Theorem qirropth 43526

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 4094	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ∖ ℚ) → ¬ 𝐴 ∈ ℚ)
2	1	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ¬ 𝐴 ∈ ℚ)
3	2	adantr 485	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → ¬ 𝐴 ∈ ℚ)
4		simpll1 1229	. . . . . . . . . . . 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 3925	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ ℂ)
6		simp2r 1217	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐶 ∈ ℚ)
7	6	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐶 ∈ ℚ)
8		qcn 12986	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
9	7, 8	syl 18	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐶 ∈ ℂ)
10		simp3r 1219	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐸 ∈ ℚ)
11	10	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐸 ∈ ℚ)
12		qcn 12986	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℚ → 𝐸 ∈ ℂ)
13	11, 12	syl 18	. . . . . . . . . . 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 11669	. . . . . . . . . 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 12991	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℚ ∧ 𝐸 ∈ ℚ) → (𝐶 − 𝐸) ∈ ℚ)
16	7, 11, 15	syl2anc 595	. . . . . . . . . . . 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 12986	. . . . . . . . . . . 12 ⊢ ((𝐶 − 𝐸) ∈ ℚ → (𝐶 − 𝐸) ∈ ℂ)
18	16, 17	syl 18	. . . . . . . . . . 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 11229	. . . . . . . . . 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 780	. . . . . . . . . . 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 1216	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐵 ∈ ℚ)
22	21	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐵 ∈ ℚ)
23		qcn 12986	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
24	22, 23	syl 18	. . . . . . . . . . . 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 11228	. . . . . . . . . . . 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 1218	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐷 ∈ ℚ)
27	26	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐷 ∈ ℚ)
28		qcn 12986	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ ℚ → 𝐷 ∈ ℂ)
29	27, 28	syl 18	. . . . . . . . . . . 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 11228	. . . . . . . . . . . 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 11619	. . . . . . . . . . 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 2814	. . . . . . . . 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 12991	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐷 − 𝐵) ∈ ℚ)
35	27, 22, 34	syl2anc 595	. . . . . . . . . . 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 12986	. . . . . . . . . . 11 ⊢ ((𝐷 − 𝐵) ∈ ℚ → (𝐷 − 𝐵) ∈ ℂ)
37	35, 36	syl 18	. . . . . . . . . 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 489	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ¬ 𝐶 = 𝐸)
39		subeq0 11483	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶 − 𝐸) = 0 ↔ 𝐶 = 𝐸))
40	39	necon3abid 3000	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶 − 𝐸) ≠ 0 ↔ ¬ 𝐶 = 𝐸))
41	9, 13, 40	syl2anc 595	. . . . . . . . . . 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 12012	. . . . . . . . 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 12993	. . . . . . . . 9 ⊢ (((𝐷 − 𝐵) ∈ ℚ ∧ (𝐶 − 𝐸) ∈ ℚ ∧ (𝐶 − 𝐸) ≠ 0) → ((𝐷 − 𝐵) / (𝐶 − 𝐸)) ∈ ℚ)
46	35, 16, 42, 45	syl3anc 1396	. . . . . . . 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 2870	. . . . . . 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 417	. . . . . 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 149	. . . . 5 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐶 = 𝐸)
50		simpl2l 1243	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐵 ∈ ℚ)
51	50, 23	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐵 ∈ ℂ)
52	51	adantr 485	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐵 ∈ ℂ)
53		simpl3l 1245	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐷 ∈ ℚ)
54	53, 28	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐷 ∈ ℂ)
55	54	adantr 485	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐷 ∈ ℂ)
56		simpl1 1208	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ (ℂ ∖ ℚ))
57	56	eldifad 3925	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ ℂ)
58		simpl3r 1246	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐸 ∈ ℚ)
59	58, 12	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐸 ∈ ℂ)
60	57, 59	mulcld 11228	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐴 · 𝐸) ∈ ℂ)
61	60	adantr 485	. . . . . . 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 489	. . . . . . . . . . 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 2775	. . . . . . . . . 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 7427	. . . . . . . . 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 7427	. . . . . . . 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 780	. . . . . . . 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 2804	. . . . . . 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 11415	. . . . . 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 417	. . . . 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 525	. . . 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 466	. . 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 417	. 2 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))
73		id 23	. . 3 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷)
74		oveq2 7419	. . 3 ⊢ (𝐶 = 𝐸 → (𝐴 · 𝐶) = (𝐴 · 𝐸))
75	73, 74	oveqan12d 7430	. 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 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∖ cdif 3910  (class class class)co 7411  ℂcc 11097  0cc0 11099   + caddc 11102   · cmul 11104   − cmin 11440   / cdiv 11870  ℚcq 12971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-n0 12504  df-z 12591  df-q 12972

This theorem is referenced by:  rmxypairf1o  43529  rmxycomplete  43535  rmxyneg  43538  rmxyadd  43539  rmxy1  43540  rmxy0  43541  jm2.22  43613