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

Theorem sharhght 40358
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 1073 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
31simp1d 1071 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
42, 3subcld 10336 . . . . . . 7 (𝜑 → (𝐶𝐴) ∈ ℂ)
54adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐷) → (𝐶𝐴) ∈ ℂ)
6 sharhght.b . . . . . . . . 9 (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴𝐷)𝐺(𝐵𝐷)) = 0))
76simpld 475 . . . . . . . 8 (𝜑𝐷 ∈ ℂ)
87, 3subcld 10336 . . . . . . 7 (𝜑 → (𝐷𝐴) ∈ ℂ)
98adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐷) → (𝐷𝐴) ∈ ℂ)
10 sharhght.sigar . . . . . . 7 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
1110sigarim 40344 . . . . . 6 (((𝐶𝐴) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℝ)
125, 9, 11syl2anc 692 . . . . 5 ((𝜑𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℝ)
1312recnd 10012 . . . 4 ((𝜑𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) ∈ ℂ)
1413mul01d 10179 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · 0) = 0)
151simp2d 1072 . . . . . 6 (𝜑𝐵 ∈ ℂ)
1615adantr 481 . . . . 5 ((𝜑𝐵 = 𝐷) → 𝐵 ∈ ℂ)
17 simpr 477 . . . . 5 ((𝜑𝐵 = 𝐷) → 𝐵 = 𝐷)
1816, 17subeq0bd 10400 . . . 4 ((𝜑𝐵 = 𝐷) → (𝐵𝐷) = 0)
1918oveq2d 6620 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐴)𝐺(𝐷𝐴)) · 0))
202, 15subcld 10336 . . . . . . . 8 (𝜑 → (𝐶𝐵) ∈ ℂ)
2120adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐷) → (𝐶𝐵) ∈ ℂ)
227, 15subcld 10336 . . . . . . . 8 (𝜑 → (𝐷𝐵) ∈ ℂ)
2322adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐷) → (𝐷𝐵) ∈ ℂ)
2410sigarval 40343 . . . . . . 7 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) = (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))))
2521, 23, 24syl2anc 692 . . . . . 6 ((𝜑𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) = (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))))
267adantr 481 . . . . . . . . . 10 ((𝜑𝐵 = 𝐷) → 𝐷 ∈ ℂ)
2717eqcomd 2627 . . . . . . . . . 10 ((𝜑𝐵 = 𝐷) → 𝐷 = 𝐵)
2826, 27subeq0bd 10400 . . . . . . . . 9 ((𝜑𝐵 = 𝐷) → (𝐷𝐵) = 0)
2928oveq2d 6620 . . . . . . . 8 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · (𝐷𝐵)) = ((∗‘(𝐶𝐵)) · 0))
3021cjcld 13870 . . . . . . . . 9 ((𝜑𝐵 = 𝐷) → (∗‘(𝐶𝐵)) ∈ ℂ)
3130mul01d 10179 . . . . . . . 8 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · 0) = 0)
3229, 31eqtrd 2655 . . . . . . 7 ((𝜑𝐵 = 𝐷) → ((∗‘(𝐶𝐵)) · (𝐷𝐵)) = 0)
3332fveq2d 6152 . . . . . 6 ((𝜑𝐵 = 𝐷) → (ℑ‘((∗‘(𝐶𝐵)) · (𝐷𝐵))) = (ℑ‘0))
34 0red 9985 . . . . . . 7 ((𝜑𝐵 = 𝐷) → 0 ∈ ℝ)
3534reim0d 13899 . . . . . 6 ((𝜑𝐵 = 𝐷) → (ℑ‘0) = 0)
3625, 33, 353eqtrd 2659 . . . . 5 ((𝜑𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) = 0)
3736oveq1d 6619 . . . 4 ((𝜑𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)) = (0 · (𝐴𝐷)))
383adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐷) → 𝐴 ∈ ℂ)
3938, 26subcld 10336 . . . . 5 ((𝜑𝐵 = 𝐷) → (𝐴𝐷) ∈ ℂ)
4039mul02d 10178 . . . 4 ((𝜑𝐵 = 𝐷) → (0 · (𝐴𝐷)) = 0)
4137, 40eqtrd 2655 . . 3 ((𝜑𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)) = 0)
4214, 19, 413eqtr4d 2665 . 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 10360 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵) + (𝐵𝐴)) = (𝐶𝐴))
4746oveq1d 6619 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = ((𝐶𝐴)𝐺(𝐷𝐴)))
4843, 44subcld 10336 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐶𝐵) ∈ ℂ)
498adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐴) ∈ ℂ)
5044, 45subcld 10336 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐴) ∈ ℂ)
5110sigaraf 40346 . . . . . . . 8 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ ∧ (𝐵𝐴) ∈ ℂ) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
5248, 49, 50, 51syl3anc 1323 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵) + (𝐵𝐴))𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
5347, 52eqtr3d 2657 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
546simprd 479 . . . . . . . . 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 40353 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((𝐴𝐷)𝐺(𝐵𝐷)) = ((𝐵𝐴)𝐺(𝐷𝐴)))
5845, 44, 56, 57syl3anc 1323 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷)𝐺(𝐵𝐷)) = ((𝐵𝐴)𝐺(𝐷𝐴)))
5955, 58eqtr3d 2657 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 0 = ((𝐵𝐴)𝐺(𝐷𝐴)))
6059oveq2d 6620 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐴)) + 0) = (((𝐶𝐵)𝐺(𝐷𝐴)) + ((𝐵𝐴)𝐺(𝐷𝐴))))
6110sigarim 40344 . . . . . . . . 9 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐴) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℝ)
