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Theorem colinearalglem1 28773

Description: Lemma for colinearalg 28777. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

**Assertion**
Ref	Expression
colinearalglem1	⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵 − 𝐴) · (𝐹 − 𝐷)) = ((𝐸 − 𝐷) · (𝐶 − 𝐴)) ↔ ((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) = ((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷)))))

**Proof of Theorem colinearalglem1**
Step	Hyp	Ref	Expression
1		simpl2 1189	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → 𝐵 ∈ ℂ)
2		simpl1 1188	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → 𝐴 ∈ ℂ)
3	1, 2	subcld 11601	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐵 − 𝐴) ∈ ℂ)
4		simpr3 1193	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → 𝐹 ∈ ℂ)
5		simpr1 1191	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → 𝐷 ∈ ℂ)
6	3, 4, 5	subdid 11700	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐵 − 𝐴) · (𝐹 − 𝐷)) = (((𝐵 − 𝐴) · 𝐹) − ((𝐵 − 𝐴) · 𝐷)))
7	1, 2, 4	subdird 11701	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐵 − 𝐴) · 𝐹) = ((𝐵 · 𝐹) − (𝐴 · 𝐹)))
8	1, 2, 5	subdird 11701	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐵 − 𝐴) · 𝐷) = ((𝐵 · 𝐷) − (𝐴 · 𝐷)))
9	7, 8	oveq12d 7435	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵 − 𝐴) · 𝐹) − ((𝐵 − 𝐴) · 𝐷)) = (((𝐵 · 𝐹) − (𝐴 · 𝐹)) − ((𝐵 · 𝐷) − (𝐴 · 𝐷))))
10		simp2 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ)
11		simp3 1135	. . . . . . . 8 ⊢ ((𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ) → 𝐹 ∈ ℂ)
12		mulcl 11222	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝐹 ∈ ℂ) → (𝐵 · 𝐹) ∈ ℂ)
13	10, 11, 12	syl2an 594	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐵 · 𝐹) ∈ ℂ)
14		simp1 1133	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ)
15		mulcl 11222	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ ℂ) → (𝐴 · 𝐹) ∈ ℂ)
16	14, 11, 15	syl2an 594	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐴 · 𝐹) ∈ ℂ)
17	13, 16	subcld 11601	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐵 · 𝐹) − (𝐴 · 𝐹)) ∈ ℂ)
18		simp1 1133	. . . . . . 7 ⊢ ((𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ) → 𝐷 ∈ ℂ)
19		mulcl 11222	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐵 · 𝐷) ∈ ℂ)
20	10, 18, 19	syl2an 594	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐵 · 𝐷) ∈ ℂ)
21		mulcl 11222	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐴 · 𝐷) ∈ ℂ)
22	14, 18, 21	syl2an 594	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐴 · 𝐷) ∈ ℂ)
23	17, 20, 22	subsub3d 11631	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵 · 𝐹) − (𝐴 · 𝐹)) − ((𝐵 · 𝐷) − (𝐴 · 𝐷))) = ((((𝐵 · 𝐹) − (𝐴 · 𝐹)) + (𝐴 · 𝐷)) − (𝐵 · 𝐷)))
24	17, 22, 20	addsubd 11622	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((((𝐵 · 𝐹) − (𝐴 · 𝐹)) + (𝐴 · 𝐷)) − (𝐵 · 𝐷)) = ((((𝐵 · 𝐹) − (𝐴 · 𝐹)) − (𝐵 · 𝐷)) + (𝐴 · 𝐷)))
25	9, 23, 24	3eqtrrd 2770	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((((𝐵 · 𝐹) − (𝐴 · 𝐹)) − (𝐵 · 𝐷)) + (𝐴 · 𝐷)) = (((𝐵 − 𝐴) · 𝐹) − ((𝐵 − 𝐴) · 𝐷)))
26	13, 16, 20	subsub4d 11632	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵 · 𝐹) − (𝐴 · 𝐹)) − (𝐵 · 𝐷)) = ((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))))
27	26	oveq1d 7432	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((((𝐵 · 𝐹) − (𝐴 · 𝐹)) − (𝐵 · 𝐷)) + (𝐴 · 𝐷)) = (((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) + (𝐴 · 𝐷)))
28	6, 25, 27	3eqtr2d 2771	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐵 − 𝐴) · (𝐹 − 𝐷)) = (((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) + (𝐴 · 𝐷)))
29		simpr2 1192	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → 𝐸 ∈ ℂ)
30	29, 5	subcld 11601	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐸 − 𝐷) ∈ ℂ)
31		simpl3 1190	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → 𝐶 ∈ ℂ)
32	31, 2	subcld 11601	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐶 − 𝐴) ∈ ℂ)
33	30, 32	mulcomd 11265	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐸 − 𝐷) · (𝐶 − 𝐴)) = ((𝐶 − 𝐴) · (𝐸 − 𝐷)))
