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Theorem genpelvl 6494
 Description: Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-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
genpelvl ((A P B P) → (𝐶 (1st ‘(A𝐹B)) ↔ g (1stA) (1stB)𝐶 = (g𝐺)))
Distinct variable groups:   x,y,z,g,,w,v,A   x,B,y,z,g,,w,v   x,𝐺,y,z,g,,w,v   g,𝐹   𝐶,g,
Allowed substitution hints:   𝐶(x,y,z,w,v)   𝐹(x,y,z,w,v,)

Proof of Theorem genpelvl
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . 7 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))}⟩)
2 genpelvl.2 . . . . . . 7 ((y Q z Q) → (y𝐺z) Q)
31, 2genipv 6491 . . . . . 6 ((A P B P) → (A𝐹B) = ⟨{f Qg (1stA) (1stB)f = (g𝐺)}, {f Qg (2ndA) (2ndB)f = (g𝐺)}⟩)
43fveq2d 5125 . . . . 5 ((A P B P) → (1st ‘(A𝐹B)) = (1st ‘⟨{f Qg (1stA) (1stB)f = (g𝐺)}, {f Qg (2ndA) (2ndB)f = (g𝐺)}⟩))
5 nqex 6347 . . . . . . 7 Q V
65rabex 3892 . . . . . 6 {f Qg (1stA) (1stB)f = (g𝐺)} V
75rabex 3892 . . . . . 6 {f Qg (2ndA) (2ndB)f = (g𝐺)} V
86, 7op1st 5715 . . . . 5 (1st ‘⟨{f Qg (1stA) (1stB)f = (g𝐺)}, {f Qg (2ndA) (2ndB)f = (g𝐺)}⟩) = {f Qg (1stA) (1stB)f = (g𝐺)}
94, 8syl6eq 2085 . . . 4 ((A P B P) → (1st ‘(A𝐹B)) = {f Qg (1stA) (1stB)f = (g𝐺)})
109eleq2d 2104 . . 3 ((A P B P) → (𝐶 (1st ‘(A𝐹B)) ↔ 𝐶 {f Qg (1stA) (1stB)f = (g𝐺)}))
11 elrabi 2689 . . 3 (𝐶 {f Qg (1stA) (1stB)f = (g𝐺)} → 𝐶 Q)
1210, 11syl6bi 152 . 2 ((A P B P) → (𝐶 (1st ‘(A𝐹B)) → 𝐶 Q))
13 prop 6457 . . . . . . 7 (A P → ⟨(1stA), (2ndA)⟩ P)
14 elprnql 6463 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P g (1stA)) → g Q)
1513, 14sylan 267 . . . . . 6 ((A P g (1stA)) → g Q)
16 prop 6457 . . . . . . 7 (B P → ⟨(1stB), (2ndB)⟩ P)
17 elprnql 6463 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P (1stB)) → Q)
1816, 17sylan 267 . . . . . 6 ((B P (1stB)) → Q)
192caovcl 5597 . . . . . 6 ((g Q Q) → (g𝐺) Q)
2015, 18, 19syl2an 273 . . . . 5 (((A P g (1stA)) (B P (1stB))) → (g𝐺) Q)
2120an4s 522 . . . 4 (((A P B P) (g (1stA) (1stB))) → (g𝐺) Q)
22 eleq1 2097 . . . 4 (𝐶 = (g𝐺) → (𝐶 Q ↔ (g𝐺) Q))
2321, 22syl5ibrcom 146 . . 3 (((A P B P) (g (1stA) (1stB))) → (𝐶 = (g𝐺) → 𝐶 Q))
2423rexlimdvva 2434 . 2 ((A P B P) → (g (1stA) (1stB)𝐶 = (g𝐺) → 𝐶 Q))
25 eqeq1 2043 . . . . . 6 (f = 𝐶 → (f = (g𝐺) ↔ 𝐶 = (g𝐺)))
26252rexbidv 2343 . . . . 5 (f = 𝐶 → (g (1stA) (1stB)f = (g𝐺) ↔ g (1stA) (1stB)𝐶 = (g𝐺)))
2726elrab3 2693 . . . 4 (𝐶 Q → (𝐶 {f Qg (1stA) (1stB)f = (g𝐺)} ↔ g (1stA) (1stB)𝐶 = (g𝐺)))
2810, 27sylan9bb 435 . . 3 (((A P B P) 𝐶 Q) → (𝐶 (1st ‘(A𝐹B)) ↔ g (1stA) (1stB)𝐶 = (g𝐺)))
2928ex 108 . 2 ((A P B P) → (𝐶 Q → (𝐶 (1st ‘(A𝐹B)) ↔ g (1stA) (1stB)𝐶 = (g𝐺))))
3012, 24, 29pm5.21ndd 620 1 ((A P B P) → (𝐶 (1st ‘(A𝐹B)) ↔ g (1stA) (1stB)𝐶 = (g𝐺)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  {crab 2304  ⟨cop 3370  ‘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-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6448 This theorem is referenced by:  genpprecll  6496  genpcdl  6501  genprndl  6503  genpdisj  6505  genpassl  6506  addnqprlemrl  6537  distrlem1prl  6557  distrlem5prl  6561  1idprl  6565  ltexprlemfl  6582  recexprlem1ssl  6604  recexprlemss1l  6606  cauappcvgprlemladdfl  6626
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