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Theorem sharhght 42999
Description: Let 𝐴𝐵𝐶 be a triangle, and let 𝐷 lie on the line 𝐴𝐵. Then (doubled) areas of triangles 𝐴𝐷𝐶 and 𝐶𝐷𝐵 relate as lengths of corresponding bases 𝐴𝐷 and 𝐷𝐵. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
Hypotheses
Ref Expression
sharhght.sigar 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
sharhght.a (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))
sharhght.b (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴𝐷)𝐺(𝐵𝐷)) = 0))
Assertion
Ref Expression
sharhght (𝜑 → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem sharhght
StepHypRef Expression
1 sharhght.a . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))
21simp3d 1136 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
31simp1d 1134 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
42, 3subcld 10985 . . . . . . 7 (𝜑 → (𝐶𝐴) ∈ ℂ)
54adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐷) → (𝐶𝐴) ∈ ℂ)
6 sharhght.b . . . . . . . . 9 (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴𝐷)𝐺(𝐵𝐷)) = 0))
76simpld 495 . . . . . . . 8 (𝜑𝐷 ∈ ℂ)
87, 3subcld 10985 . . . . . . 7 (𝜑 → (𝐷𝐴) ∈ ℂ)
98adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐷) → (𝐷𝐴) ∈ ℂ)
10 sharhght.sigar . . . . . . 7 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
1110sigarim 42985 . . . . . 6 (((𝐶𝐴) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℝ)
125, 9, 11syl2anc 584 . . . . 5 ((𝜑𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℝ)
1312recnd 10657 . . . 4 ((𝜑𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℂ)
1413mul01d 10827 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · 0) = 0)
151simp2d 1135 . . . . . 6 (𝜑𝐵 ∈ ℂ)
1615adantr 481 . . . . 5 ((𝜑𝐵 = 𝐷) → 𝐵 ∈ ℂ)
17 simpr 485 . . . . 5 ((𝜑𝐵 = 𝐷) → 𝐵 = 𝐷)
1816, 17subeq0bd 11054 . . . 4 ((𝜑𝐵 = 𝐷) → (𝐵𝐷) = 0)
1918oveq2d 7161 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐴)𝐺(𝐷𝐴)) · 0))
202, 15subcld 10985 . . . . . . . 8 (𝜑 → (𝐶𝐵) ∈ ℂ)
2120adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐷) → (𝐶𝐵) ∈ ℂ)
227, 15subcld 10985 . . . . . . . 8 (𝜑 → (𝐷𝐵) ∈ ℂ)
2322adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐷) → (𝐷𝐵) ∈ ℂ)
2410sigarval 42984 . . . . . . 7 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) = (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))))
2521, 23, 24syl2anc 584 . . . . . 6 ((𝜑𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) = (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))))
267adantr 481 . . . . . . . . . 10 ((𝜑𝐵 = 𝐷) → 𝐷 ∈ ℂ)
2717eqcomd 2824 . . . . . . . . . 10 ((𝜑𝐵 = 𝐷) → 𝐷 = 𝐵)
2826, 27subeq0bd 11054 . . . . . . . . 9 ((𝜑𝐵 = 𝐷) → (𝐷𝐵) = 0)
2928oveq2d 7161 . . . . . . . 8 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · (𝐷𝐵)) = ((∗‘(𝐶𝐵)) · 0))
3021cjcld 14543 . . . . . . . . 9 ((𝜑𝐵 = 𝐷) → (∗‘(𝐶𝐵)) ∈ ℂ)
3130mul01d 10827 . . . . . . . 8 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · 0) = 0)
3229, 31eqtrd 2853 . . . . . . 7 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · (𝐷𝐵)) = 0)
3332fveq2d 6667 . . . . . 6 ((𝜑𝐵 = 𝐷) → (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))) = (ℑ‘0))
34 0red 10632 . . . . . . 7 ((𝜑𝐵 = 𝐷) → 0 ∈ ℝ)
3534reim0d 14572 . . . . . 6 ((𝜑𝐵 = 𝐷) → (ℑ‘0) = 0)
3625, 33, 353eqtrd 2857 . . . . 5 ((𝜑𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) = 0)
3736oveq1d 7160 . . . 4 ((𝜑𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)) = (0 · (𝐴𝐷)))
383adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐷) → 𝐴 ∈ ℂ)
3938, 26subcld 10985 . . . . 5 ((𝜑𝐵 = 𝐷) → (𝐴𝐷) ∈ ℂ)
4039mul02d 10826 . . . 4 ((𝜑𝐵 = 𝐷) → (0 · (𝐴𝐷)) = 0)
4137, 40eqtrd 2853 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)) = 0)
4214, 19, 413eqtr4d 2863 . 2 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
432adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐶 ∈ ℂ)
4415adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐵 ∈ ℂ)
453adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐴 ∈ ℂ)
4643, 44, 45npncand 11009 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵) + (𝐵𝐴)) = (𝐶𝐴))
4746oveq1d 7160 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = ((𝐶𝐴)𝐺(𝐷𝐴)))
4843, 44subcld 10985 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐶𝐵) ∈ ℂ)
498adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐴) ∈ ℂ)
5044, 45subcld 10985 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐴) ∈ ℂ)
5110sigaraf 42987 . . . . . . . 8 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ ∧ (𝐵𝐴) ∈ ℂ) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
5248, 49, 50, 51syl3anc 1363 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
5347, 52eqtr3d 2855 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
