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Theorem sineq0ALT 43698
Description: A complex number whose sine is zero is an integer multiple of Ο€. The Virtual Deduction form of the proof is https://us.metamath.org/other/completeusersproof/sineq0altvd.html. The Metamath form of the proof is sineq0ALT 43698. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 26033. The Virtual Deduction proof is verified by automatically transforming it into the Metamath form of the proof using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/sineq0altro.html 26033 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sineq0ALT (𝐴 ∈ β„‚ β†’ ((sinβ€˜π΄) = 0 ↔ (𝐴 / Ο€) ∈ β„€))

Proof of Theorem sineq0ALT
StepHypRef Expression
1 pire 25968 . . . . 5 Ο€ ∈ ℝ
2 pipos 25970 . . . . 5 0 < Ο€
31, 2elrpii 12977 . . . 4 Ο€ ∈ ℝ+
4 2ne0 12316 . . . . . 6 2 β‰  0
54a1i 11 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 2 β‰  0)
6 2cn 12287 . . . . . . 7 2 ∈ β„‚
7 2re 12286 . . . . . . . 8 2 ∈ ℝ
87a1i 11 . . . . . . 7 (2 ∈ β„‚ β†’ 2 ∈ ℝ)
96, 8ax-mp 5 . . . . . 6 2 ∈ ℝ
109a1i 11 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 2 ∈ ℝ)
11 id 22 . . . . . 6 (𝐴 ∈ β„‚ β†’ 𝐴 ∈ β„‚)
1211adantr 482 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 𝐴 ∈ β„‚)
136a1i 11 . . . . . . 7 (𝐴 ∈ β„‚ β†’ 2 ∈ β„‚)
1413, 11mulcld 11234 . . . . . 6 (𝐴 ∈ β„‚ β†’ (2 Β· 𝐴) ∈ β„‚)
15 ax-icn 11169 . . . . . . . . . . . . . . 15 i ∈ β„‚
1615a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ i ∈ β„‚)
1713, 16, 11mul12d 11423 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (2 Β· (i Β· 𝐴)) = (i Β· (2 Β· 𝐴)))
1816, 11mulcld 11234 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (i Β· 𝐴) ∈ β„‚)
19182timesd 12455 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (2 Β· (i Β· 𝐴)) = ((i Β· 𝐴) + (i Β· 𝐴)))
2017, 19eqtr3d 2775 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (i Β· (2 Β· 𝐴)) = ((i Β· 𝐴) + (i Β· 𝐴)))
2120fveq2d 6896 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = (expβ€˜((i Β· 𝐴) + (i Β· 𝐴))))
22 efadd 16037 . . . . . . . . . . . 12 (((i Β· 𝐴) ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (expβ€˜((i Β· 𝐴) + (i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
2318, 18, 22syl2anc 585 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (expβ€˜((i Β· 𝐴) + (i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
2421, 23eqtrd 2773 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
2524adantr 482 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
26 sinval 16065 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ (sinβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)))
27 id 22 . . . . . . . . . . . . . . 15 ((sinβ€˜π΄) = 0 β†’ (sinβ€˜π΄) = 0)
2826, 27sylan9req 2794 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)) = 0)
29 efcl 16026 . . . . . . . . . . . . . . . . . 18 ((i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
3018, 29syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
31 negicn 11461 . . . . . . . . . . . . . . . . . . . 20 -i ∈ β„‚
3231a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ -i ∈ β„‚)
3332, 11mulcld 11234 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„‚ β†’ (-i Β· 𝐴) ∈ β„‚)
34 efcl 16026 . . . . . . . . . . . . . . . . . 18 ((-i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜(-i Β· 𝐴)) ∈ β„‚)
