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0))
7574simprd 483 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)
7675adantr 466 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐺𝑓 · (𝐹 quot 𝐺))‘𝐴) = 0)
7749, 61, 59, 59, 60, 62, 76ofval 7053 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) ∧ 𝐴 ∈ ℂ) → ((𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7877anabss3 654 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))‘𝐴) = ((𝐹𝐴) − 0))
7945, 78syl5eq 2817 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = ((𝐹𝐴) − 0))
8046ffvelrnda 6502 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) ∈ ℂ)
8180subid1d 10583 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((𝐹𝐴) − 0) = (𝐹𝐴))
8279, 81eqtrd 2805 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝑅𝐴) = (𝐹𝐴))
83 fvex 6342 . . . . . . 7 (𝑅‘0) ∈ V
8483fvconst2 6613 . . . . . 6 (𝐴 ∈ ℂ → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8584adantl 467 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(𝑅‘0)})‘𝐴) = (𝑅‘0))
8644, 82, 853eqtr3d 2813 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹𝐴) = (𝑅‘0))
8786sneqd 4328 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → {(𝐹𝐴)} = {(𝑅‘0)})
8887xpeq2d 5279 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (ℂ × {(𝐹𝐴)}) = (ℂ × {(𝑅‘0)}))
8943, 88eqtr4d 2808 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  cun 3721  wss 3723  {csn 4316   class class class wbr 4786   × cxp 5247  ccnv 5248  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  𝑓 cof 7042  cc 10136  0cc0 10138  1c1 10139   · cmul 10143   < clt 10276  cmin 10468  0cn0 11494  0𝑝c0p 23656  Polycply 24160  Xpcidp 24161  degcdgr 24163   quot cquot 24265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-0p 23657  df-ply 24164  df-idp 24165  df-coe 24166  df-dgr 24167  df-quot 24266
This theorem is referenced by:  facth  24281
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