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Theorem sineq0ALT 44000
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 44000. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 26269. 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 26269 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 26204 . . . . 5 Ο€ ∈ ℝ
2 pipos 26206 . . . . 5 0 < Ο€
31, 2elrpii 12981 . . . 4 Ο€ ∈ ℝ+
4 2ne0 12320 . . . . . 6 2 β‰  0
54a1i 11 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 2 β‰  0)
6 2cn 12291 . . . . . . 7 2 ∈ β„‚
7 2re 12290 . . . . . . . 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 479 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 𝐴 ∈ β„‚)
136a1i 11 . . . . . . 7 (𝐴 ∈ β„‚ β†’ 2 ∈ β„‚)
1413, 11mulcld 11238 . . . . . 6 (𝐴 ∈ β„‚ β†’ (2 Β· 𝐴) ∈ β„‚)
15 ax-icn 11171 . . . . . . . . . . . . . . 15 i ∈ β„‚
1615a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ i ∈ β„‚)
1713, 16, 11mul12d 11427 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (2 Β· (i Β· 𝐴)) = (i Β· (2 Β· 𝐴)))
1816, 11mulcld 11238 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (i Β· 𝐴) ∈ β„‚)
19182timesd 12459 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (2 Β· (i Β· 𝐴)) = ((i Β· 𝐴) + (i Β· 𝐴)))
2017, 19eqtr3d 2772 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (i Β· (2 Β· 𝐴)) = ((i Β· 𝐴) + (i Β· 𝐴)))
2120fveq2d 6894 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = (expβ€˜((i Β· 𝐴) + (i Β· 𝐴))))
22 efadd 16041 . . . . . . . . . . . 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 582 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (expβ€˜((i Β· 𝐴) + (i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
2421, 23eqtrd 2770 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
2524adantr 479 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
26 sinval 16069 . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)) = 0)
29 efcl 16030 . . . . . . . . . . . . . . . . . 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 11465 . . . . . . . . . . . . . . . . . . . 20 -i ∈ β„‚
3231a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ -i ∈ β„‚)
3332, 11mulcld 11238 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„‚ β†’ (-i Β· 𝐴) ∈ β„‚)
34 efcl 16030 . . . . . . . . . . . . . . . . . 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 11575 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) ∈ β„‚)
37 2mulicn 12439 . . . . . . . . . . . . . . . . 17 (2 Β· i) ∈ β„‚
3837a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ (2 Β· i) ∈ β„‚)
39 2muline0 12440 . . . . . . . . . . . . . . . . 17 (2 Β· i) β‰  0
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ (2 Β· i) β‰  0)
4136, 38, 40diveq0ad 12004 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 11585 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) = 0 ↔ (expβ€˜(i Β· 𝐴)) = (expβ€˜(-i Β· 𝐴))))
4544adantr 479 . . . . . . . . . . . . 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 7427 . . . . . . . . . . 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 16041 . . . . . . . . . . . . 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 582 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(-i Β· 𝐴))))
5049adantr 479 . . . . . . . . . . 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 2773 . . . . . . . . . 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 11243 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ ((i + -i) Β· 𝐴) = ((i Β· 𝐴) + (-i Β· 𝐴)))
5315negidi 11533 . . . . . . . . . . . . . . 15 (i + -i) = 0
5453oveq1i 7421 . . . . . . . . . . . . . 14 ((i + -i) Β· 𝐴) = (0 Β· 𝐴)
5552, 54eqtr3di 2785 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ ((i Β· 𝐴) + (-i Β· 𝐴)) = (0 Β· 𝐴))
5611mul02d 11416 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (0 Β· 𝐴) = 0)
