qirropth - Mathbox for Stefan O'Rear

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Theorem qirropth 43336

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 4072	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ∖ ℚ) → ¬ 𝐴 ∈ ℚ)
2	1	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ¬ 𝐴 ∈ ℚ)
3	2	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → ¬ 𝐴 ∈ ℚ)
4		simpll1 1214	. . . . . . . . . . . 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 3901	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ ℂ)
6		simp2r 1202	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐶 ∈ ℚ)
7	6	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐶 ∈ ℚ)
8		qcn 12913	. . . . . . . . . . . 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 1204	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐸 ∈ ℚ)
11	10	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐸 ∈ ℚ)
12		qcn 12913	. . . . . . . . . . . 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 11606	. . . . . . . . . 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 12918	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℚ ∧ 𝐸 ∈ ℚ) → (𝐶 − 𝐸) ∈ ℚ)
16	7, 11, 15	syl2anc 585	. . . . . . . . . . . 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 12913	. . . . . . . . . . . 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 11166	. . . . . . . . . 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 769	. . . . . . . . . . 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 1201	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐵 ∈ ℚ)
22	21	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐵 ∈ ℚ)
23		qcn 12913	. . . . . . . . . . . . 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 11165	. . . . . . . . . . . 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 1203	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐷 ∈ ℚ)
27	26	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐷 ∈ ℚ)
28		qcn 12913	. . . . . . . . . . . . 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 11165	. . . . . . . . . . . 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 11556	. . . . . . . . . . 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 232	. . . . . . . . . 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 2781	. . . . . . . . 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 12918	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐷 − 𝐵) ∈ ℚ)
35	27, 22, 34	syl2anc 585	. . . . . . . . . . 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 12913	. . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ¬ 𝐶 = 𝐸)
39		subeq0 11420	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶 − 𝐸) = 0 ↔ 𝐶 = 𝐸))
40	39	necon3abid 2968	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶 − 𝐸) ≠ 0 ↔ ¬ 𝐶 = 𝐸))
41	9, 13, 40	syl2anc 585	. . . . . . . . . . 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 257	. . . . . . . . . 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 11953	. . . . . . . . 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 257	. . . . . . . 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 12920	. . . . . . . . 9 ⊢ (((𝐷 − 𝐵) ∈ ℚ ∧ (𝐶 − 𝐸) ∈ ℚ ∧ (𝐶 − 𝐸) ≠ 0) → ((𝐷 − 𝐵) / (𝐶 − 𝐸)) ∈ ℚ)
46	35, 16, 42, 45	syl3anc 1374	. . . . . . . 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 2837	. . . . . . 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 412	. . . . . 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 148	. . . . 5 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐶 = 𝐸)
50		simpl2l 1228	. . . . . . . . 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 480	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐵 ∈ ℂ)
53		simpl3l 1230	. . . . . . . . 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 480	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐷 ∈ ℂ)
56		simpl1 1193	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ (ℂ ∖ ℚ))
57	56	eldifad 3901	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ ℂ)
58		simpl3r 1231	. . . . . . . . . 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 11165	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐴 · 𝐸) ∈ ℂ)
61	60	adantr 480	. . . . . . 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 484	. . . . . . . . . . 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 2742	. . . . . . . . . 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 7383	. . . . . . . . 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 7383	. . . . . . . 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 769	. . . . . . . 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 2771	. . . . . . 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 11352	. . . . . 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 412	. . . . 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 516	. . . 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 461	. . 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 412	. 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 7375	. . 3 ⊢ (𝐶 = 𝐸 → (𝐴 · 𝐶) = (𝐴 · 𝐸))
75	73, 74	oveqan12d 7386	. 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 = 𝐸) → (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)))
76	72, 75	impbid1 225	1 ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∖ cdif 3886 (class class class)co 7367 ℂcc 11036 0cc0 11038 + caddc 11041 · cmul 11043 − cmin 11377 / cdiv 11807 ℚcq 12898

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-n0 12438 df-z 12525 df-q 12899

This theorem is referenced by: rmxypairf1o 43339 rmxycomplete 43345 rmxyneg 43348 rmxyadd 43349 rmxy1 43350 rmxy0 43351 jm2.22 43423

W3C validator