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Theorem pige3ALT 26266
Description: Alternate proof of pige3 26265. This proof is based on the geometric observation that a hexagon of unit side length has perimeter 6, which is less than the unit-radius circumcircle, of perimeter 2Ο€. We translate this to algebra by looking at the function e↑(iπ‘₯) as π‘₯ goes from 0 to Ο€ / 3; it moves at unit speed and travels distance 1, hence 1 ≀ Ο€ / 3. (Contributed by Mario Carneiro, 21-May-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
pige3ALT 3 ≀ Ο€

Proof of Theorem pige3ALT
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3cn 12298 . . 3 3 ∈ β„‚
21mullidi 11224 . 2 (1 Β· 3) = 3
3 tru 1544 . . . . . 6 ⊀
4 0xr 11266 . . . . . . . 8 0 ∈ ℝ*
5 pirp 26208 . . . . . . . . . 10 Ο€ ∈ ℝ+
6 3rp 12985 . . . . . . . . . 10 3 ∈ ℝ+
7 rpdivcl 13004 . . . . . . . . . 10 ((Ο€ ∈ ℝ+ ∧ 3 ∈ ℝ+) β†’ (Ο€ / 3) ∈ ℝ+)
85, 6, 7mp2an 689 . . . . . . . . 9 (Ο€ / 3) ∈ ℝ+
9 rpxr 12988 . . . . . . . . 9 ((Ο€ / 3) ∈ ℝ+ β†’ (Ο€ / 3) ∈ ℝ*)
108, 9ax-mp 5 . . . . . . . 8 (Ο€ / 3) ∈ ℝ*
11 rpge0 12992 . . . . . . . . 9 ((Ο€ / 3) ∈ ℝ+ β†’ 0 ≀ (Ο€ / 3))
128, 11ax-mp 5 . . . . . . . 8 0 ≀ (Ο€ / 3)
13 lbicc2 13446 . . . . . . . 8 ((0 ∈ ℝ* ∧ (Ο€ / 3) ∈ ℝ* ∧ 0 ≀ (Ο€ / 3)) β†’ 0 ∈ (0[,](Ο€ / 3)))
144, 10, 12, 13mp3an 1460 . . . . . . 7 0 ∈ (0[,](Ο€ / 3))
15 ubicc2 13447 . . . . . . . 8 ((0 ∈ ℝ* ∧ (Ο€ / 3) ∈ ℝ* ∧ 0 ≀ (Ο€ / 3)) β†’ (Ο€ / 3) ∈ (0[,](Ο€ / 3)))
164, 10, 12, 15mp3an 1460 . . . . . . 7 (Ο€ / 3) ∈ (0[,](Ο€ / 3))
1714, 16pm3.2i 470 . . . . . 6 (0 ∈ (0[,](Ο€ / 3)) ∧ (Ο€ / 3) ∈ (0[,](Ο€ / 3)))
18 0re 11221 . . . . . . . 8 0 ∈ ℝ
1918a1i 11 . . . . . . 7 (⊀ β†’ 0 ∈ ℝ)
20 pire 26205 . . . . . . . . 9 Ο€ ∈ ℝ
21 3re 12297 . . . . . . . . 9 3 ∈ ℝ
22 3ne0 12323 . . . . . . . . 9 3 β‰  0
2320, 21, 22redivcli 11986 . . . . . . . 8 (Ο€ / 3) ∈ ℝ
2423a1i 11 . . . . . . 7 (⊀ β†’ (Ο€ / 3) ∈ ℝ)
25 efcn 26192 . . . . . . . . 9 exp ∈ (ℂ–cnβ†’β„‚)
2625a1i 11 . . . . . . . 8 (⊀ β†’ exp ∈ (ℂ–cnβ†’β„‚))
27 iccssre 13411 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ (Ο€ / 3) ∈ ℝ) β†’ (0[,](Ο€ / 3)) βŠ† ℝ)
2818, 23, 27mp2an 689 . . . . . . . . . . 11 (0[,](Ο€ / 3)) βŠ† ℝ
29 ax-resscn 11170 . . . . . . . . . . 11 ℝ βŠ† β„‚
3028, 29sstri 3991 . . . . . . . . . 10 (0[,](Ο€ / 3)) βŠ† β„‚
31 resmpt 6037 . . . . . . . . . 10 ((0[,](Ο€ / 3)) βŠ† β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) β†Ύ (0[,](Ο€ / 3))) = (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (i Β· π‘₯)))
3230, 31mp1i 13 . . . . . . . . 9 (⊀ β†’ ((π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) β†Ύ (0[,](Ο€ / 3))) = (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (i Β· π‘₯)))
