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Theorem isosctrlem2 25384
Description: Lemma for isosctr 25386. Corresponds to the case where one vertex is at 0, another at 1 and the third lies on the unit circle. (Contributed by Saveliy Skresanov, 31-Dec-2016.)
Assertion
Ref Expression
isosctrlem2 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))

Proof of Theorem isosctrlem2
StepHypRef Expression
1 1cnd 10614 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℂ)
2 simpl1 1187 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℂ)
31, 2negsubd 10981 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) = (1 − 𝐴))
4 1rp 12372 . . . . . . . 8 1 ∈ ℝ+
54a1i 11 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ+)
6 simpl3 1189 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ¬ 1 = 𝐴)
7 simpl2 1188 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (abs‘𝐴) = 1)
81, 2, 1sub32d 11007 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = ((1 − 1) − 𝐴))
9 1m1e0 11688 . . . . . . . . . . . . . . . . 17 (1 − 1) = 0
109oveq1i 7143 . . . . . . . . . . . . . . . 16 ((1 − 1) − 𝐴) = (0 − 𝐴)
11 df-neg 10851 . . . . . . . . . . . . . . . 16 -𝐴 = (0 − 𝐴)
1210, 11eqtr4i 2846 . . . . . . . . . . . . . . 15 ((1 − 1) − 𝐴) = -𝐴
138, 12syl6eq 2871 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = -𝐴)
14 1cnd 10614 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 1 ∈ ℂ)
15 simp1 1132 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ∈ ℂ)
1614, 15subcld 10975 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
1716adantr 483 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℂ)
18 ax-1cn 10573 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
19 subeq0 10890 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
2018, 19mpan 688 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
2120biimpd 231 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴))
2221con3dimp 411 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0)
2322neqned 3013 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
24233adant2 1127 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
2524adantr 483 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ≠ 0)
2617, 25recrecd 11391 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) = (1 − 𝐴))
2714, 16, 24div2negd 11409 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
2827adantr 483 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
2915negcld 10962 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ∈ ℂ)
3029, 16, 24cjdivd 14562 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = ((∗‘-𝐴) / (∗‘(1 − 𝐴))))
3115cjnegd 14550 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(∗‘𝐴))
32 fveq2 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
33 abs0 14625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (abs‘0) = 0
3432, 33syl6eq 2871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = 0 → (abs‘𝐴) = 0)
35 eqtr2 2841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((abs‘𝐴) = 1 ∧ (abs‘𝐴) = 0) → 1 = 0)
3634, 35sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → 1 = 0)
37 ax-1ne0 10584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1 ≠ 0
38 neneq 3012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (1 ≠ 0 → ¬ 1 = 0)
3937, 38mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → ¬ 1 = 0)
4036, 39pm2.65da 815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((abs‘𝐴) = 1 → ¬ 𝐴 = 0)
4140adantl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ¬ 𝐴 = 0)
42 df-ne 3007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
43 oveq1 7140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = (1↑2))
44 sq1 13543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (1↑2) = 1
4543, 44syl6eq 2871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = 1)
4645adantl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
47 absvalsq 14620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
4847adantr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
4946, 48eqtr3d 2857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 1 = (𝐴 · (∗‘𝐴)))
50493adant3 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 1 = (𝐴 · (∗‘𝐴)))
5150oveq1d 7148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((𝐴 · (∗‘𝐴)) / 𝐴))
52 simp1 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
5352cjcld 14535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ)
54 simp3 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
5553, 52, 54divcan3d 11399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
5651, 55eqtrd 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = (∗‘𝐴))
5742, 56syl3an3br 1404 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 𝐴 = 0) → (1 / 𝐴) = (∗‘𝐴))
5841, 57mpd3an3 1458 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = (∗‘𝐴))
5958eqcomd 2826 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘𝐴) = (1 / 𝐴))
60593adant3 1128 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘𝐴) = (1 / 𝐴))
6160negeqd 10858 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(∗‘𝐴) = -(1 / 𝐴))
6231, 61eqtrd 2855 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(1 / 𝐴))
6362oveq1d 7148 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((∗‘-𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (∗‘(1 − 𝐴))))
64 cjsub 14488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
6518, 64mpan 688 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
66 1red 10620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐴 ∈ ℂ → 1 ∈ ℝ)
6766cjred 14565 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐴 ∈ ℂ → (∗‘1) = 1)
6867oveq1d 7148 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ ℂ → ((∗‘1) − (∗‘𝐴)) = (1 − (∗‘𝐴)))
