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Theorem genpml 6500
 Description: The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-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
genpml ((A P B P) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
Distinct variable groups:   x,y,z,w,v,𝑞,A   x,B,y,z,w,v,𝑞   x,𝐺,y,z,w,v,𝑞   𝐹,𝑞
Allowed substitution hints:   𝐹(x,y,z,w,v)

Proof of Theorem genpml
Dummy variables f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6457 . . . 4 (A P → ⟨(1stA), (2ndA)⟩ P)
2 prml 6459 . . . 4 (⟨(1stA), (2ndA)⟩ Pf Q f (1stA))
3 rexex 2362 . . . 4 (f Q f (1stA) → f f (1stA))
41, 2, 33syl 17 . . 3 (A Pf f (1stA))
54adantr 261 . 2 ((A P B P) → f f (1stA))
6 prop 6457 . . . . 5 (B P → ⟨(1stB), (2ndB)⟩ P)
7 prml 6459 . . . . 5 (⟨(1stB), (2ndB)⟩ Pg Q g (1stB))
8 rexex 2362 . . . . 5 (g Q g (1stB) → g g (1stB))
96, 7, 83syl 17 . . . 4 (B Pg g (1stB))
109ad2antlr 458 . . 3 (((A P B P) f (1stA)) → g g (1stB))
11 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))}⟩)
12 genpelvl.2 . . . . . . 7 ((y Q z Q) → (y𝐺z) Q)
1311, 12genpprecll 6496 . . . . . 6 ((A P B P) → ((f (1stA) g (1stB)) → (f𝐺g) (1st ‘(A𝐹B))))
1413imp 115 . . . . 5 (((A P B P) (f (1stA) g (1stB))) → (f𝐺g) (1st ‘(A𝐹B)))
15 elprnql 6463 . . . . . . . . . 10 ((⟨(1stA), (2ndA)⟩ P f (1stA)) → f Q)
161, 15sylan 267 . . . . . . . . 9 ((A P f (1stA)) → f Q)
17 elprnql 6463 . . . . . . . . . 10 ((⟨(1stB), (2ndB)⟩ P g (1stB)) → g Q)
186, 17sylan 267 . . . . . . . . 9 ((B P g (1stB)) → g Q)
1916, 18anim12i 321 . . . . . . . 8 (((A P f (1stA)) (B P g (1stB))) → (f Q g Q))
2019an4s 522 . . . . . . 7 (((A P B P) (f (1stA) g (1stB))) → (f Q g Q))
2112caovcl 5597 . . . . . . 7 ((f Q g Q) → (f𝐺g) Q)
2220, 21syl 14 . . . . . 6 (((A P B P) (f (1stA) g (1stB))) → (f𝐺g) Q)
23 simpr 103 . . . . . . 7 ((((A P B P) (f (1stA) g (1stB))) 𝑞 = (f𝐺g)) → 𝑞 = (f𝐺g))
2423eleq1d 2103 . . . . . 6 ((((A P B P) (f (1stA) g (1stB))) 𝑞 = (f𝐺g)) → (𝑞 (1st ‘(A𝐹B)) ↔ (f𝐺g) (1st ‘(A𝐹B))))
2522, 24rspcedv 2654 . . . . 5 (((A P B P) (f (1stA) g (1stB))) → ((f𝐺g) (1st ‘(A𝐹B)) → 𝑞 Q 𝑞 (1st ‘(A𝐹B))))
2614, 25mpd 13 . . . 4 (((A P B P) (f (1stA) g (1stB))) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
2726anassrs 380 . . 3 ((((A P B P) f (1stA)) g (1stB)) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
2810, 27exlimddv 1775 . 2 (((A P B P) f (1stA)) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
295, 28exlimddv 1775 1 ((A P B P) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ 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:  addclpr  6520  mulclpr  6551
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