34	32, 29, 5	subdid 11700	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐶 − 𝐴) · (𝐸 − 𝐷)) = (((𝐶 − 𝐴) · 𝐸) − ((𝐶 − 𝐴) · 𝐷)))
35	31, 2, 29	subdird 11701	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐶 − 𝐴) · 𝐸) = ((𝐶 · 𝐸) − (𝐴 · 𝐸)))
36	31, 2, 5	subdird 11701	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐶 − 𝐴) · 𝐷) = ((𝐶 · 𝐷) − (𝐴 · 𝐷)))
37	35, 36	oveq12d 7435	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐶 − 𝐴) · 𝐸) − ((𝐶 − 𝐴) · 𝐷)) = (((𝐶 · 𝐸) − (𝐴 · 𝐸)) − ((𝐶 · 𝐷) − (𝐴 · 𝐷))))
38		simp3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ)
39		simp2 1134	. . . . . . . . 9 ⊢ ((𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ) → 𝐸 ∈ ℂ)
40		mulcl 11222	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → (𝐶 · 𝐸) ∈ ℂ)
41	38, 39, 40	syl2an 594	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐶 · 𝐸) ∈ ℂ)
42		mulcl 11222	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐸 ∈ ℂ) → (𝐴 · 𝐸) ∈ ℂ)
43	14, 39, 42	syl2an 594	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐴 · 𝐸) ∈ ℂ)
44	41, 43	subcld 11601	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐶 · 𝐸) − (𝐴 · 𝐸)) ∈ ℂ)
45		mulcl 11222	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐶 · 𝐷) ∈ ℂ)
46	38, 18, 45	syl2an 594	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (𝐶 · 𝐷) ∈ ℂ)
47	44, 46, 22	subsub3d 11631	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐶 · 𝐸) − (𝐴 · 𝐸)) − ((𝐶 · 𝐷) − (𝐴 · 𝐷))) = ((((𝐶 · 𝐸) − (𝐴 · 𝐸)) + (𝐴 · 𝐷)) − (𝐶 · 𝐷)))
48	44, 22, 46	addsubd 11622	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((((𝐶 · 𝐸) − (𝐴 · 𝐸)) + (𝐴 · 𝐷)) − (𝐶 · 𝐷)) = ((((𝐶 · 𝐸) − (𝐴 · 𝐸)) − (𝐶 · 𝐷)) + (𝐴 · 𝐷)))
49	37, 47, 48	3eqtrrd 2770	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((((𝐶 · 𝐸) − (𝐴 · 𝐸)) − (𝐶 · 𝐷)) + (𝐴 · 𝐷)) = (((𝐶 − 𝐴) · 𝐸) − ((𝐶 − 𝐴) · 𝐷)))
50	41, 43, 46	subsub4d 11632	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐶 · 𝐸) − (𝐴 · 𝐸)) − (𝐶 · 𝐷)) = ((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷))))
51	50	oveq1d 7432	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((((𝐶 · 𝐸) − (𝐴 · 𝐸)) − (𝐶 · 𝐷)) + (𝐴 · 𝐷)) = (((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷))) + (𝐴 · 𝐷)))
52	49, 51	eqtr3d 2767	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐶 − 𝐴) · 𝐸) − ((𝐶 − 𝐴) · 𝐷)) = (((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷))) + (𝐴 · 𝐷)))
53	33, 34, 52	3eqtrd 2769	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐸 − 𝐷) · (𝐶 − 𝐴)) = (((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷))) + (𝐴 · 𝐷)))
54	28, 53	eqeq12d 2741	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵 − 𝐴) · (𝐹 − 𝐷)) = ((𝐸 − 𝐷) · (𝐶 − 𝐴)) ↔ (((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) + (𝐴 · 𝐷)) = (((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷))) + (𝐴 · 𝐷))))
55	16, 20	addcld 11263	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐴 · 𝐹) + (𝐵 · 𝐷)) ∈ ℂ)
56	13, 55	subcld 11601	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) ∈ ℂ)
57	43, 46	addcld 11263	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐴 · 𝐸) + (𝐶 · 𝐷)) ∈ ℂ)
58	41, 57	subcld 11601	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷))) ∈ ℂ)
59	56, 58, 22	addcan2d 11448	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → ((((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) + (𝐴 · 𝐷)) = (((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷))) + (𝐴 · 𝐷)) ↔ ((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) = ((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷)))))
60	54, 59	bitrd 278	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵 − 𝐴) · (𝐹 − 𝐷)) = ((𝐸 − 𝐷) · (𝐶 − 𝐴)) ↔ ((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) = ((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 (class class class)co 7417 ℂcc 11136 + caddc 11141 · cmul 11143 − cmin 11474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-sub 11476

This theorem is referenced by: colinearalglem2 28774

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