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Theorem qirropth 41417
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 = 𝐷𝐶 = 𝐸)))

Proof of Theorem qirropth
StepHypRef Expression
1 eldifn 4123 . . . . . . . 8 (𝐴 ∈ (ℂ ∖ ℚ) → ¬ 𝐴 ∈ ℚ)
213ad2ant1 1133 . . . . . . 7 ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ¬ 𝐴 ∈ ℚ)
32adantr 481 . . . . . 6 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → ¬ 𝐴 ∈ ℚ)
4 simpll1 1212 . . . . . . . . . . . 12 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ (ℂ ∖ ℚ))
54eldifad 3956 . . . . . . . . . . 11 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ ℂ)
6 simp2r 1200 . . . . . . . . . . . . 13 ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐶 ∈ ℚ)
76ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐶 ∈ ℚ)
8 qcn 12929 . . . . . . . . . . . 12 (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
97, 8syl 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 1202 . . . . . . . . . . . . 13 ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐸 ∈ ℚ)
1110ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐸 ∈ ℚ)
12 qcn 12929 . . . . . . . . . . . 12 (𝐸 ∈ ℚ → 𝐸 ∈ ℂ)
1311, 12syl 17 . . . . . . . . . . 11 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐸 ∈ ℂ)
145, 9, 13subdid 11652 . . . . . . . . . 10 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐴 · (𝐶𝐸)) = ((𝐴 · 𝐶) − (𝐴 · 𝐸)))
15 qsubcl 12934 . . . . . . . . . . . . 13 ((𝐶 ∈ ℚ ∧ 𝐸 ∈ ℚ) → (𝐶𝐸) ∈ ℚ)
167, 11, 15syl2anc 584 . . . . . . . . . . . 12 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐶𝐸) ∈ ℚ)
17 qcn 12929 . . . . . . . . . . . 12 ((𝐶𝐸) ∈ ℚ → (𝐶𝐸) ∈ ℂ)
1816, 17syl 17 . . . . . . . . . . 11 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐶𝐸) ∈ ℂ)
1918, 5mulcomd 11217 . . . . . . . . . 10 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐶𝐸) · 𝐴) = (𝐴 · (𝐶𝐸)))
20 simplr 767 . . . . . . . . . . 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 1199 . . . . . . . . . . . . . 14 ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐵 ∈ ℚ)
2221ad2antrr 724 . . . . . . . . . . . . 13 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐵 ∈ ℚ)
23 qcn 12929 . . . . . . . . . . . . 13 (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
2422, 23syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐵 ∈ ℂ)
255, 9mulcld 11216 . . . . . . . . . . . 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 1201 . . . . . . . . . . . . . 14 ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → 𝐷 ∈ ℚ)
2726ad2antrr 724 . . . . . . . . . . . . 13 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐷 ∈ ℚ)
28 qcn 12929 . . . . . . . . . . . . 13 (𝐷 ∈ ℚ → 𝐷 ∈ ℂ)
2927, 28syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐷 ∈ ℂ)
305, 13mulcld 11216 . . . . . . . . . . . 12 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐴 · 𝐸) ∈ ℂ)
3124, 25, 29, 30addsubeq4d 11604 . . . . . . . . . . 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)) ↔ (𝐷𝐵) = ((𝐴 · 𝐶) − (𝐴 · 𝐸))))
3220, 31mpbid 231 . . . . . . . . . 10 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐷𝐵) = ((𝐴 · 𝐶) − (𝐴 · 𝐸)))
3314, 19, 323eqtr4d 2781 . . . . . . . . 9 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐶𝐸) · 𝐴) = (𝐷𝐵))
34 qsubcl 12934 . . . . . . . . . . . 12 ((𝐷 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐷𝐵) ∈ ℚ)
3527, 22, 34syl2anc 584 . . . . . . . . . . 11 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐷𝐵) ∈ ℚ)
36 qcn 12929 . . . . . . . . . . 11 ((𝐷𝐵) ∈ ℚ → (𝐷𝐵) ∈ ℂ)
3735, 36syl 17 . . . . . . . . . 10 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐷𝐵) ∈ ℂ)
38 simpr 485 . . . . . . . . . . 11 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ¬ 𝐶 = 𝐸)
39 subeq0 11468 . . . . . . . . . . . . 13 ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶𝐸) = 0 ↔ 𝐶 = 𝐸))
4039necon3abid 2976 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ((𝐶𝐸) ≠ 0 ↔ ¬ 𝐶 = 𝐸))
