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Theorem isosctrlem2 24854
Description: Lemma for isosctr 24856. 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 10292 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℂ)
2 simpl1 1242 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℂ)
31, 2negsubd 10656 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) = (1 − 𝐴))
4 1rp 12037 . . . . . . . 8 1 ∈ ℝ+
54a1i 11 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ+)
6 simpl3 1246 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ¬ 1 = 𝐴)
7 simpl2 1244 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (abs‘𝐴) = 1)
81, 2, 1sub32d 10682 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = ((1 − 1) − 𝐴))
9 1m1e0 11348 . . . . . . . . . . . . . . . . 17 (1 − 1) = 0
109oveq1i 6856 . . . . . . . . . . . . . . . 16 ((1 − 1) − 𝐴) = (0 − 𝐴)
11 df-neg 10527 . . . . . . . . . . . . . . . 16 -𝐴 = (0 − 𝐴)
1210, 11eqtr4i 2790 . . . . . . . . . . . . . . 15 ((1 − 1) − 𝐴) = -𝐴
138, 12syl6eq 2815 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = -𝐴)
14 1cnd 10292 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 1 ∈ ℂ)
15 simp1 1166 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ∈ ℂ)
1614, 15subcld 10650 . . . . . . . . . . . . . . . . . 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 10251 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
19 subeq0 10565 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
2018, 19mpan 681 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
2120biimpd 220 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴))
2221con3dimp 397 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0)
2322neqned 2944 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
24233adant2 1161 . . . . . . . . . . . . . . . . . 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 11056 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) = (1 − 𝐴))
2714, 16, 24div2negd 11074 . . . . . . . . . . . . . . . . . . 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 10637 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ∈ ℂ)
3029, 16, 24cjdivd 14262 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = ((∗‘-𝐴) / (∗‘(1 − 𝐴))))
3115cjnegd 14250 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(∗‘𝐴))
32 fveq2 6379 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
33 abs0 14324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (abs‘0) = 0
3432, 33syl6eq 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = 0 → (abs‘𝐴) = 0)
35 eqtr2 2785 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((abs‘𝐴) = 1 ∧ (abs‘𝐴) = 0) → 1 = 0)
3634, 35sylan2 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → 1 = 0)
37 ax-1ne0 10262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1 ≠ 0
38 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (1 ≠ 0 → 1 ≠ 0)
3938neneqd 2942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (1 ≠ 0 → ¬ 1 = 0)
4037, 39mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → ¬ 1 = 0)
4136, 40pm2.65da 851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((abs‘𝐴) = 1 → ¬ 𝐴 = 0)
4241adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ¬ 𝐴 = 0)
43 df-ne 2938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
44 oveq1 6853 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = (1↑2))
45 sq1 13170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (1↑2) = 1
4644, 45syl6eq 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = 1)
4746adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
48 absvalsq 14319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
4948adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
5047, 49eqtr3d 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 1 = (𝐴 · (∗‘𝐴)))
51503adant3 1162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 1 = (𝐴 · (∗‘𝐴)))
5251oveq1d 6861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((𝐴 · (∗‘𝐴)) / 𝐴))
53 simp1 1166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
5453cjcld 14235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ)
55 simp3 1168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
5654, 53, 55divcan3d 11064 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
5752, 56eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = (∗‘𝐴))
5843, 57syl3an3br 1527 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 𝐴 = 0) → (1 / 𝐴) = (∗‘𝐴))
5942, 58mpd3an3 1586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = (∗‘𝐴))
6059eqcomd 2771 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘𝐴) = (1 / 𝐴))
61603adant3 1162 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘𝐴) = (1 / 𝐴))
6261negeqd 10533 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(∗‘𝐴) = -(1 / 𝐴))
6331, 62eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(1 / 𝐴))
6463oveq1d 6861 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((∗‘-𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (∗‘(1 − 𝐴))))
65 cjsub 14188 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
6618, 65mpan 681 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
67 1red 10298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐴 ∈ ℂ → 1 ∈ ℝ)
