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Theorem pige3ALT 26265
Description: Alternate proof of pige3 26264. 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 12297 . . 3 3 ∈ β„‚
21mullidi 11223 . 2 (1 Β· 3) = 3
3 tru 1543 . . . . . 6 ⊀
4 0xr 11265 . . . . . . . 8 0 ∈ ℝ*
5 pirp 26207 . . . . . . . . . 10 Ο€ ∈ ℝ+
6 3rp 12984 . . . . . . . . . 10 3 ∈ ℝ+
7 rpdivcl 13003 . . . . . . . . . 10 ((Ο€ ∈ ℝ+ ∧ 3 ∈ ℝ+) β†’ (Ο€ / 3) ∈ ℝ+)
85, 6, 7mp2an 688 . . . . . . . . 9 (Ο€ / 3) ∈ ℝ+
9 rpxr 12987 . . . . . . . . 9 ((Ο€ / 3) ∈ ℝ+ β†’ (Ο€ / 3) ∈ ℝ*)
108, 9ax-mp 5 . . . . . . . 8 (Ο€ / 3) ∈ ℝ*
11 rpge0 12991 . . . . . . . . 9 ((Ο€ / 3) ∈ ℝ+ β†’ 0 ≀ (Ο€ / 3))
128, 11ax-mp 5 . . . . . . . 8 0 ≀ (Ο€ / 3)
13 lbicc2 13445 . . . . . . . 8 ((0 ∈ ℝ* ∧ (Ο€ / 3) ∈ ℝ* ∧ 0 ≀ (Ο€ / 3)) β†’ 0 ∈ (0[,](Ο€ / 3)))
144, 10, 12, 13mp3an 1459 . . . . . . 7 0 ∈ (0[,](Ο€ / 3))
15 ubicc2 13446 . . . . . . . 8 ((0 ∈ ℝ* ∧ (Ο€ / 3) ∈ ℝ* ∧ 0 ≀ (Ο€ / 3)) β†’ (Ο€ / 3) ∈ (0[,](Ο€ / 3)))
164, 10, 12, 15mp3an 1459 . . . . . . 7 (Ο€ / 3) ∈ (0[,](Ο€ / 3))
1714, 16pm3.2i 469 . . . . . 6 (0 ∈ (0[,](Ο€ / 3)) ∧ (Ο€ / 3) ∈ (0[,](Ο€ / 3)))
18 0re 11220 . . . . . . . 8 0 ∈ ℝ
1918a1i 11 . . . . . . 7 (⊀ β†’ 0 ∈ ℝ)
20 pire 26204 . . . . . . . . 9 Ο€ ∈ ℝ
21 3re 12296 . . . . . . . . 9 3 ∈ ℝ
22 3ne0 12322 . . . . . . . . 9 3 β‰  0
2320, 21, 22redivcli 11985 . . . . . . . 8 (Ο€ / 3) ∈ ℝ
2423a1i 11 . . . . . . 7 (⊀ β†’ (Ο€ / 3) ∈ ℝ)
25 efcn 26191 . . . . . . . . 9 exp ∈ (ℂ–cnβ†’β„‚)
2625a1i 11 . . . . . . . 8 (⊀ β†’ exp ∈ (ℂ–cnβ†’β„‚))
27 iccssre 13410 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ (Ο€ / 3) ∈ ℝ) β†’ (0[,](Ο€ / 3)) βŠ† ℝ)
2818, 23, 27mp2an 688 . . . . . . . . . . 11 (0[,](Ο€ / 3)) βŠ† ℝ
29 ax-resscn 11169 . . . . . . . . . . 11 ℝ βŠ† β„‚
3028, 29sstri 3990 . . . . . . . . . 10 (0[,](Ο€ / 3)) βŠ† β„‚
31 resmpt 6036 . . . . . . . . . 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 4004 . . . . . . . . . . 11 (⊀ β†’ β„‚ βŠ† β„‚)
34 ax-icn 11171 . . . . . . . . . . . . 13 i ∈ β„‚
35 simpr 483 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ π‘₯ ∈ β„‚)
36 mulcl 11196 . . . . . . . . . . . . 13 ((i ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ (i Β· π‘₯) ∈ β„‚)
3734, 35, 36sylancr 585 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ (i Β· π‘₯) ∈ β„‚)
3837fmpttd 7115 . . . . . . . . . . 11 (⊀ β†’ (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)):β„‚βŸΆβ„‚)
39 cnelprrecn 11205 . . . . . . . . . . . . . . . 16 β„‚ ∈ {ℝ, β„‚}
4039a1i 11 . . . . . . . . . . . . . . 15 (⊀ β†’ β„‚ ∈ {ℝ, β„‚})
41 ax-1cn 11170 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
4241a1i 11 . . . . . . . . . . . . . . 15 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ 1 ∈ β„‚)
4340dvmptid 25709 . . . . . . . . . . . . . . 15 (⊀ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ π‘₯)) = (π‘₯ ∈ β„‚ ↦ 1))
4434a1i 11 . . . . . . . . . . . . . . 15 (⊀ β†’ i ∈ β„‚)