5755, 56eqtrd 2770 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ ((i Β· 𝐴) + (-i Β· 𝐴)) = 0)
5857fveq2d 6894 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = (expβ€˜0))
5958adantr 479 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜((i Β· 𝐴) + (-i Β· 𝐴))) = (expβ€˜0))
60 ef0 16038 . . . . . . . . . . 11 (expβ€˜0) = 1
6160a1i 11 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜0) = 1)
6251, 59, 613eqtrd 2774 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) = 1)
6325, 62eqtrd 2770 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (expβ€˜(i Β· (2 Β· 𝐴))) = 1)
6463fveq2d 6894 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = (absβ€˜1))
65 abs1 15248 . . . . . . 7 (absβ€˜1) = 1
6664, 65eqtrdi 2786 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1)
67 absefib 16145 . . . . . . . 8 ((2 Β· 𝐴) ∈ β„‚ β†’ ((2 Β· 𝐴) ∈ ℝ ↔ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1))
6867biimparc 478 . . . . . . 7 (((absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1 ∧ (2 Β· 𝐴) ∈ β„‚) β†’ (2 Β· 𝐴) ∈ ℝ)
6968ancoms 457 . . . . . 6 (((2 Β· 𝐴) ∈ β„‚ ∧ (absβ€˜(expβ€˜(i Β· (2 Β· 𝐴)))) = 1) β†’ (2 Β· 𝐴) ∈ ℝ)
7014, 66, 69syl2an2r 681 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (2 Β· 𝐴) ∈ ℝ)
71 mulre 15072 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ 2 ∈ ℝ ∧ 2 β‰  0) β†’ (𝐴 ∈ ℝ ↔ (2 Β· 𝐴) ∈ ℝ))
72714animp1 43560 . . . . . 6 ((((𝐴 ∈ β„‚ ∧ 2 ∈ ℝ) ∧ 2 β‰  0) ∧ (2 Β· 𝐴) ∈ ℝ) β†’ 𝐴 ∈ ℝ)
73724an31 43561 . . . . 5 ((((2 β‰  0 ∧ 2 ∈ ℝ) ∧ 𝐴 ∈ β„‚) ∧ (2 Β· 𝐴) ∈ ℝ) β†’ 𝐴 ∈ ℝ)
745, 10, 12, 70, 73syl1111anc 836 . . . 4 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 𝐴 ∈ ℝ)
753a1i 11 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ ∈ ℝ+)
7674, 75modcld 13844 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) ∈ ℝ)
7776recnd 11246 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) ∈ β„‚)
7877sincld 16077 . . . . . . . . . . . 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 11220 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
8180, 1, 2ltleii 11341 . . . . . . . . . . . . . . . . . . . . 21 0 ≀ Ο€
82 gt0ne0 11683 . . . . . . . . . . . . . . . . . . . . . . 23 ((Ο€ ∈ ℝ ∧ 0 < Ο€) β†’ Ο€ β‰  0)
83823adant3 1130 . . . . . . . . . . . . . . . . . . . . . 22 ((Ο€ ∈ ℝ ∧ 0 < Ο€ ∧ 0 ≀ Ο€) β†’ Ο€ β‰  0)
84833com23 1124 . . . . . . . . . . . . . . . . . . . . 21 ((Ο€ ∈ ℝ ∧ 0 ≀ Ο€ ∧ 0 < Ο€) β†’ Ο€ β‰  0)
851, 81, 2, 84mp3an 1459 . . . . . . . . . . . . . . . . . . . 20 Ο€ β‰  0
8685a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ β‰  0)
8774, 79, 86redivcld 12046 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 / Ο€) ∈ ℝ)
8887flcld 13767 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€)
8988znegcld 12672 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€)
90 abssinper 26266 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€) β†’ (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)))) = (absβ€˜(sinβ€˜π΄)))
9190eqcomd 2736 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€) β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)))))
9291ex 411 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„€ β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))))))
9392adantr 479 . . . . . . . . . . . . . . . 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 12671 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (βŒŠβ€˜(𝐴 / Ο€)) ∈ β„‚)
9695negcld 11562 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(βŒŠβ€˜(𝐴 / Ο€)) ∈ β„‚)
971recni 11232 . . . . . . . . . . . . . . . . . . . . 21 Ο€ ∈ β„‚
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Ο€ ∈ β„‚)
9996, 98mulcld 11238 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) ∈ β„‚)
10098, 95mulcld 11238 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚)
