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Theorem sineq0ALT 43307
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 43307. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 25896. 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 25896 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 25831 . . . . 5 Ο€ ∈ ℝ
2 pipos 25833 . . . . 5 0 < Ο€
31, 2elrpii 12923 . . . 4 Ο€ ∈ ℝ+
4 2ne0 12262 . . . . . 6 2 β‰  0
54a1i 11 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 2 β‰  0)
6 2cn 12233 . . . . . . 7 2 ∈ β„‚
7 2re 12232 . . . . . . . 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 11180 . . . . . 6 (𝐴 ∈ β„‚ β†’ (2 Β· 𝐴) ∈ β„‚)
15 ax-icn 11115 . . . . . . . . . . . . . . 15 i ∈ β„‚
1615a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ i ∈ β„‚)
1713, 16, 11mul12d 11369 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (2 Β· (i Β· 𝐴)) = (i Β· (2 Β· 𝐴)))
1816, 11mulcld 11180 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (i Β· 𝐴) ∈ β„‚)
19182timesd 12401 . . . . . . . . . . . . 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 6847 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = (expβ€˜((i Β· 𝐴) + (i Β· 𝐴))))
22 efadd 15981 . . . . . . . . . . . 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 16009 . . . . . . . . . . . . . . 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 15970 . . . . . . . . . . . . . . . . . 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 11407 . . . . . . . . . . . . . . . . . . . 20 -i ∈ β„‚
3231a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ -i ∈ β„‚)
3332, 11mulcld 11180 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„‚ β†’ (-i Β· 𝐴) ∈ β„‚)
34 efcl 15970 . . . . . . . . . . . . . . . . . 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 11517 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) ∈ β„‚)
37 2mulicn 12381 . . . . . . . . . . . . . . . . 17 (2 Β· i) ∈ β„‚
3837a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ (2 Β· i) ∈ β„‚)
39 2muline0 12382 . . . . . . . . . . . . . . . . 17 (2 Β· i) β‰  0
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ (2 Β· i) β‰  0)
4136, 38, 40diveq0ad 11946 . . . . . . . . . . . . . . 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 11527 . . . . . . . . . . . . . 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 7374 . . . . . . . . . . 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 15981 . . . . . . . . . . . . 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 11185 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ ((i + -i) Β· 𝐴) = ((i Β· 𝐴) + (-i Β· 𝐴)))
5315negidi 11475 . . . . . . . . . . . . . . 15 (i + -i) = 0
5453oveq1i 7368 . . . . . . . . . . . . . 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 11358 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (0 Β· 𝐴) = 0)
5755, 56eqtrd 2773 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ ((i Β· 𝐴) + (-i Β· 𝐴)) = 0)
5857fveq2d 6847 . . . . . . . . . . 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 15978 . . . . . . . . . . 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 6847 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = (absβ€˜1))
65 abs1 15188 . . . . . . 7 (absβ€˜1) = 1
6664, 65eqtrdi 2789 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1)
67 absefib 16085 . . . . . . . 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 15012 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ 2 ∈ ℝ ∧ 2 β‰  0) β†’ (𝐴 ∈ ℝ ↔ (2 Β· 𝐴) ∈ ℝ))
72714animp1 42867 . . . . . 6 ((((𝐴 ∈ β„‚ ∧ 2 ∈ ℝ) ∧ 2 β‰  0) ∧ (2 Β· 𝐴) ∈ ℝ) β†’ 𝐴 ∈ ℝ)
73724an31 42868 . . . . 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 13786 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) ∈ ℝ)
7776recnd 11188 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) ∈ β„‚)
7877sincld 16017 . . . . . . . . . . . 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 11162 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
8180, 1, 2ltleii 11283 . . . . . . . . . . . . . . . . . . . . 21 0 ≀ Ο€
82 gt0ne0 11625 . . . . . . . . . . . . . . . . . . . . . . 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 11988 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 / Ο€) ∈ ℝ)
8887flcld 13709 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€)
8988znegcld 12614 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€)
90 abssinper 25893 . . . . . . . . . . . . . . . . . . 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 12613 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (βŒŠβ€˜(𝐴 / Ο€)) ∈ β„‚)
