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Theorem genpelxp 6493
 Description: Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
Hypothesis
Ref Expression
genpelvl.1 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
Assertion
Ref Expression
genpelxp ((A P B P) → (A𝐹B) (𝒫 Q × 𝒫 Q))
Distinct variable groups:   x,y,z,w,v,A   x,B,y,z,w,v   x,𝐺,y,z,w,v
Allowed substitution hints:   𝐹(x,y,z,w,v)

Proof of Theorem genpelxp
StepHypRef Expression
1 ssrab2 3019 . . . . 5 {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} ⊆ Q
2 nqex 6347 . . . . . 6 Q V
32elpw2 3902 . . . . 5 ({x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} 𝒫 Q ↔ {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} ⊆ Q)
41, 3mpbir 134 . . . 4 {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} 𝒫 Q
5 ssrab2 3019 . . . . 5 {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} ⊆ Q
62elpw2 3902 . . . . 5 ({x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} 𝒫 Q ↔ {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} ⊆ Q)
75, 6mpbir 134 . . . 4 {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} 𝒫 Q
8 opelxpi 4319 . . . 4 (({x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} 𝒫 Q {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} 𝒫 Q) → ⟨{x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))}⟩ (𝒫 Q × 𝒫 Q))
94, 7, 8mp2an 402 . . 3 ⟨{x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))}⟩ (𝒫 Q × 𝒫 Q)
10 fveq2 5121 . . . . . . . . 9 (w = A → (1stw) = (1stA))
1110eleq2d 2104 . . . . . . . 8 (w = A → (y (1stw) ↔ y (1stA)))
12113anbi1d 1210 . . . . . . 7 (w = A → ((y (1stw) z (1stv) x = (y𝐺z)) ↔ (y (1stA) z (1stv) x = (y𝐺z))))
13122rexbidv 2343 . . . . . 6 (w = A → (y Q z Q (y (1stw) z (1stv) x = (y𝐺z)) ↔ y Q z Q (y (1stA) z (1stv) x = (y𝐺z))))
1413rabbidv 2543 . . . . 5 (w = A → {x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))} = {x Qy Q z Q (y (1stA) z (1stv) x = (y𝐺z))})
15 fveq2 5121 . . . . . . . . 9 (w = A → (2ndw) = (2ndA))
1615eleq2d 2104 . . . . . . . 8 (w = A → (y (2ndw) ↔ y (2ndA)))
17163anbi1d 1210 . . . . . . 7 (w = A → ((y (2ndw) z (2ndv) x = (y𝐺z)) ↔ (y (2ndA) z (2ndv) x = (y𝐺z))))
18172rexbidv 2343 . . . . . 6 (w = A → (y Q z Q (y (2ndw) z (2ndv) x = (y𝐺z)) ↔ y Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))))
1918rabbidv 2543 . . . . 5 (w = A → {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))} = {x Qy Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))})
2014, 19opeq12d 3548 . . . 4 (w = A → ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩ = ⟨{x Qy Q z Q (y (1stA) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))}⟩)
21 fveq2 5121 . . . . . . . . 9 (v = B → (1stv) = (1stB))
2221eleq2d 2104 . . . . . . . 8 (v = B → (z (1stv) ↔ z (1stB)))
23223anbi2d 1211 . . . . . . 7 (v = B → ((y (1stA) z (1stv) x = (y𝐺z)) ↔ (y (1stA) z (1stB) x = (y𝐺z))))
24232rexbidv 2343 . . . . . 6 (v = B → (y Q z Q (y (1stA) z (1stv) x = (y𝐺z)) ↔ y Q z Q (y (1stA) z (1stB) x = (y𝐺z))))
2524rabbidv 2543 . . . . 5 (v = B → {x Qy Q z Q (y (1stA) z (1stv) x = (y𝐺z))} = {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))})
26 fveq2 5121 . . . . . . . . 9 (v = B → (2ndv) = (2ndB))
2726eleq2d 2104 . . . . . . . 8 (v = B → (z (2ndv) ↔ z (2ndB)))
28273anbi2d 1211 . . . . . . 7 (v = B → ((y (2ndA) z (2ndv) x = (y𝐺z)) ↔ (y (2ndA) z (2ndB) x = (y𝐺z))))
29282rexbidv 2343 . . . . . 6 (v = B → (y Q z Q (y (2ndA) z (2ndv) x = (y𝐺z)) ↔ y Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))))
3029rabbidv 2543 . . . . 5 (v = B → {x Qy Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))} = {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))})
3125, 30opeq12d 3548 . . . 4 (v = B → ⟨{x Qy Q z Q (y (1stA) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))}⟩ = ⟨{x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))}⟩)
32 genpelvl.1 . . . 4 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
3320, 31, 32ovmpt2g 5577 . . 3 ((A P B P ⟨{x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))}⟩ (𝒫 Q × 𝒫 Q)) → (A𝐹B) = ⟨{x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))}⟩)
349, 33mp3an3 1220 . 2 ((A P B P) → (A𝐹B) = ⟨{x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))}⟩)
3534, 9syl6eqel 2125 1 ((A P B P) → (A𝐹B) (𝒫 Q × 𝒫 Q))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  {crab 2304   ⊆ wss 2911  𝒫 cpw 3351  ⟨cop 3370   × cxp 4286  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275 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-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  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-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-id 4021  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-qs 6048  df-ni 6288  df-nqqs 6332 This theorem is referenced by:  addclpr  6520  mulclpr  6551
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