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Theorem ptolemy 13878
Description: Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 11723, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. This is Metamath 100 proof #95. (Contributed by David A. Wheeler, 31-May-2015.)
Assertion
Ref Expression
ptolemy (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐶) · (sin‘𝐷))) = ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶))))

Proof of Theorem ptolemy
StepHypRef Expression
1 addcl 7914 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐶 + 𝐷) ∈ ℂ)
213ad2ant2 1019 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (𝐶 + 𝐷) ∈ ℂ)
32coscld 11690 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘(𝐶 + 𝐷)) ∈ ℂ)
43negnegd 8236 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → --(cos‘(𝐶 + 𝐷)) = (cos‘(𝐶 + 𝐷)))
5 addid2 8073 . . . . . . . . . . . . . . 15 ((𝐶 + 𝐷) ∈ ℂ → (0 + (𝐶 + 𝐷)) = (𝐶 + 𝐷))
65oveq1d 5883 . . . . . . . . . . . . . 14 ((𝐶 + 𝐷) ∈ ℂ → ((0 + (𝐶 + 𝐷)) − ((𝐴 + 𝐵) + (𝐶 + 𝐷))) = ((𝐶 + 𝐷) − ((𝐴 + 𝐵) + (𝐶 + 𝐷))))
72, 6syl 14 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((0 + (𝐶 + 𝐷)) − ((𝐴 + 𝐵) + (𝐶 + 𝐷))) = ((𝐶 + 𝐷) − ((𝐴 + 𝐵) + (𝐶 + 𝐷))))
8 0cnd 7928 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → 0 ∈ ℂ)
9 addcl 7914 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
109adantr 276 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐴 + 𝐵) ∈ ℂ)
11103adant3 1017 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (𝐴 + 𝐵) ∈ ℂ)
128, 11, 2pnpcan2d 8283 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((0 + (𝐶 + 𝐷)) − ((𝐴 + 𝐵) + (𝐶 + 𝐷))) = (0 − (𝐴 + 𝐵)))
13 simp3 999 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π)
1413oveq2d 5884 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐶 + 𝐷) − ((𝐴 + 𝐵) + (𝐶 + 𝐷))) = ((𝐶 + 𝐷) − π))
157, 12, 143eqtr3rd 2219 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐶 + 𝐷) − π) = (0 − (𝐴 + 𝐵)))
16 df-neg 8108 . . . . . . . . . . . 12 -(𝐴 + 𝐵) = (0 − (𝐴 + 𝐵))
1715, 16eqtr4di 2228 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐶 + 𝐷) − π) = -(𝐴 + 𝐵))
1817fveq2d 5514 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘((𝐶 + 𝐷) − π)) = (cos‘-(𝐴 + 𝐵)))
19 cosmpi 13870 . . . . . . . . . . 11 ((𝐶 + 𝐷) ∈ ℂ → (cos‘((𝐶 + 𝐷) − π)) = -(cos‘(𝐶 + 𝐷)))
202, 19syl 14 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘((𝐶 + 𝐷) − π)) = -(cos‘(𝐶 + 𝐷)))
21 cosneg 11706 . . . . . . . . . . 11 ((𝐴 + 𝐵) ∈ ℂ → (cos‘-(𝐴 + 𝐵)) = (cos‘(𝐴 + 𝐵)))
2211, 21syl 14 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘-(𝐴 + 𝐵)) = (cos‘(𝐴 + 𝐵)))
2318, 20, 223eqtr3d 2218 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → -(cos‘(𝐶 + 𝐷)) = (cos‘(𝐴 + 𝐵)))
2423negeqd 8129 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → --(cos‘(𝐶 + 𝐷)) = -(cos‘(𝐴 + 𝐵)))
254, 24eqtr3d 2212 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘(𝐶 + 𝐷)) = -(cos‘(𝐴 + 𝐵)))
2625oveq2d 5884 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) = ((cos‘(𝐶𝐷)) − -(cos‘(𝐴 + 𝐵))))
27 subcl 8133 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐶𝐷) ∈ ℂ)
2827adantl 277 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐶𝐷) ∈ ℂ)
2928coscld 11690 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (cos‘(𝐶𝐷)) ∈ ℂ)
30293adant3 1017 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘(𝐶𝐷)) ∈ ℂ)
3111coscld 11690 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘(𝐴 + 𝐵)) ∈ ℂ)
3230, 31subnegd 8252 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘(𝐶𝐷)) − -(cos‘(𝐴 + 𝐵))) = ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))))
3326, 32eqtrd 2210 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) = ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))))
3433oveq1d 5883 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) / 2) = (((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))) / 2))
