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Theorem genpprecll 6497
 Description: Pre-closure law for general operation on lower cuts. (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
genpprecll ((A P B P) → ((𝐶 (1stA) 𝐷 (1stB)) → (𝐶𝐺𝐷) (1st ‘(A𝐹B))))
Distinct variable groups:   x,y,z,w,v,A   x,B,y,z,w,v   x,𝐺,y,z,w,v
Allowed substitution hints:   𝐶(x,y,z,w,v)   𝐷(x,y,z,w,v)   𝐹(x,y,z,w,v)

Proof of Theorem genpprecll
Dummy variables g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . 3 (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2 rspceov 5489 . . 3 ((𝐶 (1stA) 𝐷 (1stB) (𝐶𝐺𝐷) = (𝐶𝐺𝐷)) → g (1stA) (1stB)(𝐶𝐺𝐷) = (g𝐺))
31, 2mp3an3 1220 . 2 ((𝐶 (1stA) 𝐷 (1stB)) → g (1stA) (1stB)(𝐶𝐺𝐷) = (g𝐺))
4 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))}⟩)
5 genpelvl.2 . . 3 ((y Q z Q) → (y𝐺z) Q)
64, 5genpelvl 6495 . 2 ((A P B P) → ((𝐶𝐺𝐷) (1st ‘(A𝐹B)) ↔ g (1stA) (1stB)(𝐶𝐺𝐷) = (g𝐺)))
73, 6syl5ibr 145 1 ((A P B P) → ((𝐶 (1stA) 𝐷 (1stB)) → (𝐶𝐺𝐷) (1st ‘(A𝐹B))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ 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 6449 This theorem is referenced by:  genpml  6500  genprndl  6504  addnqprl  6512  mulnqprl  6549  distrlem1prl  6558  distrlem4prl  6560  ltaddpr  6571  ltexprlemrl  6584  addcanprleml  6588  addcanprlemu  6589
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