3533, 34syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ (expβ€˜(-i Β· 𝐴)) ∈ β„‚)
3630, 35subcld 11571 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) ∈ β„‚)
37 2mulicn 12435 . . . . . . . . . . . . . . . . 17 (2 Β· i) ∈ β„‚
3837a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ (2 Β· i) ∈ β„‚)
39 2muline0 12436 . . . . . . . . . . . . . . . . 17 (2 Β· i) β‰  0
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ (2 Β· i) β‰  0)
4136, 38, 40diveq0ad 12000 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ ((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)) = 0 ↔ ((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) = 0))
4241adantr 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)) = 0 ↔ ((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) = 0))
4328, 42mpbid 231 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) = 0)
4430, 35subeq0ad 11581 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) = 0 ↔ (expβ€˜(i Β· 𝐴)) = (expβ€˜(-i Β· 𝐴))))
4544adantr 482 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) = 0 ↔ (expβ€˜(i Β· 𝐴)) = (expβ€˜(-i Β· 𝐴))))
4643, 45mpbid 231 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜(i Β· 𝐴)) = (expβ€˜(-i Β· 𝐴)))
4746oveq2d 7425 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(-i Β· 𝐴))))
48 efadd 16037 . . . . . . . . . . . . 13 (((i Β· 𝐴) ∈ β„‚ ∧ (-i Β· 𝐴) ∈ β„‚) β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(-i Β· 𝐴))))
4918, 33, 48syl2anc 585 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(-i Β· 𝐴))))
5049adantr 482 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(-i Β· 𝐴))))
5147, 50eqtr4d 2776 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) = (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))))
5216, 32, 11adddird 11239 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ ((i + -i) Β· 𝐴) = ((i Β· 𝐴) + (-i Β· 𝐴)))
5315negidi 11529 . . . . . . . . . . . . . . 15 (i + -i) = 0
5453oveq1i 7419 . . . . . . . . . . . . . 14 ((i + -i) Β· 𝐴) = (0 Β· 𝐴)
5552, 54eqtr3di 2788 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ ((i Β· 𝐴) + (-i Β· 𝐴)) = (0 Β· 𝐴))
5611mul02d 11412 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (0 Β· 𝐴) = 0)
5755, 56eqtrd 2773 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ ((i Β· 𝐴) + (-i Β· 𝐴)) = 0)
5857fveq2d 6896 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = (expβ€˜0))
5958adantr 482 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = (expβ€˜0))
60 ef0 16034 . . . . . . . . . . 11 (expβ€˜0) = 1
6160a1i 11 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜0) = 1)
6251, 59, 613eqtrd 2777 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) = 1)
6325, 62eqtrd 2773 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = 1)
6463fveq2d 6896 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = (absβ€˜1))
65 abs1 15244 . . . . . . 7 (absβ€˜1) = 1
6664, 65eqtrdi 2789 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1)
67 absefib 16141 . . . . . . . 8 ((2 Β· 𝐴) ∈ β„‚ β†’ ((2 Β· 𝐴) ∈ ℝ ↔ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1))
6867biimparc 481 . . . . . . 7 (((absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1 ∧ (2 Β· 𝐴) ∈ β„‚) β†’ (2 Β· 𝐴) ∈ ℝ)