3534oveq2d 5884 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) + (((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) / 2)) = ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) + (((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))) / 2)))
36 subcl 8133 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) ∈ ℂ)
37363ad2ant1 1018 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (𝐴𝐵) ∈ ℂ)
3837coscld 11690 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘(𝐴𝐵)) ∈ ℂ)
3938, 31subcld 8245 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) ∈ ℂ)
4030, 31addcld 7954 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))) ∈ ℂ)
41 2cn 8966 . . . . . . 7 2 ∈ ℂ
42 2ap0 8988 . . . . . . 7 2 # 0
4341, 42pm3.2i 272 . . . . . 6 (2 ∈ ℂ ∧ 2 # 0)
4443a1i 9 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (2 ∈ ℂ ∧ 2 # 0))
45 divdirap 8630 . . . . 5 ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) ∈ ℂ ∧ ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0)) → ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) + ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵)))) / 2) = ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) + (((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))) / 2)))
4639, 40, 44, 45syl3anc 1238 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) + ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵)))) / 2) = ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) + (((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))) / 2)))
4738, 31, 30nppcan3d 8272 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) + ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵)))) = ((cos‘(𝐴𝐵)) + (cos‘(𝐶𝐷))))
4847oveq1d 5883 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) + ((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵)))) / 2) = (((cos‘(𝐴𝐵)) + (cos‘(𝐶𝐷))) / 2))
4946, 48eqtr3d 2212 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) + (((cos‘(𝐶𝐷)) + (cos‘(𝐴 + 𝐵))) / 2)) = (((cos‘(𝐴𝐵)) + (cos‘(𝐶𝐷))) / 2))
5035, 49eqtrd 2210 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) + (((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) / 2)) = (((cos‘(𝐴𝐵)) + (cos‘(𝐶𝐷))) / 2))
51 sinmul 11723 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))
52513ad2ant1 1018 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))
53 sinmul 11723 . . . 4 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((sin‘𝐶) · (sin‘𝐷)) = (((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) / 2))
54533ad2ant2 1019 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((sin‘𝐶) · (sin‘𝐷)) = (((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) / 2))
5552, 54oveq12d 5886 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐶) · (sin‘𝐷))) = ((((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) + (((cos‘(𝐶𝐷)) − (cos‘(𝐶 + 𝐷))) / 2)))
56 simplr 528 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → 𝐵 ∈ ℂ)
57 simpll 527 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → 𝐴 ∈ ℂ)
58 simprl 529 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → 𝐶 ∈ ℂ)
5956, 57, 58pnpcan2d 8283 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐵 + 𝐶) − (𝐴 + 𝐶)) = (𝐵𝐴))
6059fveq2d 5514 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) = (cos‘(𝐵𝐴)))
61603adant3 1017 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) = (cos‘(𝐵𝐴)))
621adantl 277 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐶 + 𝐷) ∈ ℂ)
6310, 62, 283jca 1177 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) ∈ ℂ ∧ (𝐶 + 𝐷) ∈ ℂ ∧ (𝐶𝐷) ∈ ℂ))
64633adant3 1017 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐴 + 𝐵) ∈ ℂ ∧ (𝐶 + 𝐷) ∈ ℂ ∧ (𝐶𝐷) ∈ ℂ))
65 addass 7919 . . . . . . . . . . 11 (((𝐴 + 𝐵) ∈ ℂ ∧ (𝐶 + 𝐷) ∈ ℂ ∧ (𝐶𝐷) ∈ ℂ) → (((𝐴 + 𝐵) + (𝐶 + 𝐷)) + (𝐶𝐷)) = ((𝐴 + 𝐵) + ((𝐶 + 𝐷) + (𝐶𝐷))))