6965, 68eqtrd 2855 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
7069adantr 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
7159oveq2d 7149 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (∗‘𝐴)) = (1 − (1 / 𝐴)))
7270, 71eqtrd 2855 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
73723adant3 1128 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
7473oveq2d 7149 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
7530, 63, 743eqtrd 2859 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
76403ad2ant2 1130 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ¬ 𝐴 = 0)
7776neqned 3013 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ≠ 0)
78 1cnd 10614 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 1 ∈ ℂ)
79 simpl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
80 simpr 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
8178, 79, 80divnegd 11407 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -(1 / 𝐴) = (-1 / 𝐴))
8281oveq1d 7148 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
8315, 77, 82syl2anc 586 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
8414negcld 10962 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -1 ∈ ℂ)
8584, 15, 77divcld 11394 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / 𝐴) ∈ ℂ)
8615, 77reccld 11387 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / 𝐴) ∈ ℂ)
8714, 86subcld 10975 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ∈ ℂ)
8816, 24cjne0d 14542 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) ≠ 0)
8973, 88eqnetrrd 3074 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ≠ 0)
9085, 87, 15, 89, 77divcan5d 11420 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
9184, 15, 77divcan2d 11396 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (-1 / 𝐴)) = -1)
9215, 14, 86subdid 11074 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = ((𝐴 · 1) − (𝐴 · (1 / 𝐴))))
9315mulid1d 10636 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · 1) = 𝐴)
9415, 77recidd 11389 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 / 𝐴)) = 1)
9593, 94oveq12d 7151 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · 1) − (𝐴 · (1 / 𝐴))) = (𝐴 − 1))
9692, 95eqtrd 2855 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = (𝐴 − 1))
9791, 96oveq12d 7151 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = (-1 / (𝐴 − 1)))
9883, 90, 973eqtr2d 2861 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = (-1 / (𝐴 − 1)))
99 subcl 10863 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
10099negnegd 10966 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = (𝐴 − 1))
101 negsubdi2 10923 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
102101negeqd 10858 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = -(1 − 𝐴))
103100, 102eqtr3d 2857 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) = -(1 − 𝐴))
10415, 14, 103syl2anc 586 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 − 1) = -(1 − 𝐴))
105104oveq2d 7149 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / (𝐴 − 1)) = (-1 / -(1 − 𝐴)))
10675, 98, 1053eqtrd 2859 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
107106adantr 483 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
10829, 16, 24divcld 11394 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
109108adantr 483 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
110 simpr 487 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(-𝐴 / (1 − 𝐴))) = 0)
111109, 110reim0bd 14539 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ)
112111cjred 14565 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-𝐴 / (1 − 𝐴)))
113112, 111eqeltrd 2911 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
114107, 113eqeltrrd 2912 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) ∈ ℝ)
11528, 114eqeltrrd 2912 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ∈ ℝ)
11616, 24recne0d 11388 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ≠ 0)
117116adantr 483 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ≠ 0)
118115, 117rereccld 11445 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) ∈ ℝ)
11926, 118eqeltrrd 2912 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ)
120 1red 10620 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ)
121119, 120resubcld 11046 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) ∈ ℝ)
12213, 121eqeltrrd 2912 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ)
1232, 122negrebd 10974 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℝ)
124123absord 14755 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))
125 eqeq1 2824 . . . . . . . . . . . . 13 ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 ↔ 1 = 𝐴))
126125biimpd 231 . . . . . . . . . . . 12 ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 → 1 = 𝐴))
127 eqeq1 2824 . . . . . . . . . . . . 13 ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 ↔ 1 = -𝐴))
128127biimpd 231 . . . . . . . . . . . 12 ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 → 1 = -𝐴))
129126, 128orim12d 961 . . . . . . . . . . 11 ((abs‘𝐴) = 1 → (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → (1 = 𝐴 ∨ 1 = -𝐴)))
1307, 124, 129sylc 65 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 = 𝐴 ∨ 1 = -𝐴))
131130ord 860 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (¬ 1 = 𝐴 → 1 = -𝐴))
1326, 131mpd 15 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 = -𝐴)
133132, 5eqeltrrd 2912 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ+)
1345, 133rpaddcld 12425 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) ∈ ℝ+)
1353, 134eqeltrrd 2912 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ+)