419, 13, 40syl2anc 584 . . . . . . . . . . 11 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐶𝐸) ≠ 0 ↔ ¬ 𝐶 = 𝐸))
4238, 41mpbird 256 . . . . . . . . . 10 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (𝐶𝐸) ≠ 0)
4337, 18, 5, 42divmuld 11994 . . . . . . . . 9 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → (((𝐷𝐵) / (𝐶𝐸)) = 𝐴 ↔ ((𝐶𝐸) · 𝐴) = (𝐷𝐵)))
4433, 43mpbird 256 . . . . . . . 8 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐷𝐵) / (𝐶𝐸)) = 𝐴)
45 qdivcl 12936 . . . . . . . . 9 (((𝐷𝐵) ∈ ℚ ∧ (𝐶𝐸) ∈ ℚ ∧ (𝐶𝐸) ≠ 0) → ((𝐷𝐵) / (𝐶𝐸)) ∈ ℚ)
4635, 16, 42, 45syl3anc 1371 . . . . . . . 8 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → ((𝐷𝐵) / (𝐶𝐸)) ∈ ℚ)
4744, 46eqeltrrd 2833 . . . . . . 7 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ ¬ 𝐶 = 𝐸) → 𝐴 ∈ ℚ)
4847ex 413 . . . . . 6 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (¬ 𝐶 = 𝐸𝐴 ∈ ℚ))
493, 48mt3d 148 . . . . 5 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐶 = 𝐸)
50 simpl2l 1226 . . . . . . . . 9 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐵 ∈ ℚ)
5150, 23syl 17 . . . . . . . 8 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐵 ∈ ℂ)
5251adantr 481 . . . . . . 7 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐵 ∈ ℂ)
53 simpl3l 1228 . . . . . . . . 9 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐷 ∈ ℚ)
5453, 28syl 17 . . . . . . . 8 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐷 ∈ ℂ)
5554adantr 481 . . . . . . 7 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐷 ∈ ℂ)
56 simpl1 1191 . . . . . . . . . 10 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ (ℂ ∖ ℚ))
5756eldifad 3956 . . . . . . . . 9 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐴 ∈ ℂ)
58 simpl3r 1229 . . . . . . . . . 10 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐸 ∈ ℚ)
5958, 12syl 17 . . . . . . . . 9 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → 𝐸 ∈ ℂ)
6057, 59mulcld 11216 . . . . . . . 8 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐴 · 𝐸) ∈ ℂ)
6160adantr 481 . . . . . . 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 485 . . . . . . . . . . 11 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐶 = 𝐸)
6362eqcomd 2737 . . . . . . . . . 10 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐸 = 𝐶)
6463oveq2d 7409 . . . . . . . . 9 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → (𝐴 · 𝐸) = (𝐴 · 𝐶))
6564oveq2d 7409 . . . . . . . 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 767 . . . . . . . 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)))
6765, 66eqtrd 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)))
6852, 55, 61, 67addcan2ad 11402 . . . . . 6 ((((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) ∧ 𝐶 = 𝐸) → 𝐵 = 𝐷)
6968ex 413 . . . . 5 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐶 = 𝐸𝐵 = 𝐷))
7049, 69jcai 517 . . . 4 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐶 = 𝐸𝐵 = 𝐷))
7170ancomd 462 . . 3 (((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) ∧ (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸))) → (𝐵 = 𝐷𝐶 = 𝐸))
7271ex 413 . 2 ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) → (𝐵 = 𝐷𝐶 = 𝐸)))
73 id 22 . . 3 (𝐵 = 𝐷𝐵 = 𝐷)
74 oveq2 7401 . . 3 (𝐶 = 𝐸 → (𝐴 · 𝐶) = (𝐴 · 𝐸))
7573, 74oveqan12d 7412 . 2 ((𝐵 = 𝐷𝐶 = 𝐸) → (𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)))
7672, 75impbid1 224 1 ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐵 = 𝐷𝐶 = 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  cdif 3941  (class class class)co 7393  cc 11090  0cc0 11092   + caddc 11095   · cmul 11097  cmin 11426   / cdiv 11853  cq 12914
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-n0 12455  df-z 12541  df-q 12915
This theorem is referenced by:  rmxypairf1o  41421  rmxycomplete  41427  rmxyneg  41430  rmxyadd  41431  rmxy1  41432  rmxy0  41433  jm2.22  41505
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