6968ancoms 460 . . . . . 6 (((2 Β· 𝐴) ∈ β„‚ ∧ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1) β†’ (2 Β· 𝐴) ∈ ℝ)
7014, 66, 69syl2an2r 684 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (2 Β· 𝐴) ∈ ℝ)
71 mulre 15068 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ 2 ∈ ℝ ∧ 2 β‰  0) β†’ (𝐴 ∈ ℝ ↔ (2 Β· 𝐴) ∈ ℝ))
72714animp1 43258 . . . . . 6 ((((𝐴 ∈ β„‚ ∧ 2 ∈ ℝ) ∧ 2 β‰  0) ∧ (2 Β· 𝐴) ∈ ℝ) β†’ 𝐴 ∈ ℝ)
73724an31 43259 . . . . 5 ((((2 β‰  0 ∧ 2 ∈ ℝ) ∧ 𝐴 ∈ β„‚) ∧ (2 Β· 𝐴) ∈ ℝ) β†’ 𝐴 ∈ ℝ)
745, 10, 12, 70, 73syl1111anc 839 . . . 4 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 𝐴 ∈ ℝ)
753a1i 11 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ ∈ ℝ+)
7674, 75modcld 13840 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) ∈ ℝ)
7776recnd 11242 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) ∈ β„‚)
7877sincld 16073 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (sinβ€˜(𝐴 mod Ο€)) ∈ β„‚)
791a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ ∈ ℝ)
80 0re 11216 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
8180, 1, 2ltleii 11337 . . . . . . . . . . . . . . . . . . . . 21 0 ≀ Ο€
82 gt0ne0 11679 . . . . . . . . . . . . . . . . . . . . . . 23 ((Ο€ ∈ ℝ ∧ 0 < Ο€) β†’ Ο€ β‰  0)
83823adant3 1133 . . . . . . . . . . . . . . . . . . . . . 22 ((Ο€ ∈ ℝ ∧ 0 < Ο€ ∧ 0 ≀ Ο€) β†’ Ο€ β‰  0)
84833com23 1127 . . . . . . . . . . . . . . . . . . . . 21 ((Ο€ ∈ ℝ ∧ 0 ≀ Ο€ ∧ 0 < Ο€) β†’ Ο€ β‰  0)
851, 81, 2, 84mp3an 1462 . . . . . . . . . . . . . . . . . . . 20 Ο€ β‰  0
8685a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ β‰  0)
8774, 79, 86redivcld 12042 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 / Ο€) ∈ ℝ)
8887flcld 13763 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€)
8988znegcld 12668 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€)
90 abssinper 26030 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€) β†’ (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)))) = (absβ€˜(sinβ€˜π΄)))
9190eqcomd 2739 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€) β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)))))
9291ex 414 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€ β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))))))
9392adantr 482 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€ β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))))))
9489, 93mpd 15 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)))))
9588zcnd 12667 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (βŒŠβ€˜(𝐴 / Ο€)) ∈ β„‚)
9695negcld 11558 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„‚)
971recni 11228 . . . . . . . . . . . . . . . . . . . . 21 Ο€ ∈ β„‚
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ ∈ β„‚)
9996, 98mulcld 11234 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) ∈ β„‚)
10098, 95mulcld 11234 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚)
101100negcld 11558 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚)
10295, 98mulneg1d 11667 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))
10395, 98mulcomd 11235 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))
104103negeqd 11454 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))
105102, 104eqtrd 2773 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))