4540, 35, 42, 43, 44dvmptcmul 25716 . . . . . . . . . . . . . 14 (⊀ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = (π‘₯ ∈ β„‚ ↦ (i Β· 1)))
4634mulridi 11222 . . . . . . . . . . . . . . 15 (i Β· 1) = i
4746mpteq2i 5252 . . . . . . . . . . . . . 14 (π‘₯ ∈ β„‚ ↦ (i Β· 1)) = (π‘₯ ∈ β„‚ ↦ i)
4845, 47eqtrdi 2786 . . . . . . . . . . . . 13 (⊀ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = (π‘₯ ∈ β„‚ ↦ i))
4948dmeqd 5904 . . . . . . . . . . . 12 (⊀ β†’ dom (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = dom (π‘₯ ∈ β„‚ ↦ i))
5034elexi 3492 . . . . . . . . . . . . 13 i ∈ V
51 eqid 2730 . . . . . . . . . . . . 13 (π‘₯ ∈ β„‚ ↦ i) = (π‘₯ ∈ β„‚ ↦ i)
5250, 51dmmpti 6693 . . . . . . . . . . . 12 dom (π‘₯ ∈ β„‚ ↦ i) = β„‚
5349, 52eqtrdi 2786 . . . . . . . . . . 11 (⊀ β†’ dom (β„‚ D (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯))) = β„‚)
54 dvcn 25671 . . . . . . . . . . 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 1371 . . . . . . . . . 10 (⊀ β†’ (π‘₯ ∈ β„‚ ↦ (i Β· π‘₯)) ∈ (ℂ–cnβ†’β„‚))
56 rescncf 24637 . . . . . . . . . 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 2832 . . . . . . . 8 (⊀ β†’ (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (i Β· π‘₯)) ∈ ((0[,](Ο€ / 3))–cnβ†’β„‚))
5926, 58cncfmpt1f 24654 . . . . . . 7 (⊀ β†’ (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))) ∈ ((0[,](Ο€ / 3))–cnβ†’β„‚))
60 reelprrecn 11204 . . . . . . . . . . 11 ℝ ∈ {ℝ, β„‚}
6160a1i 11 . . . . . . . . . 10 (⊀ β†’ ℝ ∈ {ℝ, β„‚})
62 recn 11202 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ π‘₯ ∈ β„‚)
63 efcl 16030 . . . . . . . . . . . 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 591 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ ℝ) β†’ (expβ€˜(i Β· π‘₯)) ∈ β„‚)
66 mulcl 11196 . . . . . . . . . . . 12 (((expβ€˜(i Β· π‘₯)) ∈ β„‚ ∧ i ∈ β„‚) β†’ ((expβ€˜(i Β· π‘₯)) Β· i) ∈ β„‚)
6764, 34, 66sylancl 584 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ β„‚) β†’ ((expβ€˜(i Β· π‘₯)) Β· i) ∈ β„‚)
6862, 67sylan2 591 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ ℝ) β†’ ((expβ€˜(i Β· π‘₯)) Β· i) ∈ β„‚)
69 eqid 2730 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
7069cnfldtopon 24519 . . . . . . . . . . . 12 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
71 toponmax 22648 . . . . . . . . . . . 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 3964 . . . . . . . . . . . 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 16030 . . . . . . . . . . . . 13 (𝑦 ∈ β„‚ β†’ (expβ€˜π‘¦) ∈ β„‚)
7877adantl 480 . . . . . . . . . . . 12 ((⊀ ∧ 𝑦 ∈ β„‚) β†’ (expβ€˜π‘¦) ∈ β„‚)
79 dvef 25732 . . . . . . . . . . . . 13 (β„‚ D exp) = exp
80 eff 16029 . . . . . . . . . . . . . . . 16 exp:β„‚βŸΆβ„‚
8180a1i 11 . . . . . . . . . . . . . . 15 (⊀ β†’ exp:β„‚βŸΆβ„‚)
8281feqmptd 6959 . . . . . . . . . . . . . 14 (⊀ β†’ exp = (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦)))
8382oveq2d 7427 . . . . . . . . . . . . 13 (⊀ β†’ (β„‚ D exp) = (β„‚ D (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦))))
