Mathbox for Saveliy Skresanov < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sharhght Structured version   Visualization version   GIF version

Theorem sharhght 43405
 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 1141 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
31simp1d 1139 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
42, 3subcld 10995 . . . . . . 7 (𝜑 → (𝐶𝐴) ∈ ℂ)
54adantr 484 . . . . . 6 ((𝜑𝐵 = 𝐷) → (𝐶𝐴) ∈ ℂ)
6 sharhght.b . . . . . . . . 9 (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴𝐷)𝐺(𝐵𝐷)) = 0))
76simpld 498 . . . . . . . 8 (𝜑𝐷 ∈ ℂ)
87, 3subcld 10995 . . . . . . 7 (𝜑 → (𝐷𝐴) ∈ ℂ)
98adantr 484 . . . . . 6 ((𝜑𝐵 = 𝐷) → (𝐷𝐴) ∈ ℂ)
10 sharhght.sigar . . . . . . 7 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
1110sigarim 43391 . . . . . 6 (((𝐶𝐴) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℝ)
125, 9, 11syl2anc 587 . . . . 5 ((𝜑𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℝ)
1312recnd 10667 . . . 4 ((𝜑𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℂ)
1413mul01d 10837 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · 0) = 0)
151simp2d 1140 . . . . . 6 (𝜑𝐵 ∈ ℂ)
1615adantr 484 . . . . 5 ((𝜑𝐵 = 𝐷) → 𝐵 ∈ ℂ)
17 simpr 488 . . . . 5 ((𝜑𝐵 = 𝐷) → 𝐵 = 𝐷)
1816, 17subeq0bd 11064 . . . 4 ((𝜑𝐵 = 𝐷) → (𝐵𝐷) = 0)
1918oveq2d 7165 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐴)𝐺(𝐷𝐴)) · 0))
202, 15subcld 10995 . . . . . . . 8 (𝜑 → (𝐶𝐵) ∈ ℂ)
2120adantr 484 . . . . . . 7 ((𝜑𝐵 = 𝐷) → (𝐶𝐵) ∈ ℂ)
227, 15subcld 10995 . . . . . . . 8 (𝜑 → (𝐷𝐵) ∈ ℂ)
2322adantr 484 . . . . . . 7 ((𝜑𝐵 = 𝐷) → (𝐷𝐵) ∈ ℂ)
2410sigarval 43390 . . . . . . 7 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) = (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))))
2521, 23, 24syl2anc 587 . . . . . 6 ((𝜑𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) = (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))))
267adantr 484 . . . . . . . . . 10 ((𝜑𝐵 = 𝐷) → 𝐷 ∈ ℂ)
2717eqcomd 2830 . . . . . . . . . 10 ((𝜑𝐵 = 𝐷) → 𝐷 = 𝐵)
2826, 27subeq0bd 11064 . . . . . . . . 9 ((𝜑𝐵 = 𝐷) → (𝐷𝐵) = 0)
2928oveq2d 7165 . . . . . . . 8 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · (𝐷𝐵)) = ((∗‘(𝐶𝐵)) · 0))
3021cjcld 14555 . . . . . . . . 9 ((𝜑𝐵 = 𝐷) → (∗‘(𝐶𝐵)) ∈ ℂ)
3130mul01d 10837 . . . . . . . 8 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · 0) = 0)
3229, 31eqtrd 2859 . . . . . . 7 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · (𝐷𝐵)) = 0)
3332fveq2d 6665 . . . . . 6 ((𝜑𝐵 = 𝐷) → (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))) = (ℑ‘0))
34 0red 10642 . . . . . . 7 ((𝜑𝐵 = 𝐷) → 0 ∈ ℝ)
3534reim0d 14584 . . . . . 6 ((𝜑𝐵 = 𝐷) → (ℑ‘0) = 0)
3625, 33, 353eqtrd 2863 . . . . 5 ((𝜑𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) = 0)
3736oveq1d 7164 . . . 4 ((𝜑𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)) = (0 · (𝐴𝐷)))
383adantr 484 . . . . . 6 ((𝜑𝐵 = 𝐷) → 𝐴 ∈ ℂ)
3938, 26subcld 10995 . . . . 5 ((𝜑𝐵 = 𝐷) → (𝐴𝐷) ∈ ℂ)
4039mul02d 10836 . . . 4 ((𝜑𝐵 = 𝐷) → (0 · (𝐴𝐷)) = 0)
4137, 40eqtrd 2859 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)) = 0)
4214, 19, 413eqtr4d 2869 . 2 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
432adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐶 ∈ ℂ)
4415adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐵 ∈ ℂ)
453adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐴 ∈ ℂ)
4643, 44, 45npncand 11019 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵) + (𝐵𝐴)) = (𝐶𝐴))
4746oveq1d 7164 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = ((𝐶𝐴)𝐺(𝐷𝐴)))
4843, 44subcld 10995 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐶𝐵) ∈ ℂ)
498adantr 484 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐴) ∈ ℂ)
5044, 45subcld 10995 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐴) ∈ ℂ)
5110sigaraf 43393 . . . . . . . 8 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ ∧ (𝐵𝐴) ∈ ℂ) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
5248, 49, 50, 51syl3anc 1368 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
5347, 52eqtr3d 2861 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
546simprd 499 . . . . . . . . 9 (𝜑 → ((𝐴𝐷)𝐺(𝐵𝐷)) = 0)
5554adantr 484 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷)𝐺(𝐵𝐷)) = 0)
567adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐷 ∈ ℂ)
5710sigarperm 43400 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((𝐴𝐷)𝐺(𝐵𝐷)) = ((𝐵𝐴)𝐺(𝐷𝐴)))
5845, 44, 56, 57syl3anc 1368 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷)𝐺(𝐵𝐷)) = ((𝐵𝐴)𝐺(𝐷𝐴)))
