Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  addnq0mo Structured version   GIF version

 Description: There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
addnq0mo ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → ∃*zwvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))
Distinct variable groups:   𝑡,A,u,v,w,z   𝑡,B,u,v,w,z

Dummy variables f g 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enq0er 6290 . . . . . . . . . . . . . 14 ~Q0 Er (𝜔 × N)
21a1i 9 . . . . . . . . . . . . 13 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))) → ~Q0 Er (𝜔 × N))
3 nnnq0lem1 6301 . . . . . . . . . . . . . 14 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))) → ((((w 𝜔 v N) (𝑠 𝜔 f N)) ((u 𝜔 𝑡 N) (g 𝜔 N))) ((w ·𝑜 f) = (v ·𝑜 𝑠) (u ·𝑜 ) = (𝑡 ·𝑜 g))))
4 addcmpblnq0 6298 . . . . . . . . . . . . . . 15 ((((w 𝜔 v N) (𝑠 𝜔 f N)) ((u 𝜔 𝑡 N) (g 𝜔 N))) → (((w ·𝑜 f) = (v ·𝑜 𝑠) (u ·𝑜 ) = (𝑡 ·𝑜 g)) → ⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩))
54imp 115 . . . . . . . . . . . . . 14 (((((w 𝜔 v N) (𝑠 𝜔 f N)) ((u 𝜔 𝑡 N) (g 𝜔 N))) ((w ·𝑜 f) = (v ·𝑜 𝑠) (u ·𝑜 ) = (𝑡 ·𝑜 g))) → ⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩)
63, 5syl 14 . . . . . . . . . . . . 13 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))) → ⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩)
72, 6erthi 6063 . . . . . . . . . . . 12 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))) → [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )
8 simprlr 478 . . . . . . . . . . . 12 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))) → z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )
9 simprrr 480 . . . . . . . . . . . 12 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))) → 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )
107, 8, 93eqtr4d 2064 . . . . . . . . . . 11 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))) → z = 𝑞)
1110expr 357 . . . . . . . . . 10 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) ((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) → (((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ) → z = 𝑞))
1211exlimdvv 1759 . . . . . . . . 9 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) ((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) → (g((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ) → z = 𝑞))
1312exlimdvv 1759 . . . . . . . 8 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) ((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) → (𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ) → z = 𝑞))
1413ex 108 . . . . . . 7 ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) → (𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ) → z = 𝑞)))
1514exlimdvv 1759 . . . . . 6 ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → (u𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) → (𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ) → z = 𝑞)))
1615exlimdvv 1759 . . . . 5 ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → (wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) → (𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ) → z = 𝑞)))
1716impd 242 . . . 4 ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → ((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) 𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )) → z = 𝑞))
1817alrimivv 1737 . . 3 ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → z𝑞((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) 𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )) → z = 𝑞))
19 opeq12 3525 . . . . . . . . . . 11 ((w = 𝑠 v = f) → ⟨w, v⟩ = ⟨𝑠, f⟩)
2019eceq1d 6053 . . . . . . . . . 10 ((w = 𝑠 v = f) → [⟨w, v⟩] ~Q0 = [⟨𝑠, f⟩] ~Q0 )
2120eqeq2d 2033 . . . . . . . . 9 ((w = 𝑠 v = f) → (A = [⟨w, v⟩] ~Q0A = [⟨𝑠, f⟩] ~Q0 ))
2221anbi1d 441 . . . . . . . 8 ((w = 𝑠 v = f) → ((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) ↔ (A = [⟨𝑠, f⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 )))
23 ax-ia1 99 . . . . . . . . . . . . 13 ((w = 𝑠 v = f) → w = 𝑠)
2423oveq1d 5451 . . . . . . . . . . . 12 ((w = 𝑠 v = f) → (w ·𝑜 𝑡) = (𝑠 ·𝑜 𝑡))
25 ax-ia2 100 . . . . . . . . . . . . 13 ((w = 𝑠 v = f) → v = f)
2625oveq1d 5451 . . . . . . . . . . . 12 ((w = 𝑠 v = f) → (v ·𝑜 u) = (f ·𝑜 u))