546simprd 496 . . . . . . . . 9 (𝜑 → ((𝐴𝐷)𝐺(𝐵𝐷)) = 0)
5554adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷)𝐺(𝐵𝐷)) = 0)
567adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐷 ∈ ℂ)
5710sigarperm 42994 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((𝐴𝐷)𝐺(𝐵𝐷)) = ((𝐵𝐴)𝐺(𝐷𝐴)))
5845, 44, 56, 57syl3anc 1363 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷)𝐺(𝐵𝐷)) = ((𝐵𝐴)𝐺(𝐷𝐴)))
5955, 58eqtr3d 2855 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 0 = ((𝐵𝐴)𝐺(𝐷𝐴)))
6059oveq2d 7161 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐴)) + 0) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
6110sigarim 42985 . . . . . . . . 9 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℝ)
6248, 49, 61syl2anc 584 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℝ)
6362recnd 10657 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℂ)
6463addid1d 10828 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐴)) + 0) = ((𝐶𝐵)𝐺(𝐷𝐴)))
6553, 60, 643eqtr2d 2859 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = ((𝐶𝐵)𝐺(𝐷𝐴)))
6644, 56negsubdi2d 11001 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐵𝐷) = (𝐷𝐵))
6766eqcomd 2824 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐵) = -(𝐵𝐷))
6867oveq1d 7160 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷𝐵) / (𝐵𝐷)) = (-(𝐵𝐷) / (𝐵𝐷)))
6944, 56subcld 10985 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐷) ∈ ℂ)
70 simpr 485 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ¬ 𝐵 = 𝐷)
7170neqned 3020 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐵𝐷)
7244, 56, 71subne0d 10994 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐷) ≠ 0)
7369, 69, 72divnegd 11417 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵𝐷) / (𝐵𝐷)) = (-(𝐵𝐷) / (𝐵𝐷)))
7469, 72dividd 11402 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐵𝐷) / (𝐵𝐷)) = 1)
7574negeqd 10868 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵𝐷) / (𝐵𝐷)) = -1)
7668, 73, 753eqtr2d 2859 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷𝐵) / (𝐵𝐷)) = -1)
7776oveq1d 7160 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = (-1 · (𝐴𝐷)))
7845, 56subcld 10985 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐴𝐷) ∈ ℂ)
7978mulm1d 11080 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (-1 · (𝐴𝐷)) = -(𝐴𝐷))
8045, 56negsubdi2d 11001 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐴𝐷) = (𝐷𝐴))
8177, 79, 803eqtrd 2857 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = (𝐷𝐴))
8256, 44subcld 10985 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐵) ∈ ℂ)
8382, 69, 78, 72div32d 11427 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = ((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷))))
8481, 83eqtr3d 2855 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐴) = ((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷))))
8584oveq2d 7161 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) = ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))))
8656, 45, 443jca 1120 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
8710, 86, 70, 55sigardiv 42995 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷) / (𝐵𝐷)) ∈ ℝ)
8810sigarls 42991 . . . . . 6 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ ∧ ((𝐴𝐷) / (𝐵𝐷)) ∈ ℝ) → ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
8948, 82, 87, 88syl3anc 1363 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
9065, 85, 893eqtrd 2857 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
9190oveq1d 7160 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = ((((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))) · (𝐵𝐷)))
9210sigarim 42985 . . . . . 6 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℝ)
9392recnd 10657 . . . . 5 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℂ)
9448, 82, 93syl2anc 584 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℂ)
9578, 69, 72divcld 11404 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷) / (𝐵𝐷)) ∈ ℂ)
9694, 95, 69mulassd 10652 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷))))
9778, 69, 72divcan1d 11405 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷)) = (𝐴𝐷))
9897oveq2d 7161 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
9991, 96, 983eqtrd 2857 . 2 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
10042, 99pm2.61dan 809 1 (𝜑 → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  cmpo 7147  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  cmin 10858  -cneg 10859   / cdiv 11285  ccj 14443  cim 14445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-2 11688  df-cj 14446  df-re 14447  df-im 14448
This theorem is referenced by:  cevathlem2  43002
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