5955, 58eqtr3d 2861 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 0 = ((𝐵𝐴)𝐺(𝐷𝐴)))
6059oveq2d 7165 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐴)) + 0) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
6110sigarim 43391 . . . . . . . . 9 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℝ)
6248, 49, 61syl2anc 587 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℝ)
6362recnd 10667 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℂ)
6463addid1d 10838 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐴)) + 0) = ((𝐶𝐵)𝐺(𝐷𝐴)))
6553, 60, 643eqtr2d 2865 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = ((𝐶𝐵)𝐺(𝐷𝐴)))
6644, 56negsubdi2d 11011 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐵𝐷) = (𝐷𝐵))
6766eqcomd 2830 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐵) = -(𝐵𝐷))
6867oveq1d 7164 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷𝐵) / (𝐵𝐷)) = (-(𝐵𝐷) / (𝐵𝐷)))
6944, 56subcld 10995 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐷) ∈ ℂ)
70 simpr 488 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ¬ 𝐵 = 𝐷)
7170neqned 3021 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐵𝐷)
7244, 56, 71subne0d 11004 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐷) ≠ 0)
7369, 69, 72divnegd 11427 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵𝐷) / (𝐵𝐷)) = (-(𝐵𝐷) / (𝐵𝐷)))
7469, 72dividd 11412 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐵𝐷) / (𝐵𝐷)) = 1)
7574negeqd 10878 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵𝐷) / (𝐵𝐷)) = -1)
7668, 73, 753eqtr2d 2865 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷𝐵) / (𝐵𝐷)) = -1)
7776oveq1d 7164 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = (-1 · (𝐴𝐷)))
7845, 56subcld 10995 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐴𝐷) ∈ ℂ)
7978mulm1d 11090 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (-1 · (𝐴𝐷)) = -(𝐴𝐷))
8045, 56negsubdi2d 11011 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐴𝐷) = (𝐷𝐴))
8177, 79, 803eqtrd 2863 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = (𝐷𝐴))
8256, 44subcld 10995 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐵) ∈ ℂ)
8382, 69, 78, 72div32d 11437 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = ((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷))))
8481, 83eqtr3d 2861 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐴) = ((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷))))
8584oveq2d 7165 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) = ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))))
8656, 45, 443jca 1125 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
8710, 86, 70, 55sigardiv 43401 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷) / (𝐵𝐷)) ∈ ℝ)
8810sigarls 43397 . . . . . 6 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ ∧ ((𝐴𝐷) / (𝐵𝐷)) ∈ ℝ) → ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
8948, 82, 87, 88syl3anc 1368 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
9065, 85, 893eqtrd 2863 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
9190oveq1d 7164 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = ((((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))) · (𝐵𝐷)))
9210sigarim 43391 . . . . . 6 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℝ)
9392recnd 10667 . . . . 5 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℂ)
9448, 82, 93syl2anc 587 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℂ)
9578, 69, 72divcld 11414 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷) / (𝐵𝐷)) ∈ ℂ)
9694, 95, 69mulassd 10662 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷))))
9778, 69, 72divcan1d 11415 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷)) = (𝐴𝐷))
9897oveq2d 7165 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
9991, 96, 983eqtrd 2863 . 2 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
10042, 99pm2.61dan 812 1 (𝜑 → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ‘cfv 6343  (class class class)co 7149   ∈ cmpo 7151  ℂcc 10533  ℝcr 10534  0cc0 10535  1c1 10536   + caddc 10538   · cmul 10540   − cmin 10868  -cneg 10869   / cdiv 11295  ∗ccj 14455  ℑcim 14457 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-po 5461  df-so 5462  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-2 11697  df-cj 14458  df-re 14459  df-im 14460 This theorem is referenced by:  cevathlem2  43408
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