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Theorem genpmu 6367
Description: The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-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
genpmu ((A P B P) → 𝑞 Q 𝑞 (2nd ‘(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 genpmu
Dummy variables f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6323 . . . 4 (A P → ⟨(1stA), (2ndA)⟩ P)
2 prmu 6326 . . . 4 (⟨(1stA), (2ndA)⟩ Pf Q f (2ndA))
3 rexex 2342 . . . 4 (f Q f (2ndA) → f f (2ndA))
41, 2, 33syl 17 . . 3 (A Pf f (2ndA))
54adantr 261 . 2 ((A P B P) → f f (2ndA))
6 prop 6323 . . . . 5 (B P → ⟨(1stB), (2ndB)⟩ P)
7 prmu 6326 . . . . 5 (⟨(1stB), (2ndB)⟩ Pg Q g (2ndB))
8 rexex 2342 . . . . 5 (g Q g (2ndB) → g g (2ndB))
96, 7, 83syl 17 . . . 4 (B Pg g (2ndB))
109ad2antlr 462 . . 3 (((A P B P) f (2ndA)) → g g (2ndB))
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, 12genppreclu 6363 . . . . . 6 ((A P B P) → ((f (2ndA) g (2ndB)) → (f𝐺g) (2nd ‘(A𝐹B))))
1413imp 115 . . . . 5 (((A P B P) (f (2ndA) g (2ndB))) → (f𝐺g) (2nd ‘(A𝐹B)))
15 elprnqu 6330 . . . . . . . . . 10 ((⟨(1stA), (2ndA)⟩ P f (2ndA)) → f Q)
161, 15sylan 267 . . . . . . . . 9 ((A P f (2ndA)) → f Q)
17 elprnqu 6330 . . . . . . . . . 10 ((⟨(1stB), (2ndB)⟩ P g (2ndB)) → g Q)
186, 17sylan 267 . . . . . . . . 9 ((B P g (2ndB)) → g Q)
1916, 18anim12i 321 . . . . . . . 8 (((A P f (2ndA)) (B P g (2ndB))) → (f Q g Q))
2019an4s 509 . . . . . . 7 (((A P B P) (f (2ndA) g (2ndB))) → (f Q g Q))
2112caovcl 5574 . . . . . . 7 ((f Q g Q) → (f𝐺g) Q)
2220, 21syl 14 . . . . . 6 (((A P B P) (f (2ndA) g (2ndB))) → (f𝐺g) Q)
23 simpr 103 . . . . . . 7 ((((A P B P) (f (2ndA) g (2ndB))) 𝑞 = (f𝐺g)) → 𝑞 = (f𝐺g))
2423eleq1d 2084 . . . . . 6 ((((A P B P) (f (2ndA) g (2ndB))) 𝑞 = (f𝐺g)) → (𝑞 (2nd ‘(A𝐹B)) ↔ (f𝐺g) (2nd ‘(A𝐹B))))
2522, 24rspcedv 2633 . . . . 5 (((A P B P) (f (2ndA) g (2ndB))) → ((f𝐺g) (2nd ‘(A𝐹B)) → 𝑞 Q 𝑞 (2nd ‘(A𝐹B))))
2614, 25mpd 13 . . . 4 (((A P B P) (f (2ndA) g (2ndB))) → 𝑞 Q 𝑞 (2nd ‘(A𝐹B)))
2726anassrs 382 . . 3 ((((A P B P) f (2ndA)) g (2ndB)) → 𝑞 Q 𝑞 (2nd ‘(A𝐹B)))
2810, 27exlimddv 1756 . 2 (((A P B P) f (2ndA)) → 𝑞 Q 𝑞 (2nd ‘(A𝐹B)))
295, 28exlimddv 1756 1 ((A P B P) → 𝑞 Q 𝑞 (2nd ‘(A𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 871   = wceq 1226  wex 1358   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:  addclpr  6386  mulclpr  6410
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