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Theorem mulnnnq0 6956
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 4444 . . . 4 ((𝐴 ∈ ω ∧ 𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
2 enq0ex 6945 . . . . 5 ~Q0 ∈ V
32ecelqsi 6300 . . . 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 4444 . . . 4 ((𝐶 ∈ ω ∧ 𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
62ecelqsi 6300 . . . 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 331 . 2 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
9 eqid 2085 . . . 4 [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0
10 eqid 2085 . . . 4 [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0
119, 10pm3.2i 266 . . 3 ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
12 eqid 2085 . . 3 [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0
13 opeq12 3609 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14 eceq1 6281 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
1514eqeq2d 2096 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
1615anbi1d 453 . . . . . . 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 2618 . . . . . . . . . . 11 𝑤 ∈ V
18 vex 2618 . . . . . . . . . . 11 𝑣 ∈ V
1917, 18opth 4040 . . . . . . . . . 10 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑤 = 𝐴𝑣 = 𝐵))
20 oveq1 5622 . . . . . . . . . . . 12 (𝑤 = 𝐴 → (𝑤 ·𝑜 𝐶) = (𝐴 ·𝑜 𝐶))
2120adantr 270 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 ·𝑜 𝐶) = (𝐴 ·𝑜 𝐶))
22 oveq1 5622 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
2322adantl 271 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
2421, 23opeq12d 3615 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩ = ⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩)
2519, 24sylbi 119 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩ = ⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩)
2625eceq1d 6282 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
2726eqeq2d 2096 . . . . . . 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 457 . . . . . 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 2701 . . . 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 3609 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
32 eceq1 6281 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
3332eqeq2d 2096 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
3433anbi2d 452 . . . . . . . 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 2618 . . . . . . . . . . . 12 𝑢 ∈ V
36 vex 2618 . . . . . . . . . . . 12 𝑡 ∈ V
3735, 36opth 4040 . . . . . . . . . . 11 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑢 = 𝐶𝑡 = 𝐷))
38 oveq2 5623 . . . . . . . . . . . . 13 (𝑢 = 𝐶 → (𝑤 ·𝑜 𝑢) = (𝑤 ·𝑜 𝐶))
3938adantr 270 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 ·𝑜 𝑢) = (𝑤 ·𝑜 𝐶))
40 oveq2 5623 . . . . . . . . . . . . 13 (𝑡 = 𝐷 → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
4140adantl 271 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
4239, 41opeq12d 3615 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩ = ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩)
4337, 42sylbi 119 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩ = ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩)
4443eceq1d 6282 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )
4544eqeq2d 2096 . . . . . . . 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 457 . . . . . . 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 2701 . . . . 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 1807 . . . 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 401 . . 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 423 . 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 6250 . . . 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 2091 . . . . . . . 8 (𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
55 eqeq1 2091 . . . . . . . 8 (𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
5654, 55bi2anan9 571 . . . . . . 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 2091 . . . . . . 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 571 . . . . . 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 1136 . . . . 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 1795 . . . 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 6954 . . . 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 6935 . . . 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 5725 . . 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 1260 . 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 61 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 102  wb 103  w3a 922   = wceq 1287  wex 1424  wcel 1436  Vcvv 2615  cop 3434  ωcom 4380   × cxp 4411  (class class class)co 5615   ·𝑜 comu 6135  [cec 6244   / cqs 6245  Ncnpi 6778   ~Q0 ceq0 6792   ·Q0 cmq0 6796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-mi 6812  df-enq0 6930  df-nq0 6931  df-mq0 6934
This theorem is referenced by:  mulclnq0  6958  nqnq0m  6961  nq0m0r  6962  distrnq0  6965  mulcomnq0  6966  nq02m  6971
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