9695negcld 11504 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„‚)
971recni 11174 . . . . . . . . . . . . . . . . . . . . 21 Ο€ ∈ β„‚
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ ∈ β„‚)
9996, 98mulcld 11180 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) ∈ β„‚)
10098, 95mulcld 11180 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚)
101100negcld 11504 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚)
10295, 98mulneg1d 11613 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))
10395, 98mulcomd 11181 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))
104103negeqd 11400 . . . . . . . . . . . . . . . . . . . 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 7366 . . . . . . . . . . . . . . . . . . . . 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 42869 . . . . . . . . . . . . . . . . . . 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 11523 . . . . . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))) = (sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))))
113112fveq2d 6847 . . . . . . . . . . . . . . 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 13782 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (𝐴 mod Ο€) = (𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
116115fveq2d 6847 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (sinβ€˜(𝐴 mod Ο€)) = (sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))))
117116fveq2d 6847 . . . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . 15 ((sinβ€˜π΄) = 0 β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜0))
122 abs0 15176 . . . . . . . . . . . . . . 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 15337 . . . . . . . . . . 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 11257 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 0 < (sinβ€˜(𝐴 mod Ο€))) β†’ (sinβ€˜(𝐴 mod Ο€)) β‰  0)
130129neneqd 2945 . . . . . . . . . . . . . . 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 25880 . . . . . . . . . 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 11218 . . . . . . . . . . 11 0 ∈ ℝ*
1391rexri 11218 . . . . . . . . . . 11 Ο€ ∈ ℝ*
140 elioo2 13311 . . . . . . . . . . 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 13791 . . . . . . . . . 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 42938 . . . . . . 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 13790 . . . . . . . . 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 11246 . . . . . . . . 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 42847 . . . . . . 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 13787 . . . . . 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 11937 . . . . 5 (𝐴 ∈ β„‚ β†’ ((𝐴 / Ο€) Β· Ο€) = 𝐴)
173172fveq2d 6847 . . . 4 (𝐴 ∈ β„‚ β†’ (sinβ€˜((𝐴 / Ο€) Β· Ο€)) = (sinβ€˜π΄))
174 id 22 . . . . 5 ((𝐴 / Ο€) ∈ β„€ β†’ (𝐴 / Ο€) ∈ β„€)
175 sinkpi 25894 . . . . 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 2940   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  β„cr 11055  0cc0 11056  1c1 11057  ici 11058   + caddc 11059   Β· cmul 11061  β„*cxr 11193   < clt 11194   ≀ cle 11195   βˆ’ cmin 11390  -cneg 11391   / cdiv 11817  2c2 12213  β„€cz 12504  β„+crp 12920  (,)cioo 13270  βŒŠcfl 13701   mod cmo 13780  abscabs 15125  expce 15949  sincsin 15951  Ο€cpi 15954
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-fi 9352  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13274  df-ioc 13275  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-fl 13703  df-mod 13781  df-seq 13913  df-exp 13974  df-fac 14180  df-bc 14209  df-hash 14237  df-shft 14958  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-limsup 15359  df-clim 15376  df-rlim 15377  df-sum 15577  df-ef 15955  df-sin 15957  df-cos 15958  df-pi 15960  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-hom 17162  df-cco 17163  df-rest 17309  df-topn 17310  df-0g 17328  df-gsum 17329  df-topgen 17330  df-pt 17331  df-prds 17334  df-xrs 17389  df-qtop 17394  df-imas 17395  df-xps 17397  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-mulg 18878  df-cntz 19102  df-cmn 19569  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-fbas 20809  df-fg 20810  df-cnfld 20813  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-lp 22503  df-perf 22504  df-cn 22594  df-cnp 22595  df-haus 22682  df-tx 22929  df-hmeo 23122  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307  df-xms 23689  df-ms 23690  df-tms 23691  df-cncf 24257  df-limc 25246  df-dv 25247
This theorem is referenced by: (None)
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