6664, 65syl 14 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((𝐴 + 𝐵) + (𝐶 + 𝐷)) + (𝐶𝐷)) = ((𝐴 + 𝐵) + ((𝐶 + 𝐷) + (𝐶𝐷))))
67 oveq1 5875 . . . . . . . . . . 11 (((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π → (((𝐴 + 𝐵) + (𝐶 + 𝐷)) + (𝐶𝐷)) = (π + (𝐶𝐷)))
68673ad2ant3 1020 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((𝐴 + 𝐵) + (𝐶 + 𝐷)) + (𝐶𝐷)) = (π + (𝐶𝐷)))
69 simpl 109 . . . . . . . . . . . . . 14 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → 𝐶 ∈ ℂ)
70 simpr 110 . . . . . . . . . . . . . 14 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → 𝐷 ∈ ℂ)
7169, 70, 693jca 1177 . . . . . . . . . . . . 13 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ))
72713ad2ant2 1019 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ))
73 ppncan 8176 . . . . . . . . . . . . 13 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐷) + (𝐶𝐷)) = (𝐶 + 𝐶))
7473oveq2d 5884 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + ((𝐶 + 𝐷) + (𝐶𝐷))) = ((𝐴 + 𝐵) + (𝐶 + 𝐶)))
7572, 74syl 14 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐴 + 𝐵) + ((𝐶 + 𝐷) + (𝐶𝐷))) = ((𝐴 + 𝐵) + (𝐶 + 𝐶)))
76 simp1 997 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
7769, 69jca 306 . . . . . . . . . . . . 13 ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐶 ∈ ℂ ∧ 𝐶 ∈ ℂ))
78773ad2ant2 1019 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (𝐶 ∈ ℂ ∧ 𝐶 ∈ ℂ))
79 add4 8095 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐶)) = ((𝐴 + 𝐶) + (𝐵 + 𝐶)))
8076, 78, 79syl2anc 411 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐴 + 𝐵) + (𝐶 + 𝐶)) = ((𝐴 + 𝐶) + (𝐵 + 𝐶)))
81 addcl 7914 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + 𝐶) ∈ ℂ)
8281ad2ant2r 509 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐴 + 𝐶) ∈ ℂ)
83 addcl 7914 . . . . . . . . . . . . . . 15 ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 + 𝐶) ∈ ℂ)
8483ad2ant2lr 510 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐵 + 𝐶) ∈ ℂ)
8582, 84jca 306 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐶) ∈ ℂ ∧ (𝐵 + 𝐶) ∈ ℂ))
86853adant3 1017 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐴 + 𝐶) ∈ ℂ ∧ (𝐵 + 𝐶) ∈ ℂ))
87 addcom 8071 . . . . . . . . . . . 12 (((𝐴 + 𝐶) ∈ ℂ ∧ (𝐵 + 𝐶) ∈ ℂ) → ((𝐴 + 𝐶) + (𝐵 + 𝐶)) = ((𝐵 + 𝐶) + (𝐴 + 𝐶)))
8886, 87syl 14 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐴 + 𝐶) + (𝐵 + 𝐶)) = ((𝐵 + 𝐶) + (𝐴 + 𝐶)))
8975, 80, 883eqtrd 2214 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐴 + 𝐵) + ((𝐶 + 𝐷) + (𝐶𝐷))) = ((𝐵 + 𝐶) + (𝐴 + 𝐶)))
9066, 68, 893eqtr3rd 2219 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐵 + 𝐶) + (𝐴 + 𝐶)) = (π + (𝐶𝐷)))
91 picn 13841 . . . . . . . . . . 11 π ∈ ℂ
92 addcom 8071 . . . . . . . . . . 11 ((π ∈ ℂ ∧ (𝐶𝐷) ∈ ℂ) → (π + (𝐶𝐷)) = ((𝐶𝐷) + π))
9391, 28, 92sylancr 414 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (π + (𝐶𝐷)) = ((𝐶𝐷) + π))
94933adant3 1017 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (π + (𝐶𝐷)) = ((𝐶𝐷) + π))
9590, 94eqtrd 2210 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((𝐵 + 𝐶) + (𝐴 + 𝐶)) = ((𝐶𝐷) + π))
9695fveq2d 5514 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶))) = (cos‘((𝐶𝐷) + π)))
97 cosppi 13872 . . . . . . . . 9 ((𝐶𝐷) ∈ ℂ → (cos‘((𝐶𝐷) + π)) = -(cos‘(𝐶𝐷)))
9828, 97syl 14 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (cos‘((𝐶𝐷) + π)) = -(cos‘(𝐶𝐷)))
99983adant3 1017 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘((𝐶𝐷) + π)) = -(cos‘(𝐶𝐷)))
10096, 99eqtrd 2210 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶))) = -(cos‘(𝐶𝐷)))
10161, 100oveq12d 5886 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) − (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶)))) = ((cos‘(𝐵𝐴)) − -(cos‘(𝐶𝐷))))
102 subcl 8133 . . . . . . . . . 10 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵𝐴) ∈ ℂ)
103102ancoms 268 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵𝐴) ∈ ℂ)