33 ssidd 4005 . . . . . . . . . . 11 (⊀ β†’ β„‚ βŠ† β„‚)
34 ax-icn 11172 . . . . . . . . . . . . 13 i ∈ β„‚
35 simpr 484 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ π‘₯ ∈ β„‚)
36 mulcl 11197 . . . . . . . . . . . . 13 ((i ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ (i Β· π‘₯) ∈ β„‚)
3734, 35, 36sylancr 586 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ (i Β· π‘₯) ∈ β„‚)
3837fmpttd 7116 . . . . . . . . . . 11 (⊀ β†’ (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)):β„‚βŸΆβ„‚)
39 cnelprrecn 11206 . . . . . . . . . . . . . . . 16 β„‚ ∈ {ℝ, β„‚}
4039a1i 11 . . . . . . . . . . . . . . 15 (⊀ β†’ β„‚ ∈ {ℝ, β„‚})
41 ax-1cn 11171 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
4241a1i 11 . . . . . . . . . . . . . . 15 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ 1 ∈ β„‚)
4340dvmptid 25710 . . . . . . . . . . . . . . 15 (⊀ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ π‘₯)) = (π‘₯ ∈ β„‚ ↦ 1))
4434a1i 11 . . . . . . . . . . . . . . 15 (⊀ β†’ i ∈ β„‚)
4540, 35, 42, 43, 44dvmptcmul 25717 . . . . . . . . . . . . . 14 (⊀ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = (π‘₯ ∈ β„‚ ↦ (i Β· 1)))
4634mulridi 11223 . . . . . . . . . . . . . . 15 (i Β· 1) = i
4746mpteq2i 5253 . . . . . . . . . . . . . 14 (π‘₯ ∈ β„‚ ↦ (i Β· 1)) = (π‘₯ ∈ β„‚ ↦ i)
4845, 47eqtrdi 2787 . . . . . . . . . . . . 13 (⊀ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = (π‘₯ ∈ β„‚ ↦ i))
4948dmeqd 5905 . . . . . . . . . . . 12 (⊀ β†’ dom (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = dom (π‘₯ ∈ β„‚ ↦ i))
5034elexi 3493 . . . . . . . . . . . . 13 i ∈ V
51 eqid 2731 . . . . . . . . . . . . 13 (π‘₯ ∈ β„‚ ↦ i) = (π‘₯ ∈ β„‚ ↦ i)
5250, 51dmmpti 6694 . . . . . . . . . . . 12 dom (π‘₯ ∈ β„‚ ↦ i) = β„‚
5349, 52eqtrdi 2787 . . . . . . . . . . 11 (⊀ β†’ dom (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = β„‚)
54 dvcn 25672 . . . . . . . . . . 11 (((β„‚ βŠ† β„‚ ∧ (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)):β„‚βŸΆβ„‚ ∧ β„‚ βŠ† β„‚) ∧ dom (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = β„‚) β†’ (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) ∈ (ℂ–cnβ†’β„‚))
5533, 38, 33, 53, 54syl31anc 1372 . . . . . . . . . 10 (⊀ β†’ (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) ∈ (ℂ–cnβ†’β„‚))
56 rescncf 24638 . . . . . . . . . 10 ((0[,](Ο€ / 3)) βŠ† β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) ∈ (ℂ–cnβ†’β„‚) β†’ ((π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) β†Ύ (0[,](Ο€ / 3))) ∈ ((0[,](Ο€ / 3))–cnβ†’β„‚)))
