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Theorem addnnnq0 6304
Description: Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
Assertion
Ref Expression
addnnnq0 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨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 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 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0
13 opeq12 3525 . . . . . . . . 9 ((w = A v = B) → ⟨w, v⟩ = ⟨A, B⟩)
1413eceq1d 6053 . . . . . . . 8 ((w = A v = B) → [⟨w, v⟩] ~Q0 = [⟨A, B⟩] ~Q0 )
1514eqeq2d 2033 . . . . . . 7 ((w = A v = B) → ([⟨A, B⟩] ~Q0 = [⟨w, v⟩] ~Q0 ↔ [⟨A, B⟩] ~Q0 = [⟨A, B⟩] ~Q0 ))
1615anbi1d 441 . . . . . 6 ((w = A v = 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 ax-ia1 99 . . . . . . . . . . 11 ((w = A v = B) → w = A)
1817oveq1d 5451 . . . . . . . . . 10 ((w = A v = B) → (w ·𝑜 𝐷) = (A ·𝑜 𝐷))
19 ax-ia2 100 . . . . . . . . . . 11 ((w = A v = B) → v = B)
2019oveq1d 5451 . . . . . . . . . 10 ((w = A v = B) → (v ·𝑜 𝐶) = (B ·𝑜 𝐶))
2118, 20oveq12d 5454 . . . . . . . . 9 ((w = A v = B) → ((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)) = ((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)))
2219oveq1d 5451 . . . . . . . . 9 ((w = A v = B) → (v ·𝑜 𝐷) = (B ·𝑜 𝐷))
2321, 22opeq12d 3531 . . . . . . . 8 ((w = A v = B) → ⟨((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)), (v ·𝑜 𝐷)⟩ = ⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩)
2423eceq1d 6053 . . . . . . 7 ((w = A v = B) → [⟨((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)), (v ·𝑜 𝐷)⟩] ~Q0 = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 )
2524eqeq2d 2033 . . . . . 6 ((w = A v = 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 445 . . . . 5 ((w = A v = 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 2619 . . . 4 ((A 𝜔 B N) → ((([⟨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 3525 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → ⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
2928eceq1d 6053 . . . . . . . . 9 ((u = 𝐶 𝑡 = 𝐷) → [⟨u, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
3029eqeq2d 2033 . . . . . . . 8 ((u = 𝐶 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨u, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
3130anbi2d 440 . . . . . . 7 ((u = 𝐶 𝑡 = 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 ax-ia2 100 . . . . . . . . . . . 12 ((u = 𝐶 𝑡 = 𝐷) → 𝑡 = 𝐷)
3332oveq2d 5452 . . . . . . . . . . 11 ((u = 𝐶 𝑡 = 𝐷) → (w ·𝑜 𝑡) = (w ·𝑜 𝐷))
34 ax-ia1 99 . . . . . . . . . . . 12 ((u = 𝐶 𝑡 = 𝐷) → u = 𝐶)
3534oveq2d 5452 . . . . . . . . . . 11 ((u = 𝐶 𝑡 = 𝐷) → (v ·𝑜 u) = (v ·𝑜 𝐶))
3633, 35oveq12d 5454 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → ((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)) = ((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)))
3732oveq2d 5452 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → (v ·𝑜 𝑡) = (v ·𝑜 𝐷))
3836, 37opeq12d 3531 . . . . . . . . 9 ((u = 𝐶 𝑡 = 𝐷) → ⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩ = ⟨((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)), (v ·𝑜 𝐷)⟩)
3938eceq1d 6053 . . . . . . . 8 ((u = 𝐶 𝑡 = 𝐷) → [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 = [⟨((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)), (v ·𝑜 𝐷)⟩] ~Q0 )
4039eqeq2d 2033 . . . . . . 7 ((u = 𝐶 𝑡 = 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 445 . . . . . 6 ((u = 𝐶 𝑡 = 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 2619 . . . . 5 ((𝐶 𝜔 𝐷 N) → ((([⟨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 1744 . . . 4 ((𝐶 𝜔 𝐷 N) → (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 391 . . 3 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ((([⟨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 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 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))
46 ecexg 6021 . . . 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 892 . . . . . . . 8 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → x = [⟨A, B⟩] ~Q0 )
4948eqeq1d 2030 . . . . . . 7 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → (x = [⟨w, v⟩] ~Q0 ↔ [⟨A, B⟩] ~Q0 = [⟨w, v⟩] ~Q0 ))
50 simp2 893 . . . . . . . 8 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → y = [⟨𝐶, 𝐷⟩] ~Q0 )
5150eqeq1d 2030 . . . . . . 7 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → (y = [⟨u, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨u, 𝑡⟩] ~Q0 ))
5249, 51anbi12d 445 . . . . . 6 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → ((x = [⟨w, v⟩] ~Q0 y = [⟨u, 𝑡⟩] ~Q0 ) ↔ ([⟨A, B⟩] ~Q0 = [⟨w, v⟩] ~Q0 [⟨𝐶, 𝐷⟩] ~Q0 = [⟨u, 𝑡⟩] ~Q0 )))
53 simp3 894 . . . . . . 7 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 )
5453eqeq1d 2030 . . . . . 6 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((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 445 . . . . 5 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → (((x = [⟨w, v⟩] ~Q0 y = [⟨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 1732 . . . 4 ((x = [⟨A, B⟩] ~Q0 y = [⟨𝐶, 𝐷⟩] ~Q0 z = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 ) → (wvu𝑡((x = [⟨w, v⟩] ~Q0 y = [⟨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 6302 . . . 4 ((x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )) → ∃*zwvu𝑡((x = [⟨w, v⟩] ~Q0 y = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))
58 dfplq0qs 6285 . . . 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 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))}
5956, 57, 58ovig 5545 . . 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 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 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 56 1 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨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 97   w3a 873   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  cop 3353  𝜔com 4240   × cxp 4270  (class class class)co 5436   +𝑜 coa 5913   ·𝑜 comu 5914  [cec 6015   / cqs 6016  Ncnpi 6130   ~Q0 ceq0 6144   +Q0 cplq0 6147
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-plq0 6282
This theorem is referenced by:  addclnq0  6306  nqpnq0nq  6308  nqnq0a  6309  nq0a0  6312  nnanq0  6313  distrnq0  6314  addassnq0  6317
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