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Theorem genpelpw 6365
Description: Result of general operation on positive reals is an ordered pair of sets of positive fractions. (Contributed by Jim Kingdon, 4-Oct-2019.)
Hypotheses
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))}⟩)
genpelvl.2 ((y Q z Q) → (y𝐺z) Q)
Assertion
Ref Expression
genpelpw ((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 genpelpw
StepHypRef Expression
1 fveq2 5099 . . . . . . . 8 (w = A → (1stw) = (1stA))
21eleq2d 2085 . . . . . . 7 (w = A → (y (1stw) ↔ y (1stA)))
323anbi1d 1194 . . . . . 6 (w = A → ((y (1stw) z (1stv) x = (y𝐺z)) ↔ (y (1stA) z (1stv) x = (y𝐺z))))
432rexbidv 2323 . . . . 5 (w = A → (y Q z Q (y (1stw) z (1stv) x = (y𝐺z)) ↔ y Q z Q (y (1stA) z (1stv) x = (y𝐺z))))
54rabbidv 2523 . . . 4 (w = A → {x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))} = {x Qy Q z Q (y (1stA) z (1stv) x = (y𝐺z))})
6 fveq2 5099 . . . . . . . 8 (w = A → (2ndw) = (2ndA))
76eleq2d 2085 . . . . . . 7 (w = A → (y (2ndw) ↔ y (2ndA)))
873anbi1d 1194 . . . . . 6 (w = A → ((y (2ndw) z (2ndv) x = (y𝐺z)) ↔ (y (2ndA) z (2ndv) x = (y𝐺z))))
982rexbidv 2323 . . . . 5 (w = A → (y Q z Q (y (2ndw) z (2ndv) x = (y𝐺z)) ↔ y Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))))
109rabbidv 2523 . . . 4 (w = A → {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))} = {x Qy Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))})
115, 10opeq12d 3527 . . 3 (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))}⟩)
12 fveq2 5099 . . . . . . . 8 (v = B → (1stv) = (1stB))
1312eleq2d 2085 . . . . . . 7 (v = B → (z (1stv) ↔ z (1stB)))
14133anbi2d 1195 . . . . . 6 (v = B → ((y (1stA) z (1stv) x = (y𝐺z)) ↔ (y (1stA) z (1stB) x = (y𝐺z))))
15142rexbidv 2323 . . . . 5 (v = B → (y Q z Q (y (1stA) z (1stv) x = (y𝐺z)) ↔ y Q z Q (y (1stA) z (1stB) x = (y𝐺z))))
1615rabbidv 2523 . . . 4 (v = B → {x Qy Q z Q (y (1stA) z (1stv) x = (y𝐺z))} = {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))})
17 fveq2 5099 . . . . . . . 8 (v = B → (2ndv) = (2ndB))
1817eleq2d 2085 . . . . . . 7 (v = B → (z (2ndv) ↔ z (2ndB)))
19183anbi2d 1195 . . . . . 6 (v = B → ((y (2ndA) z (2ndv) x = (y𝐺z)) ↔ (y (2ndA) z (2ndB) x = (y𝐺z))))
20192rexbidv 2323 . . . . 5 (v = B → (y Q z Q (y (2ndA) z (2ndv) x = (y𝐺z)) ↔ y Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))))
2120rabbidv 2523 . . . 4 (v = B → {x Qy Q z Q (y (2ndA) z (2ndv) x = (y𝐺z))} = {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))})
2216, 21opeq12d 3527 . . 3 (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))}⟩)
23 genpelvl.1 . . 3 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))}⟩)
24 nqex 6216 . . . . 5 Q V
2524rabex 3871 . . . 4 {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} V
2624rabex 3871 . . . 4 {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} V
2725, 26opex 3936 . . 3 ⟨{x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))}, {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))}⟩ V
2811, 22, 23, 27ovmpt2 5555 . 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))}⟩)
29 ssrab2 2998 . . . 4 {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} ⊆ Q
3024elpw2 3881 . . . 4 ({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)
3129, 30mpbir 134 . . 3 {x Qy Q z Q (y (1stA) z (1stB) x = (y𝐺z))} 𝒫 Q
32 ssrab2 2998 . . . 4 {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} ⊆ Q
3324elpw2 3881 . . . 4 ({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)
3432, 33mpbir 134 . . 3 {x Qy Q z Q (y (2ndA) z (2ndB) x = (y𝐺z))} 𝒫 Q
35 opelxpi 4299 . . 3 (({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))
3631, 34, 35mp2an 404 . 2 ⟨{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)
3728, 36syl6eqel 2106 1 ((A P B P) → (A𝐹B) (𝒫 Q × 𝒫 Q))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 871   = wceq 1226   wcel 1370  wrex 2281  {crab 2284  wss 2890  𝒫 cpw 3330  cop 3349   × cxp 4266  cfv 4825  (class class class)co 5432  cmpt2 5434  1st c1st 5684  2nd c2nd 5685  Qcnq 6134  Pcnp 6145
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-qs 6019  df-ni 6158  df-nqqs 6201
This theorem is referenced by: (None)
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