6867cjred 14265 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐴 ∈ ℂ → (∗‘1) = 1)
6968oveq1d 6861 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ ℂ → ((∗‘1) − (∗‘𝐴)) = (1 − (∗‘𝐴)))
7066, 69eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
7170adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
7260oveq2d 6862 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (∗‘𝐴)) = (1 − (1 / 𝐴)))
7371, 72eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
74733adant3 1162 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
7574oveq2d 6862 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
7630, 64, 753eqtrd 2803 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
77413ad2ant2 1164 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ¬ 𝐴 = 0)
7877neqned 2944 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ≠ 0)
79 1cnd 10292 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 1 ∈ ℂ)
80 simpl 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
81 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
8279, 80, 81divnegd 11072 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -(1 / 𝐴) = (-1 / 𝐴))
8382oveq1d 6861 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
8415, 78, 83syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
8514negcld 10637 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -1 ∈ ℂ)
8685, 15, 78divcld 11059 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / 𝐴) ∈ ℂ)
8715, 78reccld 11052 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / 𝐴) ∈ ℂ)
8814, 87subcld 10650 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ∈ ℂ)
8916, 24cjne0d 14242 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) ≠ 0)
9074, 89eqnetrrd 3005 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ≠ 0)
9186, 88, 15, 90, 78divcan5d 11085 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
9285, 15, 78divcan2d 11061 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (-1 / 𝐴)) = -1)
9315, 14, 87subdid 10744 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = ((𝐴 · 1) − (𝐴 · (1 / 𝐴))))
9415mulid1d 10315 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · 1) = 𝐴)
9515, 78recidd 11054 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 / 𝐴)) = 1)
9694, 95oveq12d 6864 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · 1) − (𝐴 · (1 / 𝐴))) = (𝐴 − 1))
9793, 96eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = (𝐴 − 1))
9892, 97oveq12d 6864 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = (-1 / (𝐴 − 1)))
9984, 91, 983eqtr2d 2805 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = (-1 / (𝐴 − 1)))
100 subcl 10538 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
101100negnegd 10641 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = (𝐴 − 1))
102 negsubdi2 10598 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
103102negeqd 10533 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = -(1 − 𝐴))
104101, 103eqtr3d 2801 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) = -(1 − 𝐴))
10515, 14, 104syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 − 1) = -(1 − 𝐴))
106105oveq2d 6862 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / (𝐴 − 1)) = (-1 / -(1 − 𝐴)))
10776, 99, 1063eqtrd 2803 . . . . . . . . . . . . . . . . . . . 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 11059 . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(-𝐴 / (1 − 𝐴))) = 0)
112110, 111reim0bd 14239 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ)
113112cjred 14265 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-𝐴 / (1 − 𝐴)))
114113, 112eqeltrd 2844 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
115108, 114eqeltrrd 2845 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) ∈ ℝ)
11628, 115eqeltrrd 2845 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ∈ ℝ)
11716, 24recne0d 11053 . . . . . . . . . . . . . . . . . 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 11110 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) ∈ ℝ)
12026, 119eqeltrrd 2845 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ)
121 1red 10298 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ)
122120, 121resubcld 10716 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) ∈ ℝ)
12313, 122eqeltrrd 2845 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ)
1242, 123negrebd 10649 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℝ)
125124absord 14453 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))
126 eqeq1 2769 . . . . . . . . . . . . 13 ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 ↔ 1 = 𝐴))
127126biimpd 220 . . . . . . . . . . . 12 ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 → 1 = 𝐴))
128 eqeq1 2769 . . . . . . . . . . . . 13 ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 ↔ 1 = -𝐴))
129128biimpd 220 . . . . . . . . . . . 12 ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 → 1 = -𝐴))
130127, 129orim12d 987 . . . . . . . . . . 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 890 . . . . . . . . 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 2845 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ+)