104103adantr 276 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐵𝐴) ∈ ℂ)
105104coscld 11690 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (cos‘(𝐵𝐴)) ∈ ℂ)
106105, 29subnegd 8252 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((cos‘(𝐵𝐴)) − -(cos‘(𝐶𝐷))) = ((cos‘(𝐵𝐴)) + (cos‘(𝐶𝐷))))
1071063adant3 1017 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘(𝐵𝐴)) − -(cos‘(𝐶𝐷))) = ((cos‘(𝐵𝐴)) + (cos‘(𝐶𝐷))))
108101, 107eqtrd 2210 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) − (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶)))) = ((cos‘(𝐵𝐴)) + (cos‘(𝐶𝐷))))
109108oveq1d 5883 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) − (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶)))) / 2) = (((cos‘(𝐵𝐴)) + (cos‘(𝐶𝐷))) / 2))
110 sinmul 11723 . . . . 5 (((𝐵 + 𝐶) ∈ ℂ ∧ (𝐴 + 𝐶) ∈ ℂ) → ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶))) = (((cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) − (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶)))) / 2))
11184, 82, 110syl2anc 411 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶))) = (((cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) − (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶)))) / 2))
1121113adant3 1017 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶))) = (((cos‘((𝐵 + 𝐶) − (𝐴 + 𝐶))) − (cos‘((𝐵 + 𝐶) + (𝐴 + 𝐶)))) / 2))
113 cosneg 11706 . . . . . . . 8 ((𝐴𝐵) ∈ ℂ → (cos‘-(𝐴𝐵)) = (cos‘(𝐴𝐵)))
11436, 113syl 14 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘-(𝐴𝐵)) = (cos‘(𝐴𝐵)))
115 negsubdi2 8193 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (𝐵𝐴))
116115fveq2d 5514 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘-(𝐴𝐵)) = (cos‘(𝐵𝐴)))
117114, 116eqtr3d 2212 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴𝐵)) = (cos‘(𝐵𝐴)))
1181173ad2ant1 1018 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (cos‘(𝐴𝐵)) = (cos‘(𝐵𝐴)))
119118oveq1d 5883 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((cos‘(𝐴𝐵)) + (cos‘(𝐶𝐷))) = ((cos‘(𝐵𝐴)) + (cos‘(𝐶𝐷))))
120119oveq1d 5883 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((cos‘(𝐴𝐵)) + (cos‘(𝐶𝐷))) / 2) = (((cos‘(𝐵𝐴)) + (cos‘(𝐶𝐷))) / 2))
121109, 112, 1203eqtr4d 2220 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶))) = (((cos‘(𝐴𝐵)) + (cos‘(𝐶𝐷))) / 2))
12250, 55, 1213eqtr4d 2220 1 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐶) · (sin‘𝐷))) = ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148   class class class wbr 4000  cfv 5211  (class class class)co 5868  cc 7787  0cc0 7789   + caddc 7792   · cmul 7794  cmin 8105  -cneg 8106   # cap 8515   / cdiv 8605  2c2 8946  sincsin 11623  cosccos 11624  πcpi 11626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909  ax-pre-suploc 7910  ax-addf 7911  ax-mulf 7912
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-of 6076  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-frec 6385  df-1o 6410  df-oadd 6414  df-er 6528  df-map 6643  df-pm 6644  df-en 6734  df-dom 6735  df-fin 6736  df-sup 6976  df-inf 6977  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-5 8957  df-6 8958  df-7 8959  df-8 8960  df-9 8961  df-n0 9153  df-z 9230  df-uz 9505  df-q 9596  df-rp 9628  df-xneg 9746  df-xadd 9747  df-ioo 9866  df-ioc 9867  df-ico 9868  df-icc 9869  df-fz 9983  df-fzo 10116  df-seqfrec 10419  df-exp 10493  df-fac 10677  df-bc 10699  df-ihash 10727  df-shft 10795  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979  df-clim 11258  df-sumdc 11333  df-ef 11627  df-sin 11629  df-cos 11630  df-pi 11632  df-rest 12625  df-topgen 12644  df-psmet 13120  df-xmet 13121  df-met 13122  df-bl 13123  df-mopn 13124  df-top 13129  df-topon 13142  df-bases 13174  df-ntr 13229  df-cn 13321  df-cnp 13322  df-tx 13386  df-cncf 13691  df-limced 13758  df-dvap 13759
This theorem is referenced by: (None)
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