Theorem qirropth 40250

Description: This lemma implements the concept of "equate rational and irrational parts", used to prove many arithmetical properties of the X and Y sequences. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Assertion

Ref	Expression
qirropth	⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))

Proof of Theorem qirropth

Step	Hyp	Ref	Expression
1		eldifn 4035	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ∖ ℚ) → ¬ 𝐴 ∈ ℚ)
2	1	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ¬ 𝐴 ∈ ℚ)
3	2	adantr 484	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → ¬ 𝐴 ∈ ℚ)
4		simpll1 1209	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ (ℂ ∖ ℚ))
5	4	eldifad 3872	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ ℂ)
6		simp2r 1197	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐶 ∈ ℚ)
7	6	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐶 ∈ ℚ)
8		qcn 12408	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐶 ∈ ℂ)
10		simp3r 1199	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐸 ∈ ℚ)
11	10	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐸 ∈ ℚ)
12		qcn 12408	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℚ → 𝐸 ∈ ℂ)
13	11, 12	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐸 ∈ ℂ)
14	5, 9, 13	subdid 11139	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐴 · (𝐶 − 𝐸)) = ((𝐴 · 𝐶) − (𝐴 · 𝐸)))
15		qsubcl 12413	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℚ ∧ 𝐸 ∈ ℚ) → (𝐶 − 𝐸) ∈ ℚ)
16	7, 11, 15	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐶 − 𝐸) ∈ ℚ)
17		qcn 12408	. . . . . . . . . . . 12 ⊢ ((𝐶 − 𝐸) ∈ ℚ → (𝐶 − 𝐸) ∈ ℂ)
18	16, 17	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐶 − 𝐸) ∈ ℂ)
19	18, 5	mulcomd 10705	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐶 − 𝐸) · 𝐴) = (𝐴 · (𝐶 − 𝐸)))
20		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)))
21		simp2l 1196	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐵 ∈ ℚ)
22	21	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐵 ∈ ℚ)
23		qcn 12408	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
24	22, 23	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐵 ∈ ℂ)
25	5, 9	mulcld 10704	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐴 · 𝐶) ∈ ℂ)
26		simp3l 1198	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐷 ∈ ℚ)
27	26	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐷 ∈ ℚ)
28		qcn 12408	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ ℚ → 𝐷 ∈ ℂ)
29	27, 28	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐷 ∈ ℂ)
30	5, 13	mulcld 10704	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐴 · 𝐸) ∈ ℂ)
31	24, 25, 29, 30	addsubeq4d 11091	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐷 − 𝐵) = ((𝐴 · 𝐶) − (𝐴 · 𝐸))))
32	20, 31	mpbid 235	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐷 − 𝐵) = ((𝐴 · 𝐶) − (𝐴 · 𝐸)))
33	14, 19, 32	3eqtr4d 2803	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐶 − 𝐸) · 𝐴) = (𝐷 − 𝐵))
34		qsubcl 12413	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐷 − 𝐵) ∈ ℚ)
35	27, 22, 34	syl2anc 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐷 − 𝐵) ∈ ℚ)
36		qcn 12408	. . . . . . . . . . 11 ⊢ ((𝐷 − 𝐵) ∈ ℚ → (𝐷 − 𝐵) ∈ ℂ)
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐷 − 𝐵) ∈ ℂ)
38		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ¬ 𝐶 = 𝐸)
39		subeq0 10955	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶 − 𝐸) = 0 ↔ 𝐶 = 𝐸))
40	39	necon3abid 2987	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶 − 𝐸) ≠ 0 ↔ ¬ 𝐶 = 𝐸))
41	9, 13, 40	syl2anc 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐶 − 𝐸) ≠ 0 ↔ ¬ 𝐶 = 𝐸))
42	38, 41	mpbird 260	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐶 − 𝐸) ≠ 0)
43	37, 18, 5, 42	divmuld 11481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (((𝐷 − 𝐵) / (𝐶 − 𝐸)) = 𝐴 ↔ ((𝐶 − 𝐸) · 𝐴) = (𝐷 − 𝐵)))
44	33, 43	mpbird 260	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐷 − 𝐵) / (𝐶 − 𝐸)) = 𝐴)
45		qdivcl 12415	. . . . . . . . 9 ⊢ (((𝐷 − 𝐵) ∈ ℚ ∧ (𝐶 − 𝐸) ∈ ℚ ∧ (𝐶 − 𝐸) ≠ 0) → ((𝐷 − 𝐵) / (𝐶 − 𝐸)) ∈ ℚ)
46	35, 16, 42, 45	syl3anc 1368	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐷 − 𝐵) / (𝐶 − 𝐸)) ∈ ℚ)
47	44, 46	eqeltrrd 2853	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ ℚ)
48	47	ex 416	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (¬ 𝐶 = 𝐸 → 𝐴 ∈ ℚ))
49	3, 48	mt3d 150	. . . . 5 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐶 = 𝐸)
50		simpl2l 1223	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐵 ∈ ℚ)
51	50, 23	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐵 ∈ ℂ)
52	51	adantr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐵 ∈ ℂ)
53		simpl3l 1225	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐷 ∈ ℚ)
54	53, 28	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐷 ∈ ℂ)
55	54	adantr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐷 ∈ ℂ)
56		simpl1 1188	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ (ℂ ∖ ℚ))
57	56	eldifad 3872	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ ℂ)
58		simpl3r 1226	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐸 ∈ ℚ)
59	58, 12	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐸 ∈ ℂ)
60	57, 59	mulcld 10704	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐴 · 𝐸) ∈ ℂ)
61	60	adantr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → (𝐴 · 𝐸) ∈ ℂ)
62		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐶 = 𝐸)
63	62	eqcomd 2764	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐸 = 𝐶)
64	63	oveq2d 7171	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → (𝐴 · 𝐸) = (𝐴 · 𝐶))
65	64	oveq2d 7171	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → (𝐵 + (𝐴 · 𝐸)) = (𝐵 + (𝐴 · 𝐶)))
66		simplr 768	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)))
67	65, 66	eqtrd 2793	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → (𝐵 + (𝐴 · 𝐸)) = (𝐷 + (𝐴 · 𝐸)))
68	52, 55, 61, 67	addcan2ad 10889	. . . . . 6 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐵 = 𝐷)
69	68	ex 416	. . . . 5 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐶 = 𝐸 → 𝐵 = 𝐷))
70	49, 69	jcai 520	. . . 4 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐶 = 𝐸 ∧ 𝐵 = 𝐷))
71	70	ancomd 465	. . 3 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸))
72	71	ex 416	. 2 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))
73		id 22	. . 3 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷)
74		oveq2 7163	. . 3 ⊢ (𝐶 = 𝐸 → (𝐴 · 𝐶) = (𝐴 · 𝐸))
75	73, 74	oveqan12d 7174	. 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 = 𝐸) → (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)))
76	72, 75	impbid1 228	1 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∖ cdif 3857  (class class class)co 7155  ℂcc 10578  0cc0 10580   + caddc 10583   · cmul 10585   − cmin 10913   / cdiv 11340  ℚcq 12393

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-n0 11940  df-z 12026  df-q 12394

This theorem is referenced by:  rmxypairf1o  40253  rmxycomplete  40259  rmxyneg  40262  rmxyadd  40263  rmxy1  40264  rmxy0  40265  jm2.22  40337