1355, 134rpaddcld 12090 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) ∈ ℝ+)
1363, 135eqeltrrd 2845 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ+)
137136relogcld 24674 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(1 − 𝐴)) ∈ ℝ)
138137reim0d 14264 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = 0)
139134, 136rpdivcld 12092 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ+)
140139relogcld 24674 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
141140reim0d 14264 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) = 0)
142138, 141eqtr4d 2802 . 2 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
14316, 24logcld 24622 . . . . . 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 14234 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
146145recnd 10326 . . 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 10648 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ≠ 0)
14929, 16, 148, 24divne0d 11075 . . . . . . 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 24622 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℂ)
152151imcld 14234 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℝ)
153152recnd 10326 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℂ)
154107fveq2d 6383 . . . . . . 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 24657 . . . . . . 7 (((-𝐴 / (1 − 𝐴)) ∈ ℂ ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
157109, 156sylan 575 . . . . . 6 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
15816, 24reccld 11052 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ∈ ℂ)
159158, 117logcld 24622 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
160159negnegd 10641 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → --(log‘(1 / (1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
161 isosctrlem1 24853 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)
162 logrec 24806 . . . . . . . . . 10 (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ (ℑ‘(log‘(1 − 𝐴))) ≠ π) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
16316, 24, 161, 162syl3anc 1490 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
164163negeqd 10533 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(log‘(1 − 𝐴)) = --(log‘(1 / (1 − 𝐴))))
16527fveq2d 6383 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
166160, 164, 1653eqtr4rd 2810 . . . . . . 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 2808 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(log‘(1 − 𝐴)) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
169168fveq2d 6383 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))))
170144imnegd 14249 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = -(ℑ‘(log‘(1 − 𝐴))))
171151imcjd 14244 . . . 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 2807 . . 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 10662 . 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 3024 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  cfv 6070  (class class class)co 6846  cc 10191  cr 10192  0cc0 10193  1c1 10194   + caddc 10196   · cmul 10198  cmin 10524  -cneg 10525   / cdiv 10942  2c2 11331  +crp 12033  cexp 13072  ccj 14135  cim 14137  abscabs 14273  πcpi 15093  logclog 24606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272  ax-mulf 10273
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-supp 7502  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-ixp 8118  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fsupp 8487  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-z 11629  df-dec 11746  df-uz 11892  df-q 11995  df-rp 12034  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12386  df-ioc 12387  df-ico 12388  df-icc 12389  df-fz 12539  df-fzo 12679  df-fl 12806  df-mod 12882  df-seq 13014  df-exp 13073  df-fac 13270  df-bc 13299  df-hash 13327  df-shft 14106  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-limsup 14501  df-clim 14518  df-rlim 14519  df-sum 14716  df-ef 15094  df-sin 15096  df-cos 15097  df-pi 15099  df-struct 16146  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-mulr 16242  df-starv 16243  df-sca 16244  df-vsca 16245  df-ip 16246  df-tset 16247  df-ple 16248  df-ds 16250  df-unif 16251  df-hom 16252  df-cco 16253  df-rest 16363  df-topn 16364  df-0g 16382  df-gsum 16383  df-topgen 16384  df-pt 16385  df-prds 16388  df-xrs 16442  df-qtop 16447  df-imas 16448  df-xps 16450  df-mre 16526  df-mrc 16527  df-acs 16529  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-submnd 17616  df-mulg 17822  df-cntz 18027  df-cmn 18475  df-psmet 20025  df-xmet 20026  df-met 20027  df-bl 20028  df-mopn 20029  df-fbas 20030  df-fg 20031  df-cnfld 20034  df-top 20992  df-topon 21009  df-topsp 21031  df-bases 21044  df-cld 21117  df-ntr 21118  df-cls 21119  df-nei 21196  df-lp 21234  df-perf 21235  df-cn 21325  df-cnp 21326  df-haus 21413  df-tx 21659  df-hmeo 21852  df-fil 21943  df-fm 22035  df-flim 22036  df-flf 22037  df-xms 22418  df-ms 22419  df-tms 22420  df-cncf 22974  df-limc 23935  df-dv 23936  df-log 24608
This theorem is referenced by:  isosctrlem3  24855
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