106 oveq2 7417 . . . . . . . . . . . . . . . . . . . . 21 ((-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) β†’ (𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)) = (𝐴 + -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
107106ad3antrrr 729 . . . . . . . . . . . . . . . . . . . 20 (((((-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∧ -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚) ∧ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) ∈ β„‚) ∧ 𝐴 ∈ β„‚) β†’ (𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)) = (𝐴 + -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
1081074an4132 43260 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ β„‚ ∧ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) ∈ β„‚) ∧ -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚) ∧ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))) β†’ (𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)) = (𝐴 + -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
10912, 99, 101, 105, 108syl1111anc 839 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)) = (𝐴 + -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
11012, 100negsubd 11577 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 + -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))) = (𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
111109, 110eqtrd 2773 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)) = (𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
112111fveq2d 6896 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))) = (sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))))
113112fveq2d 6896 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)))) = (absβ€˜(sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))))
11494, 113eqtrd 2773 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))))
115 modval 13836 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (𝐴 mod Ο€) = (𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
116115fveq2d 6896 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (sinβ€˜(𝐴 mod Ο€)) = (sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))))
117116fveq2d 6896 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (absβ€˜(sinβ€˜(𝐴 mod Ο€))) = (absβ€˜(sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))))
1183, 117mpan2 690 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℝ β†’ (absβ€˜(sinβ€˜(𝐴 mod Ο€))) = (absβ€˜(sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))))
11974, 118syl 17 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜(𝐴 mod Ο€))) = (absβ€˜(sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))))
120114, 119eqtr4d 2776 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 mod Ο€))))
12127fveq2d 6896 . . . . . . . . . . . . . . 15 ((sinβ€˜π΄) = 0 β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜0))
122 abs0 15232 . . . . . . . . . . . . . . 15 (absβ€˜0) = 0
123121, 122eqtrdi 2789 . . . . . . . . . . . . . 14 ((sinβ€˜π΄) = 0 β†’ (absβ€˜(sinβ€˜π΄)) = 0)
124123adantl 483 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜π΄)) = 0)
125120, 124eqtr3d 2775 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜(𝐴 mod Ο€))) = 0)
12678, 125abs00d 15393 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (sinβ€˜(𝐴 mod Ο€)) = 0)
