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Theorem isosctrlem2 24669
Description: Lemma for isosctr 24671. 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 10169 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℂ)
2 simpl1 1204 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℂ)
31, 2negsubd 10511 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) = (1 − 𝐴))
4 1rp 11950 . . . . . . . 8 1 ∈ ℝ+
54a1i 11 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ+)
6 simpl3 1208 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ¬ 1 = 𝐴)
7 simpl2 1206 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (abs‘𝐴) = 1)
81, 2, 1sub32d 10537 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = ((1 − 1) − 𝐴))
9 1m1e0 11202 . . . . . . . . . . . . . . . . 17 (1 − 1) = 0
109oveq1i 6775 . . . . . . . . . . . . . . . 16 ((1 − 1) − 𝐴) = (0 − 𝐴)
11 df-neg 10382 . . . . . . . . . . . . . . . 16 -𝐴 = (0 − 𝐴)
1210, 11eqtr4i 2749 . . . . . . . . . . . . . . 15 ((1 − 1) − 𝐴) = -𝐴
138, 12syl6eq 2774 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = -𝐴)
14 1cnd 10169 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 1 ∈ ℂ)
15 simp1 1128 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ∈ ℂ)
1614, 15subcld 10505 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
1716adantr 472 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℂ)
18 ax-1cn 10107 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
19 subeq0 10420 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
2018, 19mpan 708 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
2120biimpd 219 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴))
2221con3dimp 456 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0)
2322neqned 2903 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
24233adant2 1123 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
2524adantr 472 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ≠ 0)
2617, 25recrecd 10911 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) = (1 − 𝐴))
2714, 16, 24div2negd 10929 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
2827adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
2915negcld 10492 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ∈ ℂ)
3029, 16, 24cjdivd 14083 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = ((∗‘-𝐴) / (∗‘(1 − 𝐴))))
3115cjnegd 14071 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(∗‘𝐴))
32 fveq2 6304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
33 abs0 14145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (abs‘0) = 0
3432, 33syl6eq 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = 0 → (abs‘𝐴) = 0)
35 eqtr2 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((abs‘𝐴) = 1 ∧ (abs‘𝐴) = 0) → 1 = 0)
3634, 35sylan2 492 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → 1 = 0)
37 ax-1ne0 10118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1 ≠ 0
38 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (1 ≠ 0 → 1 ≠ 0)
3938neneqd 2901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (1 ≠ 0 → ¬ 1 = 0)
4037, 39mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → ¬ 1 = 0)
4136, 40pm2.65da 601 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((abs‘𝐴) = 1 → ¬ 𝐴 = 0)
4241adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ¬ 𝐴 = 0)
43 df-ne 2897 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
44 oveq1 6772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = (1↑2))
45 sq1 13073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (1↑2) = 1
4644, 45syl6eq 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = 1)
4746adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
48 absvalsq 14140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
4948adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
5047, 49eqtr3d 2760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 1 = (𝐴 · (∗‘𝐴)))
51503adant3 1124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 1 = (𝐴 · (∗‘𝐴)))
5251oveq1d 6780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((𝐴 · (∗‘𝐴)) / 𝐴))
53 simp1 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
5453cjcld 14056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ)
55 simp3 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
5654, 53, 55divcan3d 10919 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
5752, 56eqtrd 2758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = (∗‘𝐴))
5843, 57syl3an3br 1480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 𝐴 = 0) → (1 / 𝐴) = (∗‘𝐴))
5942, 58mpd3an3 1538 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = (∗‘𝐴))
6059eqcomd 2730 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘𝐴) = (1 / 𝐴))
61603adant3 1124 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘𝐴) = (1 / 𝐴))
6261negeqd 10388 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(∗‘𝐴) = -(1 / 𝐴))
6331, 62eqtrd 2758 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(1 / 𝐴))
6463oveq1d 6780 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((∗‘-𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (∗‘(1 − 𝐴))))
65 cjsub 14009 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
6618, 65mpan 708 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
67 1red 10168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐴 ∈ ℂ → 1 ∈ ℝ)
6867cjred 14086 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐴 ∈ ℂ → (∗‘1) = 1)
6968oveq1d 6780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ ℂ → ((∗‘1) − (∗‘𝐴)) = (1 − (∗‘𝐴)))
7066, 69eqtrd 2758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
