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Theorem genpml 6372
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 6329 . . . 4 (A P → ⟨(1stA), (2ndA)⟩ P)
2 prml 6331 . . . 4 (⟨(1stA), (2ndA)⟩ Pf Q f (1stA))
3 rexex 2346 . . . 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 6329 . . . . 5 (B P → ⟨(1stB), (2ndB)⟩ P)
7 prml 6331 . . . . 5 (⟨(1stB), (2ndB)⟩ Pg Q g (1stB))
8 rexex 2346 . . . . 5 (g Q g (1stB) → g g (1stB))
96, 7, 83syl 17 . . . 4 (B Pg g (1stB))
109ad2antlr 462 . . 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 6368 . . . . . 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 6335 . . . . . . . . . 10 ((⟨(1stA), (2ndA)⟩ P f (1stA)) → f Q)
161, 15sylan 267 . . . . . . . . 9 ((A P f (1stA)) → f Q)
17 elprnql 6335 . . . . . . . . . 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 509 . . . . . . 7 (((A P B P) (f (1stA) g (1stB))) → (f Q g Q))
2112caovcl 5578 . . . . . . 7 ((f Q g Q) → (f𝐺g) Q)
2220, 21syl 14 . . . . . 6 (((A P B P) (f (1stA) g (1stB))) → (f𝐺g) Q)
23 ax-ia2 100 . . . . . . 7 ((((A P B P) (f (1stA) g (1stB))) 𝑞 = (f𝐺g)) → 𝑞 = (f𝐺g))
2423eleq1d 2088 . . . . . 6 ((((A P B P) (f (1stA) g (1stB))) 𝑞 = (f𝐺g)) → (𝑞 (1st ‘(A𝐹B)) ↔ (f𝐺g) (1st ‘(A𝐹B))))
2522, 24rspcedv 2637 . . . . 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 382 . . 3 ((((A P B P) f (1stA)) g (1stB)) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
2810, 27exlimddv 1760 . 2 (((A P B P) f (1stA)) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
295, 28exlimddv 1760 1 ((A P B P) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   = wceq 1228  wex 1362   wcel 1374  wrex 2285  {crab 2288  cop 3353  cfv 4829  (class class class)co 5436  cmpt2 5438  1st c1st 5688  2nd c2nd 5689  Qcnq 6138  Pcnp 6149
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-qs 6023  df-ni 6164  df-nqqs 6207  df-inp 6320
This theorem is referenced by:  addclpr  6392  mulclpr  6416
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