MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  argrege0 Structured version   Visualization version   GIF version

Theorem argrege0 24578
Description: Closure of the argument of a complex number with nonnegative real part. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
argrege0 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)))

Proof of Theorem argrege0
StepHypRef Expression
1 logcl 24536 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
213adant3 1126 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘𝐴) ∈ ℂ)
32imcld 14143 . 2 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℝ)
4 simp3 1132 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘𝐴))
5 simp1 1130 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 𝐴 ∈ ℂ)
65abscld 14383 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℝ)
76recnd 10270 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℂ)
87mul01d 10437 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · 0) = 0)
9 absrpcl 14236 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ+)
1093adant3 1126 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℝ+)
1110rpne0d 12080 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ≠ 0)
125, 7, 11divcld 11003 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (𝐴 / (abs‘𝐴)) ∈ ℂ)
136, 12remul2d 14175 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘((abs‘𝐴) · (𝐴 / (abs‘𝐴)))) = ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))))
145, 7, 11divcan2d 11005 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · (𝐴 / (abs‘𝐴))) = 𝐴)
1514fveq2d 6336 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘((abs‘𝐴) · (𝐴 / (abs‘𝐴)))) = (ℜ‘𝐴))
1613, 15eqtr3d 2807 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))) = (ℜ‘𝐴))
174, 8, 163brtr4d 4818 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · 0) ≤ ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))))
18 0re 10242 . . . . . . . . . 10 0 ∈ ℝ
1918a1i 11 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ∈ ℝ)
2012recld 14142 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘(𝐴 / (abs‘𝐴))) ∈ ℝ)
2119, 20, 10lemul2d 12119 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (0 ≤ (ℜ‘(𝐴 / (abs‘𝐴))) ↔ ((abs‘𝐴) · 0) ≤ ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴))))))
2217, 21mpbird 247 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘(𝐴 / (abs‘𝐴))))
23 efiarg 24574 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
24233adant3 1126 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
2524fveq2d 6336 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))) = (ℜ‘(𝐴 / (abs‘𝐴))))
2622, 25breqtrrd 4814 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
27 recosval 15072 . . . . . . 7 ((ℑ‘(log‘𝐴)) ∈ ℝ → (cos‘(ℑ‘(log‘𝐴))) = (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
283, 27syl 17 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(ℑ‘(log‘𝐴))) = (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
2926, 28breqtrrd 4814 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (cos‘(ℑ‘(log‘𝐴))))
30 halfpire 24437 . . . . . . . . . 10 (π / 2) ∈ ℝ
31 pirp 24434 . . . . . . . . . . 11 π ∈ ℝ+
32 rphalfcl 12061 . . . . . . . . . . 11 (π ∈ ℝ+ → (π / 2) ∈ ℝ+)
33 rpge0 12048 . . . . . . . . . . 11 ((π / 2) ∈ ℝ+ → 0 ≤ (π / 2))
3431, 32, 33mp2b 10 . . . . . . . . . 10 0 ≤ (π / 2)
35 pire 24431 . . . . . . . . . . 11 π ∈ ℝ
36 rphalflt 12063 . . . . . . . . . . . 12 (π ∈ ℝ+ → (π / 2) < π)
3731, 36ax-mp 5 . . . . . . . . . . 11 (π / 2) < π
3830, 35, 37ltleii 10362 . . . . . . . . . 10 (π / 2) ≤ π
3918, 35elicc2i 12444 . . . . . . . . . 10 ((π / 2) ∈ (0[,]π) ↔ ((π / 2) ∈ ℝ ∧ 0 ≤ (π / 2) ∧ (π / 2) ≤ π))
4030, 34, 38, 39mpbir3an 1426 . . . . . . . . 9 (π / 2) ∈ (0[,]π)
413recnd 10270 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℂ)
4241abscld 14383 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ∈ ℝ)
4341absge0d 14391 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (abs‘(ℑ‘(log‘𝐴))))
44 logimcl 24537 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
45443adant3 1126 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
4645simpld 482 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π < (ℑ‘(log‘𝐴)))
4735renegcli 10544 . . . . . . . . . . . . 13 -π ∈ ℝ
48 ltle 10328 . . . . . . . . . . . . 13 ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
4947, 3, 48sylancr 575 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
5046, 49mpd 15 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴)))
5145simprd 483 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ π)
52 absle 14263 . . . . . . . . . . . 12 (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
533, 35, 52sylancl 574 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
5450, 51, 53mpbir2and 692 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
5518, 35elicc2i 12444 . . . . . . . . . 10 ((abs‘(ℑ‘(log‘𝐴))) ∈ (0[,]π) ↔ ((abs‘(ℑ‘(log‘𝐴))) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘(log‘𝐴))) ∧ (abs‘(ℑ‘(log‘𝐴))) ≤ π))