7170adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
7260oveq2d 6781 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (∗‘𝐴)) = (1 − (1 / 𝐴)))
7371, 72eqtrd 2758 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
74733adant3 1124 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
7574oveq2d 6781 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
7630, 64, 753eqtrd 2762 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
77413ad2ant2 1126 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ¬ 𝐴 = 0)
7877neqned 2903 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ≠ 0)
79 1cnd 10169 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 1 ∈ ℂ)
80 simpl 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
81 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
8279, 80, 81divnegd 10927 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -(1 / 𝐴) = (-1 / 𝐴))
8382oveq1d 6780 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
8415, 78, 83syl2anc 696 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
8514negcld 10492 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -1 ∈ ℂ)
8685, 15, 78divcld 10914 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / 𝐴) ∈ ℂ)
8715, 78reccld 10907 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / 𝐴) ∈ ℂ)
8814, 87subcld 10505 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ∈ ℂ)
8916, 24cjne0d 14063 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) ≠ 0)
9074, 89eqnetrrd 2964 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ≠ 0)
9186, 88, 15, 90, 78divcan5d 10940 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
9285, 15, 78divcan2d 10916 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (-1 / 𝐴)) = -1)
9315, 14, 87subdid 10599 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = ((𝐴 · 1) − (𝐴 · (1 / 𝐴))))
9415mulid1d 10170 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · 1) = 𝐴)
9515, 78recidd 10909 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 / 𝐴)) = 1)
9694, 95oveq12d 6783 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · 1) − (𝐴 · (1 / 𝐴))) = (𝐴 − 1))
9793, 96eqtrd 2758 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = (𝐴 − 1))
9892, 97oveq12d 6783 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = (-1 / (𝐴 − 1)))
9984, 91, 983eqtr2d 2764 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = (-1 / (𝐴 − 1)))
100 subcl 10393 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
101100negnegd 10496 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = (𝐴 − 1))
102 negsubdi2 10453 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
103102negeqd 10388 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = -(1 − 𝐴))
104101, 103eqtr3d 2760 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) = -(1 − 𝐴))
10515, 14, 104syl2anc 696 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 − 1) = -(1 − 𝐴))
106105oveq2d 6781 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / (𝐴 − 1)) = (-1 / -(1 − 𝐴)))
10776, 99, 1063eqtrd 2762 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
108107adantr 472 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
10929, 16, 24divcld 10914 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
110109adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
111 simpr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(-𝐴 / (1 − 𝐴))) = 0)
112110, 111reim0bd 14060 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ)
113112cjred 14086 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-𝐴 / (1 − 𝐴)))
114113, 112eqeltrd 2803 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
115108, 114eqeltrrd 2804 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) ∈ ℝ)
11628, 115eqeltrrd 2804 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ∈ ℝ)
11716, 24recne0d 10908 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ≠ 0)
118117adantr 472 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ≠ 0)
119116, 118rereccld 10965 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) ∈ ℝ)
12026, 119eqeltrrd 2804 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ)
121 1red 10168 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ)
122120, 121resubcld 10571 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) ∈ ℝ)
12313, 122eqeltrrd 2804 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ)
1242, 123negrebd 10504 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℝ)
125124absord 14274 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))
126 eqeq1 2728 . . . . . . . . . . . . 13 ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 ↔ 1 = 𝐴))
127126biimpd 219 . . . . . . . . . . . 12 ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 → 1 = 𝐴))
128 eqeq1 2728 . . . . . . . . . . . . 13 ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 ↔ 1 = -𝐴))
129128biimpd 219 . . . . . . . . . . . 12 ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 → 1 = -𝐴))
130127, 129orim12d 919 . . . . . . . . . . 11 ((abs‘𝐴) = 1 → (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → (1 = 𝐴 ∨ 1 = -𝐴)))
1317, 125, 130sylc 65 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 = 𝐴 ∨ 1 = -𝐴))
132131ord 391 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (¬ 1 = 𝐴 → 1 = -𝐴))
1336, 132mpd 15 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 = -𝐴)
134133, 5eqeltrrd 2804 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ+)
1355, 134rpaddcld 12001 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) ∈ ℝ+)
1363, 135eqeltrrd 2804 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ+)