127 notnotb 315 . . . . . . . . . . . . 13 ((sinβ€˜(𝐴 mod Ο€)) = 0 ↔ Β¬ Β¬ (sinβ€˜(𝐴 mod Ο€)) = 0)
128127bicomi 223 . . . . . . . . . . . 12 (Β¬ Β¬ (sinβ€˜(𝐴 mod Ο€)) = 0 ↔ (sinβ€˜(𝐴 mod Ο€)) = 0)
129 ltne 11311 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 0 < (sinβ€˜(𝐴 mod Ο€))) β†’ (sinβ€˜(𝐴 mod Ο€)) β‰  0)
130129neneqd 2946 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ 0 < (sinβ€˜(𝐴 mod Ο€))) β†’ Β¬ (sinβ€˜(𝐴 mod Ο€)) = 0)
131130expcom 415 . . . . . . . . . . . . . 14 (0 < (sinβ€˜(𝐴 mod Ο€)) β†’ (0 ∈ ℝ β†’ Β¬ (sinβ€˜(𝐴 mod Ο€)) = 0))
13280, 131mpi 20 . . . . . . . . . . . . 13 (0 < (sinβ€˜(𝐴 mod Ο€)) β†’ Β¬ (sinβ€˜(𝐴 mod Ο€)) = 0)
133132con3i 154 . . . . . . . . . . . 12 (Β¬ Β¬ (sinβ€˜(𝐴 mod Ο€)) = 0 β†’ Β¬ 0 < (sinβ€˜(𝐴 mod Ο€)))
134128, 133sylbir 234 . . . . . . . . . . 11 ((sinβ€˜(𝐴 mod Ο€)) = 0 β†’ Β¬ 0 < (sinβ€˜(𝐴 mod Ο€)))
135126, 134syl 17 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Β¬ 0 < (sinβ€˜(𝐴 mod Ο€)))
136 sinq12gt0 26017 . . . . . . . . . 10 ((𝐴 mod Ο€) ∈ (0(,)Ο€) β†’ 0 < (sinβ€˜(𝐴 mod Ο€)))
137135, 136nsyl 140 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Β¬ (𝐴 mod Ο€) ∈ (0(,)Ο€))
13880rexri 11272 . . . . . . . . . . 11 0 ∈ ℝ*
1391rexri 11272 . . . . . . . . . . 11 Ο€ ∈ ℝ*
140 elioo2 13365 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ Ο€ ∈ ℝ*) β†’ ((𝐴 mod Ο€) ∈ (0(,)Ο€) ↔ ((𝐴 mod Ο€) ∈ ℝ ∧ 0 < (𝐴 mod Ο€) ∧ (𝐴 mod Ο€) < Ο€)))
141138, 139, 140mp2an 691 . . . . . . . . . 10 ((𝐴 mod Ο€) ∈ (0(,)Ο€) ↔ ((𝐴 mod Ο€) ∈ ℝ ∧ 0 < (𝐴 mod Ο€) ∧ (𝐴 mod Ο€) < Ο€))
142141notbii 320 . . . . . . . . 9 (Β¬ (𝐴 mod Ο€) ∈ (0(,)Ο€) ↔ Β¬ ((𝐴 mod Ο€) ∈ ℝ ∧ 0 < (𝐴 mod Ο€) ∧ (𝐴 mod Ο€) < Ο€))
143137, 142sylib 217 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Β¬ ((𝐴 mod Ο€) ∈ ℝ ∧ 0 < (𝐴 mod Ο€) ∧ (𝐴 mod Ο€) < Ο€))
144 3anan12 1097 . . . . . . . . 9 (((𝐴 mod Ο€) ∈ ℝ ∧ 0 < (𝐴 mod Ο€) ∧ (𝐴 mod Ο€) < Ο€) ↔ (0 < (𝐴 mod Ο€) ∧ ((𝐴 mod Ο€) ∈ ℝ ∧ (𝐴 mod Ο€) < Ο€)))
145144notbii 320 . . . . . . . 8 (Β¬ ((𝐴 mod Ο€) ∈ ℝ ∧ 0 < (𝐴 mod Ο€) ∧ (𝐴 mod Ο€) < Ο€) ↔ Β¬ (0 < (𝐴 mod Ο€) ∧ ((𝐴 mod Ο€) ∈ ℝ ∧ (𝐴 mod Ο€) < Ο€)))
146143, 145sylib 217 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Β¬ (0 < (𝐴 mod Ο€) ∧ ((𝐴 mod Ο€) ∈ ℝ ∧ (𝐴 mod Ο€) < Ο€)))
147 modlt 13845 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (𝐴 mod Ο€) < Ο€)
148147ancoms 460 . . . . . . . . 9 ((Ο€ ∈ ℝ+ ∧ 𝐴 ∈ ℝ) β†’ (𝐴 mod Ο€) < Ο€)
1493, 74, 148sylancr 588 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) < Ο€)
15076, 149jca 513 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((𝐴 mod Ο€) ∈ ℝ ∧ (𝐴 mod Ο€) < Ο€))
151 not12an2impnot1 43329 . . . . . . 7 ((Β¬ (0 < (𝐴 mod Ο€) ∧ ((𝐴 mod Ο€) ∈ ℝ ∧ (𝐴 mod Ο€) < Ο€)) ∧ ((𝐴 mod Ο€) ∈ ℝ ∧ (𝐴 mod Ο€) < Ο€)) β†’ Β¬ 0 < (𝐴 mod Ο€))
152146, 150, 151syl2anc 585 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Β¬ 0 < (𝐴 mod Ο€))
153 modge0 13844 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ 0 ≀ (𝐴 mod Ο€))
154153ancoms 460 . . . . . . . 8 ((Ο€ ∈ ℝ+ ∧ 𝐴 ∈ ℝ) β†’ 0 ≀ (𝐴 mod Ο€))
1553, 74, 154sylancr 588 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 0 ≀ (𝐴 mod Ο€))
156 leloe 11300 . . . . . . . . 9 ((0 ∈ ℝ ∧ (𝐴 mod Ο€) ∈ ℝ) β†’ (0 ≀ (𝐴 mod Ο€) ↔ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€))))
