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Theorem mulnnnq0 6605
Description: Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
Assertion
Ref Expression
mulnnnq0 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Proof of Theorem mulnnnq0
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 4403 . . . 4 ((𝐴 ∈ ω ∧ 𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
2 enq0ex 6594 . . . . 5 ~Q0 ∈ V
32ecelqsi 6190 . . . 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 4403 . . . 4 ((𝐶 ∈ ω ∧ 𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
62ecelqsi 6190 . . . 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 2056 . . . 4 [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0
10 eqid 2056 . . . 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 2056 . . 3 [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0
13 opeq12 3578 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14 eceq1 6171 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
1514eqeq2d 2067 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
1615anbi1d 446 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 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 vex 2577 . . . . . . . . . . 11 𝑤 ∈ V
18 vex 2577 . . . . . . . . . . 11 𝑣 ∈ V
1917, 18opth 4001 . . . . . . . . . 10 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑤 = 𝐴𝑣 = 𝐵))
20 oveq1 5546 . . . . . . . . . . . 12 (𝑤 = 𝐴 → (𝑤 ·𝑜 𝐶) = (𝐴 ·𝑜 𝐶))
2120adantr 265 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 ·𝑜 𝐶) = (𝐴 ·𝑜 𝐶))
22 oveq1 5546 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
2322adantl 266 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
2421, 23opeq12d 3584 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩ = ⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩)
2519, 24sylbi 118 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩ = ⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩)
2625eceq1d 6172 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
2726eqeq2d 2067 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ↔ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
2816, 27anbi12d 450 . . . . . 6 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )))
2913, 28syl 14 . . . . 5 ((𝑤 = 𝐴𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )))
3029spc2egv 2659 . . . 4 ((𝐴 ∈ ω ∧ 𝐵N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
31 opeq12 3578 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
32 eceq1 6171 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
3332eqeq2d 2067 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
3433anbi2d 445 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 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 )))
35 vex 2577 . . . . . . . . . . . 12 𝑢 ∈ V
36 vex 2577 . . . . . . . . . . . 12 𝑡 ∈ V
3735, 36opth 4001 . . . . . . . . . . 11 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑢 = 𝐶𝑡 = 𝐷))
38 oveq2 5547 . . . . . . . . . . . . 13 (𝑢 = 𝐶 → (𝑤 ·𝑜 𝑢) = (𝑤 ·𝑜 𝐶))
3938adantr 265 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 ·𝑜 𝑢) = (𝑤 ·𝑜 𝐶))
40 oveq2 5547 . . . . . . . . . . . . 13 (𝑡 = 𝐷 → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
4140adantl 266 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
4239, 41opeq12d 3584 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩ = ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩)
4337, 42sylbi 118 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩ = ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩)
4443eceq1d 6172 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )
4544eqeq2d 2067 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ))
4634, 45anbi12d 450 . . . . . . 7 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
4731, 46syl 14 . . . . . 6 ((𝑢 = 𝐶𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
4847spc2egv 2659 . . . . 5 ((𝐶 ∈ ω ∧ 𝐷N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
49482eximdv 1778 . . . 4 ((𝐶 ∈ ω ∧ 𝐷N) → (∃𝑤𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
5030, 49sylan9 395 . . 3 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
5111, 12, 50mp2ani 416 . 2 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
52 ecexg 6140 . . . 4 ( ~Q0 ∈ V → [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V)
532, 52ax-mp 7 . . 3 [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V
54 eqeq1 2062 . . . . . . . 8 (𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
55 eqeq1 2062 . . . . . . . 8 (𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
5654, 55bi2anan9 548 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 )))
57 eqeq1 2062 . . . . . . 7 (𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 → (𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
5856, 57bi2anan9 548 . . . . . 6 (((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
59583impa 1110 . . . . 5 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
60594exbidv 1766 . . . 4 ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0𝑦 = [⟨𝐶, 𝐷⟩] ~Q0𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
61 mulnq0mo 6603 . . . 4 ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
62 dfmq0qs 6584 . . . 4 ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))}
6360, 61, 62ovig 5649 . . 3 (([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V) → (∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
6453, 63mp3an3 1232 . 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 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
658, 51, 64sylc 60 1 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cop 3405  ωcom 4340   × cxp 4370  (class class class)co 5539   ·𝑜 comu 6029  [cec 6134   / cqs 6135  Ncnpi 6427   ~Q0 ceq0 6441   ·Q0 cmq0 6445
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-mi 6461  df-enq0 6579  df-nq0 6580  df-mq0 6583
This theorem is referenced by:  mulclnq0  6607  nqnq0m  6610  nq0m0r  6611  distrnq0  6614  mulcomnq0  6615  nq02m  6620
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