8479, 83, 823eqtr3a 2794 . . . . . . . . . . . 12 (⊀ β†’ (β„‚ D (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦))) = (𝑦 ∈ β„‚ ↦ (expβ€˜π‘¦)))
85 fveq2 6890 . . . . . . . . . . . 12 (𝑦 = (i Β· π‘₯) β†’ (expβ€˜π‘¦) = (expβ€˜(i Β· π‘₯)))
8640, 40, 37, 76, 78, 78, 48, 84, 85, 85dvmptco 25724 . . . . . . . . . . 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 25708 . . . . . . . . . 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 24539 . . . . . . . . . 10 (topGenβ€˜ran (,)) = ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)
90 iccntr 24557 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ (Ο€ / 3) ∈ ℝ) β†’ ((intβ€˜(topGenβ€˜ran (,)))β€˜(0[,](Ο€ / 3))) = (0(,)(Ο€ / 3)))
9118, 24, 90sylancr 585 . . . . . . . . . 10 (⊀ β†’ ((intβ€˜(topGenβ€˜ran (,)))β€˜(0[,](Ο€ / 3))) = (0(,)(Ο€ / 3)))
9261, 65, 68, 87, 88, 89, 69, 91dvmptres2 25714 . . . . . . . . 9 (⊀ β†’ (ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))) = (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i)))
9392dmeqd 5904 . . . . . . . 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 7444 . . . . . . . . 9 ((expβ€˜(i Β· π‘₯)) Β· i) ∈ V
95 eqid 2730 . . . . . . . . 9 (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i)) = (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i))
9694, 95dmmpti 6693 . . . . . . . 8 dom (π‘₯ ∈ (0(,)(Ο€ / 3)) ↦ ((expβ€˜(i Β· π‘₯)) Β· i)) = (0(,)(Ο€ / 3))
9793, 96eqtrdi 2786 . . . . . . 7 (⊀ β†’ dom (ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))) = (0(,)(Ο€ / 3)))
98 1re 11218 . . . . . . . 8 1 ∈ ℝ
9998a1i 11 . . . . . . 7 (⊀ β†’ 1 ∈ ℝ)
10092fveq1d 6892 . . . . . . . . . . 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 7419 . . . . . . . . . . . . . 14 (π‘₯ = 𝑦 β†’ (i Β· π‘₯) = (i Β· 𝑦))
102101fveq2d 6894 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ (expβ€˜(i Β· π‘₯)) = (expβ€˜(i Β· 𝑦)))
103102oveq1d 7426 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((expβ€˜(i Β· π‘₯)) Β· i) = ((expβ€˜(i Β· 𝑦)) Β· i))
104103, 95, 94fvmpt3i 7002 . . . . . . . . . . 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 2790 . . . . . . . . . 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 6894 . . . . . . . . 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 13389 . . . . . . . . . . . . . . 15 (0(,)(Ο€ / 3)) βŠ† ℝ
108107a1i 11 . . . . . . . . . . . . . 14 (⊀ β†’ (0(,)(Ο€ / 3)) βŠ† ℝ)
109108sselda 3981 . . . . . . . . . . . . 13 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ 𝑦 ∈ ℝ)
110109recnd 11246 . . . . . . . . . . . 12 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ 𝑦 ∈ β„‚)
111 mulcl 11196 . . . . . . . . . . . 12 ((i ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (i Β· 𝑦) ∈ β„‚)
11234, 110, 111sylancr 585 . . . . . . . . . . 11 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (i Β· 𝑦) ∈ β„‚)