137136relogcld 24489 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(1 − 𝐴)) ∈ ℝ)
138137reim0d 14085 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = 0)
139134, 136rpdivcld 12003 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ+)
140139relogcld 24489 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
141140reim0d 14085 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) = 0)
142138, 141eqtr4d 2761 . 2 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
14316, 24logcld 24437 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
144143adantr 472 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(1 − 𝐴)) ∈ ℂ)
145144imcld 14055 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
146145recnd 10181 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℂ)
147109adantr 472 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
14815, 78negne0d 10503 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ≠ 0)
14929, 16, 148, 24divne0d 10930 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ≠ 0)
150149adantr 472 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ≠ 0)
151147, 150logcld 24437 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℂ)
152151imcld 14055 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℝ)
153152recnd 10181 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℂ)
154107fveq2d 6308 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
155154adantr 472 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
156 logcj 24472 . . . . . . 7 (((-𝐴 / (1 − 𝐴)) ∈ ℂ ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
157109, 156sylan 489 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
15816, 24reccld 10907 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ∈ ℂ)
159158, 117logcld 24437 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
160159negnegd 10496 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → --(log‘(1 / (1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
161 isosctrlem1 24668 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)
162 logrec 24621 . . . . . . . . . 10 (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ (ℑ‘(log‘(1 − 𝐴))) ≠ π) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
16316, 24, 161, 162syl3anc 1439 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
164163negeqd 10388 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(log‘(1 − 𝐴)) = --(log‘(1 / (1 − 𝐴))))
16527fveq2d 6308 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
166160, 164, 1653eqtr4rd 2769 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
167166adantr 472 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
168155, 157, 1673eqtr3rd 2767 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(log‘(1 − 𝐴)) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
169168fveq2d 6308 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))))
170144imnegd 14070 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = -(ℑ‘(log‘(1 − 𝐴))))
171151imcjd 14065 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
172169, 170, 1713eqtr3d 2766 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(ℑ‘(log‘(1 − 𝐴))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
173146, 153, 172neg11d 10517 . 2 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
174142, 173pm2.61dane 2983 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 196  wo 382  wa 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  cfv 6001  (class class class)co 6765  cc 10047  cr 10048  0cc0 10049  1c1 10050   + caddc 10052   · cmul 10054  cmin 10379  -cneg 10380   / cdiv 10797  2c2 11183  +crp 11946  cexp 12975  ccj 13956  cim 13958  abscabs 14094  πcpi 14917  logclog 24421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127  ax-addf 10128  ax-mulf 10129
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-of 7014  df-om 7183  df-1st 7285  df-2nd 7286  df-supp 7416  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fsupp 8392  df-fi 8433  df-sup 8464  df-inf 8465  df-oi 8531  df-card 8878  df-cda 9103  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-q 11903  df-rp 11947  df-xneg 12060  df-xadd 12061  df-xmul 12062  df-ioo 12293  df-ioc 12294  df-ico 12295  df-icc 12296  df-fz 12441  df-fzo 12581  df-fl 12708  df-mod 12784  df-seq 12917  df-exp 12976  df-fac 13176  df-bc 13205  df-hash 13233  df-shft 13927  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-limsup 14322  df-clim 14339  df-rlim 14340  df-sum 14537  df-ef 14918  df-sin 14920  df-cos 14921  df-pi 14923  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-mulr 16078  df-starv 16079  df-sca 16080  df-vsca 16081  df-ip 16082  df-tset 16083  df-ple 16084  df-ds 16087  df-unif 16088  df-hom 16089  df-cco 16090  df-rest 16206  df-topn 16207  df-0g 16225  df-gsum 16226  df-topgen 16227  df-pt 16228  df-prds 16231  df-xrs 16285  df-qtop 16290  df-imas 16291  df-xps 16293  df-mre 16369  df-mrc 16370  df-acs 16372  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-submnd 17458  df-mulg 17663  df-cntz 17871  df-cmn 18316  df-psmet 19861  df-xmet 19862  df-met 19863  df-bl 19864  df-mopn 19865  df-fbas 19866  df-fg 19867  df-cnfld 19870  df-top 20822  df-topon 20839  df-topsp 20860  df-bases 20873  df-cld 20946  df-ntr 20947  df-cls 20948  df-nei 21025  df-lp 21063  df-perf 21064  df-cn 21154  df-cnp 21155  df-haus 21242  df-tx 21488  df-hmeo 21681  df-fil 21772  df-fm 21864  df-flim 21865  df-flf 21866  df-xms 22247  df-ms 22248  df-tms 22249  df-cncf 22803  df-limc 23750  df-dv 23751  df-log 24423
This theorem is referenced by:  isosctrlem3  24670
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