5730, 55, 56mpsyl 68 . . . . . . . . 9 (⊀ β†’ ((π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) β†Ύ (0[,](Ο€ / 3))) ∈ ((0[,](Ο€ / 3))–cnβ†’β„‚))
5832, 57eqeltrrd 2833 . . . . . . . 8 (⊀ β†’ (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (i Β· π‘₯)) ∈ ((0[,](Ο€ / 3))–cnβ†’β„‚))
5926, 58cncfmpt1f 24655 . . . . . . 7 (⊀ β†’ (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))) ∈ ((0[,](Ο€ / 3))–cnβ†’β„‚))
60 reelprrecn 11205 . . . . . . . . . . 11 ℝ ∈ {ℝ, β„‚}
6160a1i 11 . . . . . . . . . 10 (⊀ β†’ ℝ ∈ {ℝ, β„‚})
62 recn 11203 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ π‘₯ ∈ β„‚)
63 efcl 16031 . . . . . . . . . . . 12 ((i Β· π‘₯) ∈ β„‚ β†’ (expβ€˜(i Β· π‘₯)) ∈ β„‚)
6437, 63syl 17 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ (expβ€˜(i Β· π‘₯)) ∈ β„‚)
6562, 64sylan2 592 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ ℝ) β†’ (expβ€˜(i Β· π‘₯)) ∈ β„‚)
66 mulcl 11197 . . . . . . . . . . . 12 (((expβ€˜(i Β· π‘₯)) ∈ β„‚ ∧ i ∈ β„‚) β†’ ((expβ€˜(i Β· π‘₯)) Β· i) ∈ β„‚)
6764, 34, 66sylancl 585 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ ((expβ€˜(i Β· π‘₯)) Β· i) ∈ β„‚)
6862, 67sylan2 592 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ ℝ) β†’ ((expβ€˜(i Β· π‘₯)) Β· i) ∈ β„‚)
69 eqid 2731 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
7069cnfldtopon 24520 . . . . . . . . . . . 12 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
71 toponmax 22649 . . . . . . . . . . . 12 ((TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚) β†’ β„‚ ∈ (TopOpenβ€˜β„‚fld))
7270, 71mp1i 13 . . . . . . . . . . 11 (⊀ β†’ β„‚ ∈ (TopOpenβ€˜β„‚fld))
7329a1i 11 . . . . . . . . . . . 12 (⊀ β†’ ℝ βŠ† β„‚)
74 df-ss 3965 . . . . . . . . . . . 12 (ℝ βŠ† β„‚ ↔ (ℝ ∩ β„‚) = ℝ)
7573, 74sylib 217 . . . . . . . . . . 11 (⊀ β†’ (ℝ ∩ β„‚) = ℝ)
7634a1i 11 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ i ∈ β„‚)
77 efcl 16031 . . . . . . . . . . . . 13 (𝑦 ∈ β„‚ β†’ (expβ€˜π‘¦) ∈ β„‚)
7877adantl 481 . . . . . . . . . . . 12 ((⊀ ∧ 𝑦 ∈ β„‚) β†’ (expβ€˜π‘¦) ∈ β„‚)
79 dvef 25733 . . . . . . . . . . . . 13 (β„‚ D exp) = exp
80 eff 16030 . . . . . . . . . . . . . . . 16 exp:β„‚βŸΆβ„‚
8180a1i 11 . . . . . . . . . . . . . . 15 (⊀ β†’ exp:β„‚βŸΆβ„‚)
8281feqmptd 6960 . . . . . . . . . . . . . 14 (⊀ β†’ exp = (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦)))
8382oveq2d 7428 . . . . . . . . . . . . 13 (⊀ β†’ (β„‚ D exp) = (β„‚ D (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦))))
8479, 83, 823eqtr3a 2795 . . . . . . . . . . . 12 (⊀ β†’ (β„‚ D (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦))) = (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦)))