113 efcl 16030 . . . . . . . . . . 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 15245 . . . . . . . . . 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 584 . . . . . . . . 9 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜((expβ€˜(i Β· 𝑦)) Β· i)) = ((absβ€˜(expβ€˜(i Β· 𝑦))) Β· (absβ€˜i)))
117 absefi 16143 . . . . . . . . . . . 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 15237 . . . . . . . . . . . 12 (absβ€˜i) = 1
120119a1i 11 . . . . . . . . . . 11 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜i) = 1)
121118, 120oveq12d 7429 . . . . . . . . . 10 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ ((absβ€˜(expβ€˜(i Β· 𝑦))) Β· (absβ€˜i)) = (1 Β· 1))
12241mulridi 11222 . . . . . . . . . 10 (1 Β· 1) = 1
123121, 122eqtrdi 2786 . . . . . . . . 9 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ ((absβ€˜(expβ€˜(i Β· 𝑦))) Β· (absβ€˜i)) = 1)
124106, 116, 1233eqtrd 2774 . . . . . . . 8 ((⊀ ∧ 𝑦 ∈ (0(,)(Ο€ / 3))) β†’ (absβ€˜((ℝ D (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))))β€˜π‘¦)) = 1)
125 1le1 11846 . . . . . . . 8 1 ≀ 1
126124, 125eqbrtrdi 5186 . . . . . . 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 25745 . . . . . 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 688 . . . . 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 7419 . . . . . . . . . . . . 13 (π‘₯ = 0 β†’ (i Β· π‘₯) = (i Β· 0))
130 it0e0 12438 . . . . . . . . . . . . 13 (i Β· 0) = 0
131129, 130eqtrdi 2786 . . . . . . . . . . . 12 (π‘₯ = 0 β†’ (i Β· π‘₯) = 0)
132131fveq2d 6894 . . . . . . . . . . 11 (π‘₯ = 0 β†’ (expβ€˜(i Β· π‘₯)) = (expβ€˜0))
133 ef0 16038 . . . . . . . . . . 11 (expβ€˜0) = 1
134132, 133eqtrdi 2786 . . . . . . . . . 10 (π‘₯ = 0 β†’ (expβ€˜(i Β· π‘₯)) = 1)
135 eqid 2730 . . . . . . . . . 10 (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯))) = (π‘₯ ∈ (0[,](Ο€ / 3)) ↦ (expβ€˜(i Β· π‘₯)))
136 fvex 6903 . . . . . . . . . 10 (expβ€˜(i Β· π‘₯)) ∈ V
137134, 135, 136fvmpt3i 7002 . . . . . . . . 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 7419 . . . . . . . . . . 11 (π‘₯ = (Ο€ / 3) β†’ (i Β· π‘₯) = (i Β· (Ο€ / 3)))
140139fveq2d 6894 . . . . . . . . . 10 (π‘₯ = (Ο€ / 3) β†’ (expβ€˜(i Β· π‘₯)) = (expβ€˜(i Β· (Ο€ / 3))))
141140, 135, 136fvmpt3i 7002 . . . . . . . . 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 7423 . . . . . . 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 11232 . . . . . . . . . 10 (Ο€ / 3) ∈ β„‚
14534, 144mulcli 11225 . . . . . . . . 9 (i Β· (Ο€ / 3)) ∈ β„‚
146 efcl 16030 . . . . . . . . 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 11465 . . . . . . . . . 10 -i ∈ β„‚
149148, 144mulcli 11225 . . . . . . . . 9 (-i Β· (Ο€ / 3)) ∈ β„‚
150 efcl 16030 . . . . . . . . 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 16070 . . . . . . . . . . 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 26262 . . . . . . . . . . 11 ((sinβ€˜(Ο€ / 3)) = ((βˆšβ€˜3) / 2) ∧ (cosβ€˜(Ο€ / 3)) = (1 / 2))
155154simpri 484 . . . . . . . . . 10 (cosβ€˜(Ο€ / 3)) = (1 / 2)