101100negcld 11562 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))) ∈ β„‚)
10295, 98mulneg1d 11671 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))
10395, 98mulcomd 11239 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))
104103negeqd 11458 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ -((βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))
105102, 104eqtrd 2770 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€) = -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))
106 oveq2 7419 . . . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . 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 43562 . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)) = (𝐴 + -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
11012, 100negsubd 11581 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 + -(Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))) = (𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
111109, 110eqtrd 2770 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€)) = (𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
112111fveq2d 6894 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (sinβ€˜(𝐴 + (-(βŒŠβ€˜(𝐴 / Ο€)) Β· Ο€))) = (sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))))
113112fveq2d 6894 . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))))
115 modval 13840 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (𝐴 mod Ο€) = (𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))
116115fveq2d 6894 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (sinβ€˜(𝐴 mod Ο€)) = (sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€))))))
117116fveq2d 6894 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (absβ€˜(sinβ€˜(𝐴 mod Ο€))) = (absβ€˜(sinβ€˜(𝐴 βˆ’ (Ο€ Β· (βŒŠβ€˜(𝐴 / Ο€)))))))
1183, 117mpan2 687 . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜(sinβ€˜(𝐴 mod Ο€))))
12127fveq2d 6894 . . . . . . . . . . . . . . 15 ((sinβ€˜π΄) = 0 β†’ (absβ€˜(sinβ€˜π΄)) = (absβ€˜0))
122 abs0 15236 . . . . . . . . . . . . . . 15 (absβ€˜0) = 0
123121, 122eqtrdi 2786 . . . . . . . . . . . . . 14 ((sinβ€˜π΄) = 0 β†’ (absβ€˜(sinβ€˜π΄)) = 0)
124123adantl 480 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜π΄)) = 0)
125120, 124eqtr3d 2772 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (absβ€˜(sinβ€˜(𝐴 mod Ο€))) = 0)
12678, 125abs00d 15397 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (sinβ€˜(𝐴 mod Ο€)) = 0)
127 notnotb 314 . . . . . . . . . . . . 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 11315 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 0 < (sinβ€˜(𝐴 mod Ο€))) β†’ (sinβ€˜(𝐴 mod Ο€)) β‰  0)
130129neneqd 2943 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ 0 < (sinβ€˜(𝐴 mod Ο€))) β†’ Β¬ (sinβ€˜(𝐴 mod Ο€)) = 0)
131130expcom 412 . . . . . . . . . . . . . 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 26253 . . . . . . . . . 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 11276 . . . . . . . . . . 11 0 ∈ ℝ*
1391rexri 11276 . . . . . . . . . . 11 Ο€ ∈ ℝ*
140 elioo2 13369 . . . . . . . . . . 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 688 . . . . . . . . . 10 ((𝐴 mod Ο€) ∈ (0(,)Ο€) ↔ ((𝐴 mod Ο€) ∈ ℝ ∧ 0 < (𝐴 mod Ο€) ∧ (𝐴 mod Ο€) < Ο€))
142141notbii 319 . . . . . . . . 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 1094 . . . . . . . . 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 319 . . . . . . . 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 13849 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ (𝐴 mod Ο€) < Ο€)
148147ancoms 457 . . . . . . . . 9 ((Ο€ ∈ ℝ+ ∧ 𝐴 ∈ ℝ) β†’ (𝐴 mod Ο€) < Ο€)
1493, 74, 148sylancr 585 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) < Ο€)