85 fveq2 6891 . . . . . . . . . . . 12 (𝑦 = (i Β· π‘₯) β†’ (expβ€˜π‘¦) = (expβ€˜(i Β· π‘₯)))
8640, 40, 37, 76, 78, 78, 48, 84, 85, 85dvmptco 25725 . . . . . . . . . . 11 (⊀ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (expβ€˜(i Β· π‘₯)))) = (π‘₯ ∈ β„‚ ↦ ((expβ€˜(i Β· π‘₯)) Β· i)))
8769, 61, 72, 75, 64, 67, 86dvmptres3 25709 . . . . . . . . . 10 (⊀ β†’ (ℝ D (π‘₯ ∈ ℝ ↦ (expβ€˜(i Β· π‘₯)))) = (π‘₯ ∈ ℝ ↦ ((expβ€˜(i Β· π‘₯)) Β· i)))
8828a1i 11 . . . . . . . . . 10 (⊀ β†’ (0[,](Ο€ / 3)) βŠ† ℝ)
8969tgioo2 24540 . . . . . . . . . 10 (topGenβ€˜ran (,)) = ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)
90 iccntr 24558 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ (Ο€ / 3) ∈ ℝ) β†’ ((intβ€˜(topGenβ€˜ran (,)))β€˜(0[,](Ο€ / 3))) = (0(,)(Ο€ / 3)))
9118, 24, 90sylancr 586 . . . . . . . . . 10 (⊀ β†’ ((intβ€˜(topGenβ€˜ran (,)))β€˜(0[,](Ο€ / 3))) = (0(,)(Ο€ / 3)))
9261, 65, 68, 87, 88, 89, 69, 91dvmptres2 25715 . . . . . . . . 9 (⊀ β†’ (ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))) = (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i)))
9392dmeqd 5905 . . . . . . . 8 (⊀ β†’ dom (ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))) = dom (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i)))
94 ovex 7445 . . . . . . . . 9 ((expβ€˜(i Β· π‘₯)) Β· i) ∈ V
95 eqid 2731 . . . . . . . . 9 (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i)) = (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i))
9694, 95dmmpti 6694 . . . . . . . 8 dom (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i)) = (0(,)(Ο€ / 3))
9793, 96eqtrdi 2787 . . . . . . 7 (⊀ β†’ dom (ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))) = (0(,)(Ο€ / 3)))
98 1re 11219 . . . . . . . 8 1 ∈ ℝ
9998a1i 11 . . . . . . 7 (⊀ β†’ 1 ∈ ℝ)
10092fveq1d 6893 . . . . . . . . . . 11 (⊀ β†’ ((ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))))β€˜π‘¦) = ((π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i))β€˜π‘¦))
101 oveq2 7420 . . . . . . . . . . . . . 14 (π‘₯ = 𝑦 β†’ (i Β· π‘₯) = (i Β· 𝑦))
102101fveq2d 6895 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ (expβ€˜(i Β· π‘₯)) = (expβ€˜(i Β· 𝑦)))
103102oveq1d 7427 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((expβ€˜(i Β· π‘₯)) Β· i) = ((expβ€˜(i Β· 𝑦)) Β· i))
104103, 95, 94fvmpt3i 7003 . . . . . . . . . . 11 (𝑦 ∈ (0(,)(Ο€ / 3)) β†’ ((π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i))β€˜π‘¦) = ((expβ€˜(i Β· 𝑦)) Β· i))
105100, 104sylan9eq 2791 . . . . . . . . . 10 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ ((ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))))β€˜π‘¦) = ((expβ€˜(i Β· 𝑦)) Β· i))