2724, 26oveq12d 5454 . . . . . . . . . . 11 ((w = 𝑠 v = f) → ((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)) = ((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)))
2825oveq1d 5451 . . . . . . . . . . 11 ((w = 𝑠 v = f) → (v ·𝑜 𝑡) = (f ·𝑜 𝑡))
2927, 28opeq12d 3531 . . . . . . . . . 10 ((w = 𝑠 v = f) → ⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩ = ⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩)
3029eceq1d 6053 . . . . . . . . 9 ((w = 𝑠 v = f) → [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩] ~Q0 )
3130eqeq2d 2033 . . . . . . . 8 ((w = 𝑠 v = f) → (𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩] ~Q0 ))
3222, 31anbi12d 445 . . . . . . 7 ((w = 𝑠 v = f) → (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩] ~Q0 )))
33 opeq12 3525 . . . . . . . . . . 11 ((u = g 𝑡 = ) → ⟨u, 𝑡⟩ = ⟨g, ⟩)
3433eceq1d 6053 . . . . . . . . . 10 ((u = g 𝑡 = ) → [⟨u, 𝑡⟩] ~Q0 = [⟨g, ⟩] ~Q0 )
3534eqeq2d 2033 . . . . . . . . 9 ((u = g 𝑡 = ) → (B = [⟨u, 𝑡⟩] ~Q0B = [⟨g, ⟩] ~Q0 ))
3635anbi2d 440 . . . . . . . 8 ((u = g 𝑡 = ) → ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) ↔ (A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 )))
37 ax-ia2 100 . . . . . . . . . . . . 13 ((u = g 𝑡 = ) → 𝑡 = )
3837oveq2d 5452 . . . . . . . . . . . 12 ((u = g 𝑡 = ) → (𝑠 ·𝑜 𝑡) = (𝑠 ·𝑜 ))
39 ax-ia1 99 . . . . . . . . . . . . 13 ((u = g 𝑡 = ) → u = g)
4039oveq2d 5452 . . . . . . . . . . . 12 ((u = g 𝑡 = ) → (f ·𝑜 u) = (f ·𝑜 g))
4138, 40oveq12d 5454 . . . . . . . . . . 11 ((u = g 𝑡 = ) → ((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)) = ((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)))
4237oveq2d 5452 . . . . . . . . . . 11 ((u = g 𝑡 = ) → (f ·𝑜 𝑡) = (f ·𝑜 ))
4341, 42opeq12d 3531 . . . . . . . . . 10 ((u = g 𝑡 = ) → ⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩ = ⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩)
4443eceq1d 6053 . . . . . . . . 9 ((u = g 𝑡 = ) → [⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )
4544eqeq2d 2033 . . . . . . . 8 ((u = g 𝑡 = ) → (𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩] ~Q0𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))
4636, 45anbi12d 445 . . . . . . 7 ((u = g 𝑡 = ) → (((A = [⟨𝑠, f⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )))
4732, 46cbvex4v 1787 . . . . . 6 (wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ↔ 𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 ))
4847anbi2i 433 . . . . 5 ((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) ↔ (wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) 𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )))
4948imbi1i 227 . . . 4 (((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) → z = 𝑞) ↔ ((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) 𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )) → z = 𝑞))
50492albii 1340 . . 3 (z𝑞((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) → z = 𝑞) ↔ z𝑞((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) 𝑠fg((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩] ~Q0 )) → z = 𝑞))
5118, 50sylibr 137 . 2 ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → z𝑞((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) → z = 𝑞))
52 eqeq1 2028 . . . . 5 (z = 𝑞 → (z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))
5352anbi2d 440 . . . 4 (z = 𝑞 → (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )))
54534exbidv 1732 . . 3 (z = 𝑞 → (wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ↔ wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )))
5554mo4 1943 . 2 (∃*zwvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) ↔ z𝑞((wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ) wvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) 𝑞 = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 )) → z = 𝑞))
5651, 55sylibr 137 1 ((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) → ∃*zwvu𝑡((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃*wmo 1883  ⟨cop 3353   class class class wbr 3738  𝜔com 4240   × cxp 4270  (class class class)co 5436   +𝑜 coa 5913   ·𝑜 comu 5914   Er wer 6014  [cec 6015   / cqs 6016  Ncnpi 6130   ~Q0 ceq0 6144 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 This theorem is referenced by:  addnnnq0  6304
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