15076, 149jca 510 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ ((𝐴 mod Ο€) ∈ ℝ ∧ (𝐴 mod Ο€) < Ο€))
151 not12an2impnot1 43631 . . . . . . 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 582 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ Β¬ 0 < (𝐴 mod Ο€))
153 modge0 13848 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ 0 ≀ (𝐴 mod Ο€))
154153ancoms 457 . . . . . . . 8 ((Ο€ ∈ ℝ+ ∧ 𝐴 ∈ ℝ) β†’ 0 ≀ (𝐴 mod Ο€))
1553, 74, 154sylancr 585 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 0 ≀ (𝐴 mod Ο€))
156 leloe 11304 . . . . . . . . 9 ((0 ∈ ℝ ∧ (𝐴 mod Ο€) ∈ ℝ) β†’ (0 ≀ (𝐴 mod Ο€) ↔ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€))))
157156biimp3a 1467 . . . . . . . 8 ((0 ∈ ℝ ∧ (𝐴 mod Ο€) ∈ ℝ ∧ 0 ≀ (𝐴 mod Ο€)) β†’ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)))
158157idiALT 43540 . . . . . . 7 ((0 ∈ ℝ ∧ (𝐴 mod Ο€) ∈ ℝ ∧ 0 ≀ (𝐴 mod Ο€)) β†’ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)))
15980, 76, 155, 158mp3an2i 1464 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)))
160 pm2.53 847 . . . . . . . 8 ((0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)) β†’ (Β¬ 0 < (𝐴 mod Ο€) β†’ 0 = (𝐴 mod Ο€)))
161160imp 405 . . . . . . 7 (((0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€)) ∧ Β¬ 0 < (𝐴 mod Ο€)) β†’ 0 = (𝐴 mod Ο€))
162161ancoms 457 . . . . . 6 ((Β¬ 0 < (𝐴 mod Ο€) ∧ (0 < (𝐴 mod Ο€) ∨ 0 = (𝐴 mod Ο€))) β†’ 0 = (𝐴 mod Ο€))
163152, 159, 162syl2anc 582 . . . . 5 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ 0 = (𝐴 mod Ο€))
164163eqcomd 2736 . . . 4 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 mod Ο€) = 0)
165 mod0 13845 . . . . . 6 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+) β†’ ((𝐴 mod Ο€) = 0 ↔ (𝐴 / Ο€) ∈ β„€))
166165biimp3a 1467 . . . . 5 ((𝐴 ∈ ℝ ∧ Ο€ ∈ ℝ+ ∧ (𝐴 mod Ο€) = 0) β†’ (𝐴 / Ο€) ∈ β„€)
1671663com12 1121 . . . 4 ((Ο€ ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ (𝐴 mod Ο€) = 0) β†’ (𝐴 / Ο€) ∈ β„€)
1683, 74, 164, 167mp3an2i 1464 . . 3 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) = 0) β†’ (𝐴 / Ο€) ∈ β„€)
169168ex 411 . 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 11995 . . . . 5 (𝐴 ∈ β„‚ β†’ ((𝐴 / Ο€) Β· Ο€) = 𝐴)
173172fveq2d 6894 . . . 4 (𝐴 ∈ β„‚ β†’ (sinβ€˜((𝐴 / Ο€) Β· Ο€)) = (sinβ€˜π΄))
174 id 22 . . . . 5 ((𝐴 / Ο€) ∈ β„€ β†’ (𝐴 / Ο€) ∈ β„€)
175 sinkpi 26267 . . . . 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 2791 . . 3 ((𝐴 ∈ β„‚ ∧ (𝐴 / Ο€) ∈ β„€) β†’ (sinβ€˜π΄) = 0)
178177ex 411 . 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 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113  ici 11114   + caddc 11115   Β· cmul 11117  β„*cxr 11251   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  -cneg 11449   / cdiv 11875  2c2 12271  β„€cz 12562  β„+crp 12978  (,)cioo 13328  βŒŠcfl 13759   mod cmo 13838  abscabs 15185  expce 16009  sincsin 16011  Ο€cpi 16014
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-ioc 13333  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-mod 13839  df-seq 13971  df-exp 14032  df-fac 14238  df-bc 14267  df-hash 14295  df-shft 15018  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-limsup 15419  df-clim 15436  df-rlim 15437  df-sum 15637  df-ef 16015  df-sin 16017  df-cos 16018  df-pi 16020  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-xrs 17452  df-qtop 17457  df-imas 17458  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-fbas 21141  df-fg 21142  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-lp 22860  df-perf 22861  df-cn 22951  df-cnp 22952  df-haus 23039  df-tx 23286  df-hmeo 23479  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664  df-xms 24046  df-ms 24047  df-tms 24048  df-cncf 24618  df-limc 25615  df-dv 25616
This theorem is referenced by: (None)
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