5642, 43, 54, 55syl3anbrc 1428 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ∈ (0[,]π))
57 cosord 24499 . . . . . . . . 9 (((π / 2) ∈ (0[,]π) ∧ (abs‘(ℑ‘(log‘𝐴))) ∈ (0[,]π)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(abs‘(ℑ‘(log‘𝐴)))) < (cos‘(π / 2))))
5840, 56, 57sylancr 575 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(abs‘(ℑ‘(log‘𝐴)))) < (cos‘(π / 2))))
59 fveq2 6332 . . . . . . . . . . 11 ((abs‘(ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴))))
6059a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴)))))
61 cosneg 15083 . . . . . . . . . . . 12 ((ℑ‘(log‘𝐴)) ∈ ℂ → (cos‘-(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴))))
6241, 61syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘-(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴))))
63 fveq2 6332 . . . . . . . . . . . 12 ((abs‘(ℑ‘(log‘𝐴))) = -(ℑ‘(log‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘-(ℑ‘(log‘𝐴))))
6463eqeq1d 2773 . . . . . . . . . . 11 ((abs‘(ℑ‘(log‘𝐴))) = -(ℑ‘(log‘𝐴)) → ((cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴))) ↔ (cos‘-(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴)))))
6562, 64syl5ibrcom 237 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) = -(ℑ‘(log‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴)))))
663absord 14362 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐴)) ∨ (abs‘(ℑ‘(log‘𝐴))) = -(ℑ‘(log‘𝐴))))
6760, 65, 66mpjaod 849 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴))))
68 coshalfpi 24442 . . . . . . . . . 10 (cos‘(π / 2)) = 0
6968a1i 11 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(π / 2)) = 0)
7067, 69breq12d 4799 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((cos‘(abs‘(ℑ‘(log‘𝐴)))) < (cos‘(π / 2)) ↔ (cos‘(ℑ‘(log‘𝐴))) < 0))
7158, 70bitrd 268 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(ℑ‘(log‘𝐴))) < 0))
7271notbid 307 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
73 lenlt 10318 . . . . . . 7 (((abs‘(ℑ‘(log‘𝐴))) ∈ ℝ ∧ (π / 2) ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ ¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴)))))
7442, 30, 73sylancl 574 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ ¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴)))))
753recoscld 15080 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(ℑ‘(log‘𝐴))) ∈ ℝ)
76 lenlt 10318 . . . . . . 7 ((0 ∈ ℝ ∧ (cos‘(ℑ‘(log‘𝐴))) ∈ ℝ) → (0 ≤ (cos‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
7718, 75, 76sylancr 575 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (0 ≤ (cos‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
7872, 74, 773bitr4d 300 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ 0 ≤ (cos‘(ℑ‘(log‘𝐴)))))
7929, 78mpbird 247 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2))
80 absle 14263 . . . . 5 (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ (π / 2) ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ (-(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2))))
813, 30, 80sylancl 574 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ (-(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2))))
8279, 81mpbid 222 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (-(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2)))
8382simpld 482 . 2 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -(π / 2) ≤ (ℑ‘(log‘𝐴)))
8482simprd 483 . 2 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ (π / 2))
8530renegcli 10544 . . 3 -(π / 2) ∈ ℝ
8685, 30elicc2i 12444 . 2 ((ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2)))
873, 83, 84, 86syl3anbrc 1428 1 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  cc 10136  cr 10137  0cc0 10138  ici 10140   · cmul 10143   < clt 10276  cle 10277  -cneg 10469   / cdiv 10886  2c2 11272  +crp 12035  [,]cicc 12383  cre 14045  cim 14046  abscabs 14182  expce 14998  cosccos 15001  πcpi 15003  logclog 24522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ioc 12385  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-shft 14015  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-limsup 14410  df-clim 14427  df-rlim 14428  df-sum 14625  df-ef 15004  df-sin 15006  df-cos 15007  df-pi 15009  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-0g 16310  df-gsum 16311  df-topgen 16312  df-pt 16313  df-prds 16316  df-xrs 16370  df-qtop 16375  df-imas 16376  df-xps 16378  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-mulg 17749  df-cntz 17957  df-cmn 18402  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-fbas 19958  df-fg 19959  df-cnfld 19962  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-lp 21161  df-perf 21162  df-cn 21252  df-cnp 21253  df-haus 21340  df-tx 21586  df-hmeo 21779  df-fil 21870  df-fm 21962  df-flim 21963  df-flf 21964  df-xms 22345  df-ms 22346  df-tms 22347  df-cncf 22901  df-limc 23850  df-dv 23851  df-log 24524
This theorem is referenced by:  logimul  24581  isosctrlem1  24769  asinbnd  24847  isosctrlem1ALT  39692
  Copyright terms: Public domain W3C validator