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Theorem addnnnq0 6431
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 4319 . . . 4 ((A 𝜔 B N) → ⟨A, B (𝜔 × N))
2 enq0ex 6421 . . . . 5 ~Q0 V
32ecelqsi 6096 . . . 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 4319 . . . 4 ((𝐶 𝜔 𝐷 N) → ⟨𝐶, 𝐷 (𝜔 × N))
62ecelqsi 6096 . . . 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 2037 . . . 4 [⟨A, B⟩] ~Q0 = [⟨A, B⟩] ~Q0
10 eqid 2037 . . . 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 2037 . . 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 3542 . . . . . . . . 9 ((w = A v = B) → ⟨w, v⟩ = ⟨A, B⟩)
1413eceq1d 6078 . . . . . . . 8 ((w = A v = B) → [⟨w, v⟩] ~Q0 = [⟨A, B⟩] ~Q0 )
1514eqeq2d 2048 . . . . . . 7 ((w = A v = B) → ([⟨A, B⟩] ~Q0 = [⟨w, v⟩] ~Q0 ↔ [⟨A, B⟩] ~Q0 = [⟨A, B⟩] ~Q0 ))
1615anbi1d 438 . . . . . 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 simpl 102 . . . . . . . . . . 11 ((w = A v = B) → w = A)
1817oveq1d 5470 . . . . . . . . . 10 ((w = A v = B) → (w ·𝑜 𝐷) = (A ·𝑜 𝐷))
19 simpr 103 . . . . . . . . . . 11 ((w = A v = B) → v = B)
2019oveq1d 5470 . . . . . . . . . 10 ((w = A v = B) → (v ·𝑜 𝐶) = (B ·𝑜 𝐶))
2118, 20oveq12d 5473 . . . . . . . . 9 ((w = A v = B) → ((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)) = ((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)))
2219oveq1d 5470 . . . . . . . . 9 ((w = A v = B) → (v ·𝑜 𝐷) = (B ·𝑜 𝐷))
2321, 22opeq12d 3548 . . . . . . . 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 6078 . . . . . . 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 2048 . . . . . 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 442 . . . . 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 2636 . . . 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 3542 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → ⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
2928eceq1d 6078 . . . . . . . . 9 ((u = 𝐶 𝑡 = 𝐷) → [⟨u, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
3029eqeq2d 2048 . . . . . . . 8 ((u = 𝐶 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨u, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
3130anbi2d 437 . . . . . . 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 simpr 103 . . . . . . . . . . . 12 ((u = 𝐶 𝑡 = 𝐷) → 𝑡 = 𝐷)
3332oveq2d 5471 . . . . . . . . . . 11 ((u = 𝐶 𝑡 = 𝐷) → (w ·𝑜 𝑡) = (w ·𝑜 𝐷))
34 simpl 102 . . . . . . . . . . . 12 ((u = 𝐶 𝑡 = 𝐷) → u = 𝐶)
3534oveq2d 5471 . . . . . . . . . . 11 ((u = 𝐶 𝑡 = 𝐷) → (v ·𝑜 u) = (v ·𝑜 𝐶))
3633, 35oveq12d 5473 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → ((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)) = ((w ·𝑜 𝐷) +𝑜 (v ·𝑜 𝐶)))
3732oveq2d 5471 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → (v ·𝑜 𝑡) = (v ·𝑜 𝐷))
3836, 37opeq12d 3548 . . . . . . . . 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 6078 . . . . . . . 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 2048 . . . . . . 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 442 . . . . . 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 2636 . . . . 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 1759 . . . 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 389 . . 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 408 . 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 6046 . . . 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 903 . . . . . . . 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 2045 . . . . . . 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 904 . . . . . . . 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 2045 . . . . . . 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 442 . . . . . 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 905 . . . . . . 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 2045 . . . . . 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 442 . . . . 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 1747 . . . 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 6429 . . . 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 6412 . . . 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 5564 . . 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 1220 . 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 884   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370  𝜔com 4256   × cxp 4286  (class class class)co 5455   +𝑜 coa 5937   ·𝑜 comu 5938  [cec 6040   / cqs 6041  Ncnpi 6256   ~Q0 ceq0 6270   +Q0 cplq0 6273
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-enq0 6406  df-nq0 6407  df-plq0 6409
This theorem is referenced by:  addclnq0  6433  nqpnq0nq  6435  nqnq0a  6436  nq0a0  6439  nnanq0  6440  distrnq0  6441  addassnq0  6444
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