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Theorem genpelvl 6360
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 6357 . . . . . 6 ((A P B P) → (A𝐹B) = ⟨{f Qg (1stA) (1stB)f = (g𝐺)}, {f Qg (2ndA) (2ndB)f = (g𝐺)}⟩)
43fveq2d 5103 . . . . 5 ((A P B P) → (1st ‘(A𝐹B)) = (1st ‘⟨{f Qg (1stA) (1stB)f = (g𝐺)}, {f Qg (2ndA) (2ndB)f = (g𝐺)}⟩))
5 nqex 6216 . . . . . . 7 Q V
65rabex 3871 . . . . . 6 {f Qg (1stA) (1stB)f = (g𝐺)} V
75rabex 3871 . . . . . 6 {f Qg (2ndA) (2ndB)f = (g𝐺)} V
86, 7op1st 5692 . . . . 5 (1st ‘⟨{f Qg (1stA) (1stB)f = (g𝐺)}, {f Qg (2ndA) (2ndB)f = (g𝐺)}⟩) = {f Qg (1stA) (1stB)f = (g𝐺)}
94, 8syl6eq 2066 . . . 4 ((A P B P) → (1st ‘(A𝐹B)) = {f Qg (1stA) (1stB)f = (g𝐺)})
109eleq2d 2085 . . 3 ((A P B P) → (𝐶 (1st ‘(A𝐹B)) ↔ 𝐶 {f Qg (1stA) (1stB)f = (g𝐺)}))
11 elrabi 2668 . . 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 6323 . . . . . . 7 (A P → ⟨(1stA), (2ndA)⟩ P)
14 elprnql 6329 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P g (1stA)) → g Q)
1513, 14sylan 267 . . . . . 6 ((A P g (1stA)) → g Q)
16 prop 6323 . . . . . . 7 (B P → ⟨(1stB), (2ndB)⟩ P)
17 elprnql 6329 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P (1stB)) → Q)
1816, 17sylan 267 . . . . . 6 ((B P (1stB)) → Q)
192caovcl 5574 . . . . . 6 ((g Q Q) → (g𝐺) Q)
2015, 18, 19syl2an 273 . . . . 5 (((A P g (1stA)) (B P (1stB))) → (g𝐺) Q)
2120an4s 509 . . . 4 (((A P B P) (g (1stA) (1stB))) → (g𝐺) Q)
22 eleq1 2078 . . . 4 (𝐶 = (g𝐺) → (𝐶 Q ↔ (g𝐺) Q))
2321, 22syl5ibrcom 146 . . 3 (((A P B P) (g (1stA) (1stB))) → (𝐶 = (g𝐺) → 𝐶 Q))
2423rexlimdvva 2414 . 2 ((A P B P) → (g (1stA) (1stB)𝐶 = (g𝐺) → 𝐶 Q))
25 eqeq1 2024 . . . . . 6 (f = 𝐶 → (f = (g𝐺) ↔ 𝐶 = (g𝐺)))
26252rexbidv 2323 . . . . 5 (f = 𝐶 → (g (1stA) (1stB)f = (g𝐺) ↔ g (1stA) (1stB)𝐶 = (g𝐺)))
2726elrab3 2672 . . . 4 (𝐶 Q → (𝐶 {f Qg (1stA) (1stB)f = (g𝐺)} ↔ g (1stA) (1stB)𝐶 = (g𝐺)))
2810, 27sylan9bb 438 . . 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 608 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 871   = wceq 1226   wcel 1370  wrex 2281  {crab 2284  cop 3349  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-1st 5686  df-2nd 5687  df-qs 6019  df-ni 6158  df-nqqs 6201  df-inp 6314
This theorem is referenced by:  genpprecll  6362  genpcdl  6368  genprndl  6370  genpdisj  6372  genpassl  6373  distrlem1prl  6415  distrlem5prl  6419  1idprl  6423  ltexprlemfl  6440  recexprlem1ssl  6461  recexprlemss1l  6463
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