6248, 49, 61syl2anc 692 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℝ)
6362recnd 10012 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) ∈ ℂ)
6463addid1d 10180 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐴)) + 0) = ((𝐶𝐵)𝐺(𝐷𝐴)))
6553, 60, 643eqtr2d 2661 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = ((𝐶𝐵)𝐺(𝐷𝐴)))
6644, 56negsubdi2d 10352 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐵𝐷) = (𝐷𝐵))
6766eqcomd 2627 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐵) = -(𝐵𝐷))
6867oveq1d 6619 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷𝐵) / (𝐵𝐷)) = (-(𝐵𝐷) / (𝐵𝐷)))
6944, 56subcld 10336 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐷) ∈ ℂ)
70 simpr 477 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ¬ 𝐵 = 𝐷)
7170neqned 2797 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐵𝐷)
7244, 56, 71subne0d 10345 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵𝐷) ≠ 0)
7369, 69, 72divnegd 10758 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵𝐷) / (𝐵𝐷)) = (-(𝐵𝐷) / (𝐵𝐷)))
7469, 72dividd 10743 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐵𝐷) / (𝐵𝐷)) = 1)
7574negeqd 10219 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵𝐷) / (𝐵𝐷)) = -1)
7668, 73, 753eqtr2d 2661 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷𝐵) / (𝐵𝐷)) = -1)
7776oveq1d 6619 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = (-1 · (𝐴𝐷)))
7845, 56subcld 10336 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐴𝐷) ∈ ℂ)
7978mulm1d 10426 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (-1 · (𝐴𝐷)) = -(𝐴𝐷))
8045, 56negsubdi2d 10352 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐴𝐷) = (𝐷𝐴))
8177, 79, 803eqtrd 2659 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = (𝐷𝐴))
8256, 44subcld 10336 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐵) ∈ ℂ)
8382, 69, 78, 72div32d 10768 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷𝐵) / (𝐵𝐷)) · (𝐴𝐷)) = ((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷))))
8481, 83eqtr3d 2657 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷𝐴) = ((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷))))
8584oveq2d 6620 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐴)) = ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))))
8656, 45, 443jca 1240 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
8710, 86, 70, 55sigardiv 40354 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷) / (𝐵𝐷)) ∈ ℝ)
8810sigarls 40350 . . . . . 6 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ ∧ ((𝐴𝐷) / (𝐵𝐷)) ∈ ℝ) → ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
8948, 82, 87, 88syl3anc 1323 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺((𝐷𝐵) · ((𝐴𝐷) / (𝐵𝐷)))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
9065, 85, 893eqtrd 2659 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐴)𝐺(𝐷𝐴)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))))
9190oveq1d 6619 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = ((((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))) · (𝐵𝐷)))
9210sigarim 40344 . . . . . 6 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℝ)
9392recnd 10012 . . . . 5 (((𝐶𝐵) ∈ ℂ ∧ (𝐷𝐵) ∈ ℂ) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℂ)
9448, 82, 93syl2anc 692 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶𝐵)𝐺(𝐷𝐵)) ∈ ℂ)
9578, 69, 72divcld 10745 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴𝐷) / (𝐵𝐷)) ∈ ℂ)
9694, 95, 69mulassd 10007 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((((𝐶𝐵)𝐺(𝐷𝐵)) · ((𝐴𝐷) / (𝐵𝐷))) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷))))
9778, 69, 72divcan1d 10746 . . . 4 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷)) = (𝐴𝐷))
9897oveq2d 6620 . . 3 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐵)𝐺(𝐷𝐵)) · (((𝐴𝐷) / (𝐵𝐷)) · (𝐵𝐷))) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
9991, 96, 983eqtrd 2659 . 2 ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
10042, 99pm2.61dan 831 1 (𝜑 → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5847  (class class class)co 6604  cmpt2 6606  cc 9878  cr 9879  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885  cmin 10210  -cneg 10211   / cdiv 10628  ccj 13770  cim 13772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-2 11023  df-cj 13773  df-re 13774  df-im 13775
This theorem is referenced by:  cevathlem2  40361
  Copyright terms: Public domain W3C validator