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Theorem addnnnq0 6575
Description: Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
Assertion
Ref Expression
addnnnq0 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Proof of Theorem addnnnq0
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 4401 . . . 4 ((𝐴 ∈ ω ∧ 𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
2 enq0ex 6565 . . . . 5 ~Q0 ∈ V
32ecelqsi 6188 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (ω × N) → [⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
41, 3syl 14 . . 3 ((𝐴 ∈ ω ∧ 𝐵N) → [⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
5 opelxpi 4401 . . . 4 ((𝐶 ∈ ω ∧ 𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
62ecelqsi 6188 . . . 4 (⟨𝐶, 𝐷⟩ ∈ (ω × N) → [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
75, 6syl 14 . . 3 ((𝐶 ∈ ω ∧ 𝐷N) → [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
84, 7anim12i 325 . 2 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
9 eqid 2054 . . . 4 [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0
10 eqid 2054 . . . 4 [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0
119, 10pm3.2i 261 . . 3 ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
12 eqid 2054 . . 3 [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0
13 opeq12 3576 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
1413eceq1d 6170 . . . . . . . 8 ((𝑤 = 𝐴𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
1514eqeq2d 2065 . . . . . . 7 ((𝑤 = 𝐴𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
1615anbi1d 446 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )))
17 simpl 106 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → 𝑤 = 𝐴)
1817oveq1d 5552 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 ·𝑜 𝐷) = (𝐴 ·𝑜 𝐷))
19 simpr 107 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2019oveq1d 5552 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·𝑜 𝐶) = (𝐵 ·𝑜 𝐶))
2118, 20oveq12d 5555 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)) = ((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)))
2219oveq1d 5552 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
2321, 22opeq12d 3582 . . . . . . . 8 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩ = ⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩)
2423eceq1d 6170 . . . . . . 7 ((𝑤 = 𝐴𝑣 = 𝐵) → [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
2524eqeq2d 2065 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → ([⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ↔ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
2616, 25anbi12d 450 . . . . 5 ((𝑤 = 𝐴𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )))
2726spc2egv 2657 . . . 4 ((𝐴 ∈ ω ∧ 𝐵N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
28 opeq12 3576 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
2928eceq1d 6170 . . . . . . . . 9 ((𝑢 = 𝐶𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
3029eqeq2d 2065 . . . . . . . 8 ((𝑢 = 𝐶𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
3130anbi2d 445 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )))
32 simpr 107 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → 𝑡 = 𝐷)
3332oveq2d 5553 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 ·𝑜 𝑡) = (𝑤 ·𝑜 𝐷))
34 simpl 106 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → 𝑢 = 𝐶)
3534oveq2d 5553 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·𝑜 𝑢) = (𝑣 ·𝑜 𝐶))
3633, 35oveq12d 5555 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)) = ((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)))
3732oveq2d 5553 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
3836, 37opeq12d 3582 . . . . . . . . 9 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ = ⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩)
3938eceq1d 6170 . . . . . . . 8 ((𝑢 = 𝐶𝑡 = 𝐷) → [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )
4039eqeq2d 2065 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → ([⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ))
4131, 40anbi12d 450 . . . . . 6 ((𝑢 = 𝐶𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
4241spc2egv 2657 . . . . 5 ((𝐶 ∈ ω ∧ 𝐷N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
43422eximdv 1776 . . . 4 ((𝐶 ∈ ω ∧ 𝐷N) → (∃𝑤𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
4427, 43sylan9 395 . . 3 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
4511, 12, 44mp2ani 416 . 2 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
46 ecexg 6138 . . . 4 ( ~Q0 ∈ V → [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V)
472, 46ax-mp 7 . . 3 [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V
48 simp1 913 . . . . . . . 8 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → 𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 )
4948eqeq1d 2062 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
50 simp2 914 . . . . . . . 8 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 )
5150eqeq1d 2062 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
5249, 51anbi12d 450 . . . . . 6 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 )))
53 simp3 915 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
5453eqeq1d 2062 . . . . . 6 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
5552, 54anbi12d 450 . . . . 5 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
56554exbidv 1764 . . . 4 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
57 addnq0mo 6573 . . . 4 ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
58 dfplq0qs 6556 . . . 4 +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))}
5956, 57, 58ovig 5647 . . 3 (([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V) → (∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
6047, 59mp3an3 1230 . 2 (([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 )) → (∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
618, 45, 60sylc 60 1 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 894   = wceq 1257  wex 1395  wcel 1407  Vcvv 2572  cop 3403  ωcom 4338   × cxp 4368  (class class class)co 5537   +𝑜 coa 6026   ·𝑜 comu 6027  [cec 6132   / cqs 6133  Ncnpi 6398   ~Q0 ceq0 6412   +Q0 cplq0 6415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-id 4055  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-mi 6432  df-enq0 6550  df-nq0 6551  df-plq0 6553
This theorem is referenced by:  addclnq0  6577  nqpnq0nq  6579  nqnq0a  6580  nq0a0  6583  nnanq0  6584  distrnq0  6585  addassnq0  6588
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