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Theorem mulnnnq0 6305
 Description: Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
Assertion
Ref Expression
mulnnnq0 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨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 4303 . . . 4 ((A 𝜔 B N) → ⟨A, B (𝜔 × N))
2 enq0ex 6294 . . . . 5 ~Q0 V
32ecelqsi 6071 . . . 4 (⟨A, B (𝜔 × N) → [⟨A, B⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
41, 3syl 14 . . 3 ((A 𝜔 B N) → [⟨A, B⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
5 opelxpi 4303 . . . 4 ((𝐶 𝜔 𝐷 N) → ⟨𝐶, 𝐷 (𝜔 × N))
62ecelqsi 6071 . . . 4 (⟨𝐶, 𝐷 (𝜔 × N) → [⟨𝐶, 𝐷⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
75, 6syl 14 . . 3 ((𝐶 𝜔 𝐷 N) → [⟨𝐶, 𝐷⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
84, 7anim12i 321 . 2 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨A, B⟩] ~Q0 ((𝜔 × N) / ~Q0 ) [⟨𝐶, 𝐷⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
9 eqid 2022 . . . 4 [⟨A, B⟩] ~Q0 = [⟨A, B⟩] ~Q0
10 eqid 2022 . . . 4 [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0
119, 10pm3.2i 257 . . 3 ([⟨A, B⟩] ~Q0 = [⟨A, B⟩] ~Q0 [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
12 eqid 2022 . . 3 [⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩] ~Q0 = [⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩] ~Q0
13 opeq12 3525 . . . . . 6 ((w = A v = B) → ⟨w, v⟩ = ⟨A, B⟩)
14 eceq1 6052 . . . . . . . . 9 (⟨w, v⟩ = ⟨A, B⟩ → [⟨w, v⟩] ~Q0 = [⟨A, B⟩] ~Q0 )
1514eqeq2d 2033 . . . . . . . 8 (⟨w, v⟩ = ⟨A, B⟩ → ([⟨A, B⟩] ~Q0 = [⟨w, v⟩] ~Q0 ↔ [⟨A, B⟩] ~Q0 = [⟨A, B⟩] ~Q0 ))
1615anbi1d 441 . . . . . . 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 2538 . . . . . . . . . . 11 w V
18 vex 2538 . . . . . . . . . . 11 v V
1917, 18opth 3948 . . . . . . . . . 10 (⟨w, v⟩ = ⟨A, B⟩ ↔ (w = A v = B))
20 oveq1 5443 . . . . . . . . . . . 12 (w = A → (w ·𝑜 𝐶) = (A ·𝑜 𝐶))
2120adantr 261 . . . . . . . . . . 11 ((w = A v = B) → (w ·𝑜 𝐶) = (A ·𝑜 𝐶))
22 oveq1 5443 . . . . . . . . . . . 12 (v = B → (v ·𝑜 𝐷) = (B ·𝑜 𝐷))
2322adantl 262 . . . . . . . . . . 11 ((w = A v = B) → (v ·𝑜 𝐷) = (B ·𝑜 𝐷))
2421, 23opeq12d 3531 . . . . . . . . . 10 ((w = A v = B) → ⟨(w ·𝑜 𝐶), (v ·𝑜 𝐷)⟩ = ⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩)
2519, 24sylbi 114 . . . . . . . . 9 (⟨w, v⟩ = ⟨A, B⟩ → ⟨(w ·𝑜 𝐶), (v ·𝑜 𝐷)⟩ = ⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩)
2625eceq1d 6053 . . . . . . . 8 (⟨w, v⟩ = ⟨A, B⟩ → [⟨(w ·𝑜 𝐶), (v ·𝑜 𝐷)⟩] ~Q0 = [⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩] ~Q0 )
2726eqeq2d 2033 . . . . . . 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 445 . . . . . 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 = A v = 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 2619 . . . 4 ((A 𝜔 B N) → ((([⟨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 3525 . . . . . . 7 ((u = 𝐶 𝑡 = 𝐷) → ⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
32 eceq1 6052 . . . . . . . . . 10 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨u, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
3332eqeq2d 2033 . . . . . . . . 9 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨u, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
3433anbi2d 440 . . . . . . . 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 2538 . . . . . . . . . . . 12 u V
36 vex 2538 . . . . . . . . . . . 12 𝑡 V
3735, 36opth 3948 . . . . . . . . . . 11 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ ↔ (u = 𝐶 𝑡 = 𝐷))
38 oveq2 5444 . . . . . . . . . . . . 13 (u = 𝐶 → (w ·𝑜 u) = (w ·𝑜 𝐶))