157156biimp3a 1470 . . . . . . . 8 ((0 ∈ ℝ ∧ (𝐴 mod Ο€) ∈ ℝ ∧ 0 ≀ (𝐴 mod Ο€)) β†’ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)))
158157idiALT 43238 . . . . . . 7 ((0 ∈ ℝ ∧ (𝐴 mod Ο€) ∈ ℝ ∧ 0 ≀ (𝐴 mod Ο€)) β†’ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)))
15980, 76, 155, 158mp3an2i 1467 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)))
160 pm2.53 850 . . . . . . . 8 ((0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)) β†’ (Β¬ 0 < (𝐴 mod Ο€) β†’ 0 = (𝐴 mod Ο€)))
161160imp 408 . . . . . . 7 (((0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)) ∧ Β¬ 0 < (𝐴 mod Ο€)) β†’ 0 = (𝐴 mod Ο€))
162161ancoms 460 . . . . . 6 ((Β¬ 0 < (𝐴 mod Ο€) ∧ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€))) β†’ 0 = (𝐴 mod Ο€))
163152, 159, 162syl2anc 585 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 0 = (𝐴 mod Ο€))
164163eqcomd 2739 . . . 4 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) = 0)
165 mod0 13841 . . . . . 6 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ ((𝐴 mod Ο€) = 0 ↔ (𝐴 / Ο€) ∈ β„€))
166165biimp3a 1470 . . . . 5 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+ ∧ (𝐴 mod Ο€) = 0) β†’ (𝐴 / Ο€) ∈ β„€)
1671663com12 1124 . . . 4 ((Ο€ ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ (𝐴 mod Ο€) = 0) β†’ (𝐴 / Ο€) ∈ β„€)
1683, 74, 164, 167mp3an2i 1467 . . 3 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 / Ο€) ∈ β„€)
169168ex 414 . 2 (𝐴 ∈ β„‚ β†’ ((sinβ€˜π΄) = 0 β†’ (𝐴 / Ο€) ∈ β„€))
17097a1i 11 . . . . . 6 (𝐴 ∈ β„‚ β†’ Ο€ ∈ β„‚)
17185a1i 11 . . . . . 6 (𝐴 ∈ β„‚ β†’ Ο€ β‰  0)
17211, 170, 171divcan1d 11991 . . . . 5 (𝐴 ∈ β„‚ β†’ ((𝐴 / Ο€) Β· Ο€) = 𝐴)
173172fveq2d 6896 . . . 4 (𝐴 ∈ β„‚ β†’ (sinβ€˜((𝐴 / Ο€) Β· Ο€)) = (sinβ€˜π΄))
174 id 22 . . . . 5 ((𝐴 / Ο€) ∈ β„€ β†’ (𝐴 / Ο€) ∈ β„€)
175 sinkpi 26031 . . . . 5 ((𝐴 / Ο€) ∈ β„€ β†’ (sinβ€˜((𝐴 / Ο€) Β· Ο€)) = 0)
176174, 175syl 17 . . . 4 ((𝐴 / Ο€) ∈ β„€ β†’ (sinβ€˜((𝐴 / Ο€) Β· Ο€)) = 0)
177173, 176sylan9req 2794 . . 3 ((𝐴 ∈ β„‚ ∧ (𝐴 / Ο€) ∈ β„€) β†’ (sinβ€˜π΄) = 0)
178177ex 414 . 2 (𝐴 ∈ β„‚ β†’ ((𝐴 / Ο€) ∈ β„€ β†’ (sinβ€˜π΄) = 0))
179169, 178impbid 211 1 (𝐴 ∈ β„‚ β†’ ((sinβ€˜π΄) = 0 ↔ (𝐴 / Ο€) ∈ β„€))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111  ici 11112   + caddc 11113   Β· cmul 11115  β„*cxr 11247   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  -cneg 11445   / cdiv 11871  2c2 12267  β„€cz 12558  β„+crp 12974  (,)cioo 13324  βŒŠcfl 13755   mod cmo 13834  abscabs 15181  expce 16005  sincsin 16007  Ο€cpi 16010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ioc 13329  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-fac 14234  df-bc 14263  df-hash 14291  df-shft 15014  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-limsup 15415  df-clim 15432  df-rlim 15433  df-sum 15633  df-ef 16011  df-sin 16013  df-cos 16014  df-pi 16016  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-mulg 18951  df-cntz 19181  df-cmn 19650  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lp 22640  df-perf 22641  df-cn 22731  df-cnp 22732  df-haus 22819  df-tx 23066  df-hmeo 23259  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-xms 23826  df-ms 23827  df-tms 23828  df-cncf 24394  df-limc 25383  df-dv 25384
This theorem is referenced by: (None)
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