106105fveq2d 6895 . . . . . . . . 9 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜((ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))))β€˜π‘¦)) = (absβ€˜((expβ€˜(i Β· 𝑦)) Β· i)))
107 ioossre 13390 . . . . . . . . . . . . . . 15 (0(,)(Ο€ / 3)) βŠ† ℝ
108107a1i 11 . . . . . . . . . . . . . 14 (⊀ β†’ (0(,)(Ο€ / 3)) βŠ† ℝ)
109108sselda 3982 . . . . . . . . . . . . 13 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ 𝑦 ∈ ℝ)
110109recnd 11247 . . . . . . . . . . . 12 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ 𝑦 ∈ β„‚)
111 mulcl 11197 . . . . . . . . . . . 12 ((i ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (i Β· 𝑦) ∈ β„‚)
11234, 110, 111sylancr 586 . . . . . . . . . . 11 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (i Β· 𝑦) ∈ β„‚)
113 efcl 16031 . . . . . . . . . . 11 ((i Β· 𝑦) ∈ β„‚ β†’ (expβ€˜(i Β· 𝑦)) ∈ β„‚)
114112, 113syl 17 . . . . . . . . . 10 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (expβ€˜(i Β· 𝑦)) ∈ β„‚)
115 absmul 15246 . . . . . . . . . 10 (((expβ€˜(i Β· 𝑦)) ∈ β„‚ ∧ i ∈ β„‚) β†’ (absβ€˜((expβ€˜(i Β· 𝑦)) Β· i)) = ((absβ€˜(expβ€˜(i Β· 𝑦))) Β· (absβ€˜i)))
116114, 34, 115sylancl 585 . . . . . . . . 9 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜((expβ€˜(i Β· 𝑦)) Β· i)) = ((absβ€˜(expβ€˜(i Β· 𝑦))) Β· (absβ€˜i)))
117 absefi 16144 . . . . . . . . . . . 12 (𝑦 ∈ ℝ β†’ (absβ€˜(expβ€˜(i Β· 𝑦))) = 1)
118109, 117syl 17 . . . . . . . . . . 11 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜(expβ€˜(i Β· 𝑦))) = 1)
119 absi 15238 . . . . . . . . . . . 12 (absβ€˜i) = 1
120119a1i 11 . . . . . . . . . . 11 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜i) = 1)
121118, 120oveq12d 7430 . . . . . . . . . 10 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ ((absβ€˜(expβ€˜(i Β· 𝑦))) Β· (absβ€˜i)) = (1 Β· 1))
12241mulridi 11223 . . . . . . . . . 10 (1 Β· 1) = 1
123121, 122eqtrdi 2787 . . . . . . . . 9 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ ((absβ€˜(expβ€˜(i Β· 𝑦))) Β· (absβ€˜i)) = 1)
124106, 116, 1233eqtrd 2775 . . . . . . . 8 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜((ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))))β€˜π‘¦)) = 1)
125 1le1 11847 . . . . . . . 8 1 ≀ 1
126124, 125eqbrtrdi 5187 . . . . . . 7 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜((ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))))β€˜π‘¦)) ≀ 1)
12719, 24, 59, 97, 99, 126dvlip 25746 . . . . . 6 ((⊀ ∧ (0 ∈ (0[,](Ο€ / 3)) ∧ (Ο€ / 3) ∈ (0[,](Ο€ / 3)))) β†’ (absβ€˜(((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜0) βˆ’ ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜(Ο€ / 3)))) ≀ (1 Β· (absβ€˜(0 βˆ’ (Ο€ / 3)))))