3938adantr 261 . . . . . . . . . . . 12 ((u = 𝐶 𝑡 = 𝐷) → (w ·𝑜 u) = (w ·𝑜 𝐶))
40 oveq2 5444 . . . . . . . . . . . . 13 (𝑡 = 𝐷 → (v ·𝑜 𝑡) = (v ·𝑜 𝐷))
4140adantl 262 . . . . . . . . . . . 12 ((u = 𝐶 𝑡 = 𝐷) → (v ·𝑜 𝑡) = (v ·𝑜 𝐷))
4239, 41opeq12d 3531 . . . . . . . . . . 11 ((u = 𝐶 𝑡 = 𝐷) → ⟨(w ·𝑜 u), (v ·𝑜 𝑡)⟩ = ⟨(w ·𝑜 𝐶), (v ·𝑜 𝐷)⟩)
4337, 42sylbi 114 . . . . . . . . . 10 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ⟨(w ·𝑜 u), (v ·𝑜 𝑡)⟩ = ⟨(w ·𝑜 𝐶), (v ·𝑜 𝐷)⟩)
4443eceq1d 6053 . . . . . . . . 9 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨(w ·𝑜 u), (v ·𝑜 𝑡)⟩] ~Q0 = [⟨(w ·𝑜 𝐶), (v ·𝑜 𝐷)⟩] ~Q0 )
4544eqeq2d 2033 . . . . . . . 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 445 . . . . . . 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 = 𝐶 𝑡 = 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 2619 . . . . 5 ((𝐶 𝜔 𝐷 N) → ((([⟨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 1744 . . . 4 ((𝐶 𝜔 𝐷 N) → (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 391 . . 3 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ((([⟨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 410 . 2 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → 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 6021 . . . 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 2028 . . . . . . . 8 (x = [⟨A, B⟩] ~Q0 → (x = [⟨w, v⟩] ~Q0 ↔ [⟨A, B⟩] ~Q0 = [⟨w, v⟩] ~Q0 ))
55 eqeq1 2028 . . . . . . . 8 (y = [⟨𝐶, 𝐷⟩] ~Q0 → (y = [⟨u, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨u, 𝑡⟩] ~Q0 ))
5654, 55bi2anan9 526 . . . . . . 7 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 ) → ((x = [⟨w, v⟩] ~Q0 y = [⟨u, 𝑡⟩] ~Q0 ) ↔ ([⟨A, B⟩] ~Q0 = [⟨w, v⟩] ~Q0 [⟨𝐶, 𝐷⟩] ~Q0 = [⟨u, 𝑡⟩] ~Q0 )))
57 eqeq1 2028 . . . . . . 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 526 . . . . . 6 (((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 ) z = [⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩] ~Q0 ) → (((x = [⟨w, v⟩] ~Q0 y = [⟨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 1085 . . . . 5 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩] ~Q0 ) → (((x = [⟨w, v⟩] ~Q0 y = [⟨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 1732 . . . 4 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩] ~Q0 ) → (wvu𝑡((x = [⟨w, v⟩] ~Q0 y = [⟨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 6303 . . . 4 ((x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )) → ∃*zwvu𝑡((x = [⟨w, v⟩] ~Q0 y = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 𝑡)⟩] ~Q0 ))
62 dfmq0qs 6284 . . . 4 ·Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )) wvu𝑡((x = [⟨w, v⟩] ~Q0 y = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 𝑡)⟩] ~Q0 ))}
6360, 61, 62ovig 5545 . . 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 1206 . 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 56 1 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨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 97   ↔ wb 98   ∧ w3a 873   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535  ⟨cop 3353  𝜔com 4240   × cxp 4270  (class class class)co 5436   ·𝑜 comu 5914  [cec 6015   / cqs 6016  Ncnpi 6130   ~Q0 ceq0 6144   ·Q0 cmq0 6148 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238 This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921  df-er 6017  df-ec 6019  df-qs 6023  df-ni 6164  df-mi 6166  df-enq0 6279  df-nq0 6280  df-mq0 6283 This theorem is referenced by:  mulclnq0  6307  nqnq0m  6310  nq0m0r  6311  distrnq0  6314  mulcomnq0  6315  nq02m  6319
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