156153, 155eqtr3i 2760 . . . . . . . . 9 (((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) / 2) = (1 / 2)
157147, 151addcli 11224 . . . . . . . . . 10 ((expβ€˜(i Β· (Ο€ / 3))) + (expβ€˜(-i Β· (Ο€ / 3)))) ∈ β„‚
158 2cn 12291 . . . . . . . . . 10 2 ∈ β„‚
159 2ne0 12320 . . . . . . . . . 10 2 β‰  0
160157, 41, 158, 159div11i 11977 . . . . . . . . 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 11553 . . . . . . 7 (1 βˆ’ (expβ€˜(i Β· (Ο€ / 3)))) = (expβ€˜(-i Β· (Ο€ / 3)))
163 mulneg12 11656 . . . . . . . . 9 ((i ∈ β„‚ ∧ (Ο€ / 3) ∈ β„‚) β†’ (-i Β· (Ο€ / 3)) = (i Β· -(Ο€ / 3)))
16434, 144, 163mp2an 688 . . . . . . . 8 (-i Β· (Ο€ / 3)) = (i Β· -(Ο€ / 3))
165164fveq2i 6893 . . . . . . 7 (expβ€˜(-i Β· (Ο€ / 3))) = (expβ€˜(i Β· -(Ο€ / 3)))
166143, 162, 1653eqtri 2762 . . . . . 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 6893 . . . . 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 15351 . . . . . . . 8 (absβ€˜-(Ο€ / 3)) = (absβ€˜(Ο€ / 3))
169 df-neg 11451 . . . . . . . . 9 -(Ο€ / 3) = (0 βˆ’ (Ο€ / 3))
170169fveq2i 6893 . . . . . . . 8 (absβ€˜-(Ο€ / 3)) = (absβ€˜(0 βˆ’ (Ο€ / 3)))
171168, 170eqtr3i 2760 . . . . . . 7 (absβ€˜(Ο€ / 3)) = (absβ€˜(0 βˆ’ (Ο€ / 3)))
172 rprege0 12993 . . . . . . . 8 ((Ο€ / 3) ∈ ℝ+ β†’ ((Ο€ / 3) ∈ ℝ ∧ 0 ≀ (Ο€ / 3)))
173 absid 15247 . . . . . . . 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 2760 . . . . . 6 (absβ€˜(0 βˆ’ (Ο€ / 3))) = (Ο€ / 3)
176175oveq2i 7422 . . . . 5 (1 Β· (absβ€˜(0 βˆ’ (Ο€ / 3)))) = (1 Β· (Ο€ / 3))
177128, 167, 1763brtr3i 5176 . . . 4 (absβ€˜(expβ€˜(i Β· -(Ο€ / 3)))) ≀ (1 Β· (Ο€ / 3))
17823renegcli 11525 . . . . 5 -(Ο€ / 3) ∈ ℝ
179 absefi 16143 . . . . 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 11223 . . . 4 (1 Β· (Ο€ / 3)) = (Ο€ / 3)
182177, 180, 1813brtr3i 5176 . . 3 1 ≀ (Ο€ / 3)
183 3pos 12321 . . . . 5 0 < 3
18421, 183pm3.2i 469 . . . 4 (3 ∈ ℝ ∧ 0 < 3)
185 lemuldiv 12098 . . . 4 ((1 ∈ ℝ ∧ Ο€ ∈ ℝ ∧ (3 ∈ ℝ ∧ 0 < 3)) β†’ ((1 Β· 3) ≀ Ο€ ↔ 1 ≀ (Ο€ / 3)))
18698, 20, 184, 185mp3an 1459 . . 3 ((1 Β· 3) ≀ Ο€ ↔ 1 ≀ (Ο€ / 3))
187182, 186mpbir 230 . 2 (1 Β· 3) ≀ Ο€
1882, 187eqbrtrri 5170 1 3 ≀ Ο€
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539  βŠ€wtru 1540   ∈ wcel 2104   ∩ cin 3946   βŠ† wss 3947  {cpr 4629   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676   β†Ύ cres 5677  βŸΆwf 6538  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  3c3 12272  β„+crp 12978  (,)cioo 13328  [,]cicc 13331  βˆšcsqrt 15184  abscabs 15185  expce 16009  sincsin 16011  cosccos 16012  Ο€cpi 16014  TopOpenctopn 17371  topGenctg 17387  β„‚fldccnfld 21144  TopOnctopon 22632  intcnt 22741  β€“cnβ†’ccncf 24616   D cdv 25612
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  ax-mulf 11192
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-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-cmp 23111  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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