1283, 17, 127mp2an 689 . . . . 5 (absβ€˜(((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜0) βˆ’ ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜(Ο€ / 3)))) ≀ (1 Β· (absβ€˜(0 βˆ’ (Ο€ / 3))))
129 oveq2 7420 . . . . . . . . . . . . 13 (π‘₯ = 0 β†’ (i Β· π‘₯) = (i Β· 0))
130 it0e0 12439 . . . . . . . . . . . . 13 (i Β· 0) = 0
131129, 130eqtrdi 2787 . . . . . . . . . . . 12 (π‘₯ = 0 β†’ (i Β· π‘₯) = 0)
132131fveq2d 6895 . . . . . . . . . . 11 (π‘₯ = 0 β†’ (expβ€˜(i Β· π‘₯)) = (expβ€˜0))
133 ef0 16039 . . . . . . . . . . 11 (expβ€˜0) = 1
134132, 133eqtrdi 2787 . . . . . . . . . 10 (π‘₯ = 0 β†’ (expβ€˜(i Β· π‘₯)) = 1)
135 eqid 2731 . . . . . . . . . 10 (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))) = (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))
136 fvex 6904 . . . . . . . . . 10 (expβ€˜(i Β· π‘₯)) ∈ V
137134, 135, 136fvmpt3i 7003 . . . . . . . . 9 (0 ∈ (0[,](Ο€ / 3)) β†’ ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜0) = 1)
13814, 137ax-mp 5 . . . . . . . 8 ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜0) = 1
139 oveq2 7420 . . . . . . . . . . 11 (π‘₯ = (Ο€ / 3) β†’ (i Β· π‘₯) = (i Β· (Ο€ / 3)))
140139fveq2d 6895 . . . . . . . . . 10 (π‘₯ = (Ο€ / 3) β†’ (expβ€˜(i Β· π‘₯)) = (expβ€˜(i Β· (Ο€ / 3))))
141140, 135, 136fvmpt3i 7003 . . . . . . . . 9 ((Ο€ / 3) ∈ (0[,](Ο€ / 3)) β†’ ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜(Ο€ / 3)) = (expβ€˜(i Β· (Ο€ / 3))))
14216, 141ax-mp 5 . . . . . . . 8 ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜(Ο€ / 3)) = (expβ€˜(i Β· (Ο€ / 3)))
143138, 142oveq12i 7424 . . . . . . 7 (((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜0) βˆ’ ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜(Ο€ / 3))) = (1 βˆ’ (expβ€˜(i Β· (Ο€ / 3))))
14423recni 11233 . . . . . . . . . 10 (Ο€ / 3) ∈ β„‚
14534, 144mulcli 11226 . . . . . . . . 9 (i Β· (Ο€ / 3)) ∈ β„‚
146 efcl 16031 . . . . . . . . 9 ((i Β· (Ο€ / 3)) ∈ β„‚ β†’ (expβ€˜(i Β· (Ο€ / 3))) ∈ β„‚)
147145, 146ax-mp 5 . . . . . . . 8 (expβ€˜(i Β· (Ο€ / 3))) ∈ β„‚
148 negicn 11466 . . . . . . . . . 10 -i ∈ β„‚
149148, 144mulcli 11226 . . . . . . . . 9 (-i Β· (Ο€ / 3)) ∈ β„‚
150 efcl 16031 . . . . . . . . 9 ((-i Β· (Ο€ / 3)) ∈ β„‚ β†’ (expβ€˜(-i Β· (Ο€ / 3))) ∈ β„‚)
151149, 150ax-mp 5 . . . . . . . 8 (expβ€˜(-i Β· (Ο€ / 3))) ∈ β„‚
152 cosval 16071 . . . . . . . . . . 11 ((Ο€ / 3) ∈ β„‚ β†’ (cosβ€˜(Ο€ / 3)) = (((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) / 2))
153144, 152ax-mp 5 . . . . . . . . . 10 (cosβ€˜(Ο€ / 3)) = (((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) / 2)
