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Theorem addnq0mo 6429
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

Proof of Theorem addnq0mo
Dummy variables f g 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enq0er 6417 . . . . . . . . . . . . . 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 6428 . . . . . . . . . . . . . 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 6425 . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . 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 490 . . . . . . . . . . . 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 492 . . . . . . . . . . . 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 2079 . . . . . . . . . . 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 1774 . . . . . . . . 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 1774 . . . . . . . 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 1774 . . . . . 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 1774 . . . . 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 1752 . . 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 3542 . . . . . . . . . . 11 ((w = 𝑠 v = f) → ⟨w, v⟩ = ⟨𝑠, f⟩)
2019eceq1d 6078 . . . . . . . . . 10 ((w = 𝑠 v = f) → [⟨w, v⟩] ~Q0 = [⟨𝑠, f⟩] ~Q0 )
2120eqeq2d 2048 . . . . . . . . 9 ((w = 𝑠 v = f) → (A = [⟨w, v⟩] ~Q0A = [⟨𝑠, f⟩] ~Q0 ))
2221anbi1d 438 . . . . . . . 8 ((w = 𝑠 v = f) → ((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) ↔ (A = [⟨𝑠, f⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 )))
23 simpl 102 . . . . . . . . . . . . 13 ((w = 𝑠 v = f) → w = 𝑠)
2423oveq1d 5470 . . . . . . . . . . . 12 ((w = 𝑠 v = f) → (w ·𝑜 𝑡) = (𝑠 ·𝑜 𝑡))
25 simpr 103 . . . . . . . . . . . . 13 ((w = 𝑠 v = f) → v = f)
2625oveq1d 5470 . . . . . . . . . . . 12 ((w = 𝑠 v = f) → (v ·𝑜 u) = (f ·𝑜 u))
2724, 26oveq12d 5473 . . . . . . . . . . 11 ((w = 𝑠 v = f) → ((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)) = ((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)))
2825oveq1d 5470 . . . . . . . . . . 11 ((w = 𝑠 v = f) → (v ·𝑜 𝑡) = (f ·𝑜 𝑡))
2927, 28opeq12d 3548 . . . . . . . . . 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 6078 . . . . . . . . 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 2048 . . . . . . . 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 442 . . . . . . 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 3542 . . . . . . . . . . 11 ((u = g 𝑡 = ) → ⟨u, 𝑡⟩ = ⟨g, ⟩)
3433eceq1d 6078 . . . . . . . . . 10 ((u = g 𝑡 = ) → [⟨u, 𝑡⟩] ~Q0 = [⟨g, ⟩] ~Q0 )
3534eqeq2d 2048 . . . . . . . . 9 ((u = g 𝑡 = ) → (B = [⟨u, 𝑡⟩] ~Q0B = [⟨g, ⟩] ~Q0 ))
3635anbi2d 437 . . . . . . . 8 ((u = g 𝑡 = ) → ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) ↔ (A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 )))
37 simpr 103 . . . . . . . . . . . . 13 ((u = g 𝑡 = ) → 𝑡 = )
3837oveq2d 5471 . . . . . . . . . . . 12 ((u = g 𝑡 = ) → (𝑠 ·𝑜 𝑡) = (𝑠 ·𝑜 ))
39 simpl 102 . . . . . . . . . . . . 13 ((u = g 𝑡 = ) → u = g)
4039oveq2d 5471 . . . . . . . . . . . 12 ((u = g 𝑡 = ) → (f ·𝑜 u) = (f ·𝑜 g))
4138, 40oveq12d 5473 . . . . . . . . . . 11 ((u = g 𝑡 = ) → ((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)) = ((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)))
4237oveq2d 5471 . . . . . . . . . . 11 ((u = g 𝑡 = ) → (f ·𝑜 𝑡) = (f ·𝑜 ))
4341, 42opeq12d 3548 . . . . . . . . . 10 ((u = g 𝑡 = ) → ⟨((𝑠 ·𝑜 𝑡) +𝑜 (f ·𝑜 u)), (f ·𝑜 𝑡)⟩ = ⟨((𝑠 ·𝑜 ) +𝑜 (f ·𝑜 g)), (f ·𝑜 )⟩)
4443eceq1d 6078 . . . . . . . . 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 2048 . . . . . . . 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 442 . . . . . . 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 1802 . . . . . 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 430 . . . . 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 1357 . . 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 2043 . . . . 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 437 . . . 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 1747 . . 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 1958 . 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 1240   = wceq 1242  wex 1378   wcel 1390  ∃*wmo 1898  cop 3370   class class class wbr 3755  𝜔com 4256   × cxp 4286  (class class class)co 5455   +𝑜 coa 5937   ·𝑜 comu 5938   Er wer 6039  [cec 6040   / cqs 6041  Ncnpi 6256   ~Q0 ceq0 6270
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
This theorem is referenced by:  addnnnq0  6431
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