136135relogcld 25193 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(1 − 𝐴)) ∈ ℝ)
137136reim0d 14564 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = 0)
138133, 135rpdivcld 12427 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ+)
139138relogcld 25193 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
140139reim0d 14564 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) = 0)
141137, 140eqtr4d 2858 . 2 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
14216, 24logcld 25141 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
143142adantr 483 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(1 − 𝐴)) ∈ ℂ)
144143imcld 14534 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
145144recnd 10647 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℂ)
146108adantr 483 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
14715, 77negne0d 10973 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ≠ 0)
14829, 16, 147, 24divne0d 11410 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ≠ 0)
149148adantr 483 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ≠ 0)
150146, 149logcld 25141 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℂ)
151150imcld 14534 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℝ)
152151recnd 10647 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℂ)
153106fveq2d 6650 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
154153adantr 483 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
155 logcj 25176 . . . . . . 7 (((-𝐴 / (1 − 𝐴)) ∈ ℂ ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
156108, 155sylan 582 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
15716, 24reccld 11387 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ∈ ℂ)
158157, 116logcld 25141 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
159158negnegd 10966 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → --(log‘(1 / (1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
160 isosctrlem1 25383 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)
161 logrec 25328 . . . . . . . . . 10 (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ (ℑ‘(log‘(1 − 𝐴))) ≠ π) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
16216, 24, 160, 161syl3anc 1367 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
163162negeqd 10858 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(log‘(1 − 𝐴)) = --(log‘(1 / (1 − 𝐴))))
16427fveq2d 6650 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
165159, 163, 1643eqtr4rd 2866 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
166165adantr 483 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
167154, 156, 1663eqtr3rd 2864 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(log‘(1 − 𝐴)) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
168167fveq2d 6650 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))))
169143imnegd 14549 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = -(ℑ‘(log‘(1 − 𝐴))))
170150imcjd 14544 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
171168, 169, 1703eqtr3d 2863 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(ℑ‘(log‘(1 − 𝐴))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
172145, 152, 171neg11d 10987 . 2 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
173141, 172pm2.61dane 3093 1 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1083   = wceq 1537  wcel 2114  wne 3006  cfv 6331  (class class class)co 7133  cc 10513  cr 10514  0cc0 10515  1c1 10516   + caddc 10518   · cmul 10520  cmin 10848  -cneg 10849   / cdiv 11275  2c2 11671  +crp 12368  cexp 13414  ccj 14435  cim 14437  abscabs 14573  πcpi 15400  logclog 25125
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593  ax-addf 10594  ax-mulf 10595
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-om 7559  df-1st 7667  df-2nd 7668  df-supp 7809  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-pm 8387  df-ixp 8440  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-fsupp 8812  df-fi 8853  df-sup 8884  df-inf 8885  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-z 11961  df-dec 12078  df-uz 12223  df-q 12328  df-rp 12369  df-xneg 12486  df-xadd 12487  df-xmul 12488  df-ioo 12721  df-ioc 12722  df-ico 12723  df-icc 12724  df-fz 12877  df-fzo 13018  df-fl 13146  df-mod 13222  df-seq 13354  df-exp 13415  df-fac 13619  df-bc 13648  df-hash 13676  df-shft 14406  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-limsup 14808  df-clim 14825  df-rlim 14826  df-sum 15023  df-ef 15401  df-sin 15403  df-cos 15404  df-pi 15406  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-mulr 16558  df-starv 16559  df-sca 16560  df-vsca 16561  df-ip 16562  df-tset 16563  df-ple 16564  df-ds 16566  df-unif 16567  df-hom 16568  df-cco 16569  df-rest 16675  df-topn 16676  df-0g 16694  df-gsum 16695  df-topgen 16696  df-pt 16697  df-prds 16700  df-xrs 16754  df-qtop 16759  df-imas 16760  df-xps 16762  df-mre 16836  df-mrc 16837  df-acs 16839  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-submnd 17936  df-mulg 18204  df-cntz 18426  df-cmn 18887  df-psmet 20513  df-xmet 20514  df-met 20515  df-bl 20516  df-mopn 20517  df-fbas 20518  df-fg 20519  df-cnfld 20522  df-top 21478  df-topon 21495  df-topsp 21517  df-bases 21530  df-cld 21603  df-ntr 21604  df-cls 21605  df-nei 21682  df-lp 21720  df-perf 21721  df-cn 21811  df-cnp 21812  df-haus 21899  df-tx 22146  df-hmeo 22339  df-fil 22430  df-fm 22522  df-flim 22523  df-flf 22524  df-xms 22906  df-ms 22907  df-tms 22908  df-cncf 23462  df-limc 24448  df-dv 24449  df-log 25127
This theorem is referenced by:  isosctrlem3  25385
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