154 sincos3rdpi 26263 . . . . . . . . . . 11 ((sinβ€˜(Ο€ / 3)) = ((βˆšβ€˜3) / 2) ∧ (cosβ€˜(Ο€ / 3)) = (1 / 2))
155154simpri 485 . . . . . . . . . 10 (cosβ€˜(Ο€ / 3)) = (1 / 2)
156153, 155eqtr3i 2761 . . . . . . . . 9 (((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) / 2) = (1 / 2)
157147, 151addcli 11225 . . . . . . . . . 10 ((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) ∈ β„‚
158 2cn 12292 . . . . . . . . . 10 2 ∈ β„‚
159 2ne0 12321 . . . . . . . . . 10 2 β‰  0
160157, 41, 158, 159div11i 11978 . . . . . . . . 9 ((((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) / 2) = (1 / 2) ↔ ((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) = 1)
161156, 160mpbi 229 . . . . . . . 8 ((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) = 1
16241, 147, 151, 161subaddrii 11554 . . . . . . 7 (1 βˆ’ (expβ€˜(i Β· (Ο€ / 3)))) = (expβ€˜(-i Β· (Ο€ / 3)))
163 mulneg12 11657 . . . . . . . . 9 ((i ∈ β„‚ ∧ (Ο€ / 3) ∈ β„‚) β†’ (-i Β· (Ο€ / 3)) = (i Β· -(Ο€ / 3)))
16434, 144, 163mp2an 689 . . . . . . . 8 (-i Β· (Ο€ / 3)) = (i Β· -(Ο€ / 3))
165164fveq2i 6894 . . . . . . 7 (expβ€˜(-i Β· (Ο€ / 3))) = (expβ€˜(i Β· -(Ο€ / 3)))
166143, 162, 1653eqtri 2763 . . . . . 6 (((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜0) βˆ’ ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜(Ο€ / 3))) = (expβ€˜(i Β· -(Ο€ / 3)))
167166fveq2i 6894 . . . . 5 (absβ€˜(((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜0) βˆ’ ((π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))β€˜(Ο€ / 3)))) = (absβ€˜(expβ€˜(i Β· -(Ο€ / 3))))
168144absnegi 15352 . . . . . . . 8 (absβ€˜-(Ο€ / 3)) = (absβ€˜(Ο€ / 3))
169 df-neg 11452 . . . . . . . . 9 -(Ο€ / 3) = (0 βˆ’ (Ο€ / 3))
170169fveq2i 6894 . . . . . . . 8 (absβ€˜-(Ο€ / 3)) = (absβ€˜(0 βˆ’ (Ο€ / 3)))
171168, 170eqtr3i 2761 . . . . . . 7 (absβ€˜(Ο€ / 3)) = (absβ€˜(0 βˆ’ (Ο€ / 3)))
172 rprege0 12994 . . . . . . . 8 ((Ο€ / 3) ∈ ℝ+ β†’ ((Ο€ / 3) ∈ ℝ ∧ 0 ≀ (Ο€ / 3)))
173 absid 15248 . . . . . . . 8 (((Ο€ / 3) ∈ ℝ ∧ 0 ≀ (Ο€ / 3)) β†’ (absβ€˜(Ο€ / 3)) = (Ο€ / 3))
1748, 172, 173mp2b 10 . . . . . . 7 (absβ€˜(Ο€ / 3)) = (Ο€ / 3)
175171, 174eqtr3i 2761 . . . . . 6 (absβ€˜(0 βˆ’ (Ο€ / 3))) = (Ο€ / 3)
176175oveq2i 7423 . . . . 5 (1 Β· (absβ€˜(0 βˆ’ (Ο€ / 3)))) = (1 Β· (Ο€ / 3))
177128, 167, 1763brtr3i 5177 . . . 4 (absβ€˜(expβ€˜(i Β· -(Ο€ / 3)))) ≀ (1 Β· (Ο€ / 3))
17823renegcli 11526 . . . . 5 -(Ο€ / 3) ∈ ℝ
179 absefi 16144 . . . . 5 (-(Ο€ / 3) ∈ ℝ β†’ (absβ€˜(expβ€˜(i Β· -(Ο€ / 3)))) = 1)
180178, 179ax-mp 5 . . . 4 (absβ€˜(expβ€˜(i Β· -(Ο€ / 3)))) = 1
181144mullidi 11224 . . . 4 (1 Β· (Ο€ / 3)) = (Ο€ / 3)
182177, 180, 1813brtr3i 5177 . . 3 1 ≀ (Ο€ / 3)
183 3pos 12322 . . . . 5 0 < 3
18421, 183pm3.2i 470 . . . 4 (3 ∈ ℝ ∧ 0 < 3)
185 lemuldiv 12099 . . . 4 ((1 ∈ ℝ ∧ Ο€ ∈ ℝ ∧ (3 ∈ ℝ ∧ 0 < 3)) β†’ ((1 Β· 3) ≀ Ο€ ↔ 1 ≀ (Ο€ / 3)))
18698, 20, 184, 185mp3an 1460 . . 3 ((1 Β· 3) ≀ Ο€ ↔ 1 ≀ (Ο€ / 3))
187182, 186mpbir 230 . 2 (1 Β· 3) ≀ Ο€
1882, 187eqbrtrri 5171 1 3 ≀ Ο€
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1540  βŠ€wtru 1541   ∈ wcel 2105   ∩ cin 3947   βŠ† wss 3948  {cpr 4630   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  β„‚cc 11111  β„cr 11112  0cc0 11113  1c1 11114  ici 11115   + caddc 11116   Β· cmul 11118  β„*cxr 11252   < clt 11253   ≀ cle 11254   βˆ’ cmin 11449  -cneg 11450   / cdiv 11876  2c2 12272  3c3 12273  β„+crp 12979  (,)cioo 13329  [,]cicc 13332  βˆšcsqrt 15185  abscabs 15186  expce 16010  sincsin 16012  cosccos 16013  Ο€cpi 16015  TopOpenctopn 17372  topGenctg 17388  β„‚fldccnfld 21145  TopOnctopon 22633  intcnt 22742  β€“cnβ†’ccncf 24617   D cdv 25613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191  ax-addf 11192  ax-mulf 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-er 8706  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9365  df-fi 9409  df-sup 9440  df-inf 9441  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-q 12938  df-rp 12980  df-xneg 13097  df-xadd 13098  df-xmul 13099  df-ioo 13333  df-ioc 13334  df-ico 13335  df-icc 13336  df-fz 13490  df-fzo 13633  df-fl 13762  df-seq 13972  df-exp 14033  df-fac 14239  df-bc 14268  df-hash 14296  df-shft 15019  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-limsup 15420  df-clim 15437  df-rlim 15438  df-sum 15638  df-ef 16016  df-sin 16018  df-cos 16019  df-pi 16021  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-hom 17226  df-cco 17227  df-rest 17373  df-topn 17374  df-0g 17392  df-gsum 17393  df-topgen 17394  df-pt 17395  df-prds 17398  df-xrs 17453  df-qtop 17458  df-imas 17459  df-xps 17461  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-submnd 18707  df-mulg 18988  df-cntz 19223  df-cmn 19692  df-psmet 21137  df-xmet 21138  df-met 21139  df-bl 21140  df-mopn 21141  df-fbas 21142  df-fg 21143  df-cnfld 21146  df-top 22617  df-topon 22634  df-topsp 22656  df-bases 22670  df-cld 22744  df-ntr 22745  df-cls 22746  df-nei 22823  df-lp 22861  df-perf 22862  df-cn 22952  df-cnp 22953  df-haus 23040  df-cmp 23112  df-tx 23287  df-hmeo 23480  df-fil 23571  df-fm 23663  df-flim 23664  df-flf 23665  df-xms 24047  df-ms 24048  df-tms 24049  df-cncf 24619  df-limc 25616  df-dv 25617
This theorem is referenced by: (None)
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