ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genprndl Structured version   GIF version

Theorem genprndl 6376
Description: The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-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)
genprndl.ord ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
genprndl.com ((x Q y Q) → (x𝐺y) = (y𝐺x))
genprndl.lower ((((A P g (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))
Assertion
Ref Expression
genprndl ((A P B P) → 𝑞 Q (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
Distinct variable groups:   x,y,z,g,,w,v,𝑞,A   x,B,y,z,g,,w,v,𝑞   x,𝐺,y,z,g,,w,v,𝑞   g,𝐹,𝑞   A,𝑟,𝑞,v,w,x,y,z   B,𝑟,g,   ,𝐹,𝑟,v,w,x,y,z   𝐺,𝑟

Proof of Theorem genprndl
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . . . . 10 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 . . . . . . . . . 10 ((y Q z Q) → (y𝐺z) Q)
31, 2genpelvl 6366 . . . . . . . . 9 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2322 . . . . . . . . 9 (𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏) ↔ 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
53, 4syl6bb 185 . . . . . . . 8 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))))
65biimpa 280 . . . . . . 7 (((A P B P) 𝑞 (1st ‘(A𝐹B))) → 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
76adantrl 450 . . . . . 6 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
8 prop 6329 . . . . . . . . . . . . . . . 16 (A P → ⟨(1stA), (2ndA)⟩ P)
9 prnmaxl 6342 . . . . . . . . . . . . . . . 16 ((⟨(1stA), (2ndA)⟩ P 𝑎 (1stA)) → 𝑐 (1stA)𝑎 <Q 𝑐)
108, 9sylan 267 . . . . . . . . . . . . . . 15 ((A P 𝑎 (1stA)) → 𝑐 (1stA)𝑎 <Q 𝑐)
11 prop 6329 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
12 prnmaxl 6342 . . . . . . . . . . . . . . . 16 ((⟨(1stB), (2ndB)⟩ P 𝑏 (1stB)) → 𝑑 (1stB)𝑏 <Q 𝑑)
1311, 12sylan 267 . . . . . . . . . . . . . . 15 ((B P 𝑏 (1stB)) → 𝑑 (1stB)𝑏 <Q 𝑑)
1410, 13anim12i 321 . . . . . . . . . . . . . 14 (((A P 𝑎 (1stA)) (B P 𝑏 (1stB))) → (𝑐 (1stA)𝑎 <Q 𝑐 𝑑 (1stB)𝑏 <Q 𝑑))
1514an4s 509 . . . . . . . . . . . . 13 (((A P B P) (𝑎 (1stA) 𝑏 (1stB))) → (𝑐 (1stA)𝑎 <Q 𝑐 𝑑 (1stB)𝑏 <Q 𝑑))
16 reeanv 2457 . . . . . . . . . . . . 13 (𝑐 (1stA)𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑) ↔ (𝑐 (1stA)𝑎 <Q 𝑐 𝑑 (1stB)𝑏 <Q 𝑑))
1715, 16sylibr 137 . . . . . . . . . . . 12 (((A P B P) (𝑎 (1stA) 𝑏 (1stB))) → 𝑐 (1stA)𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑))
18 genprndl.ord . . . . . . . . . . . . . . 15 ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
19 genprndl.com . . . . . . . . . . . . . . 15 ((x Q y Q) → (x𝐺y) = (y𝐺x))
2018, 19genplt2i 6364 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐 𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2120reximi 2394 . . . . . . . . . . . . 13 (𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑) → 𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2221reximi 2394 . . . . . . . . . . . 12 (𝑐 (1stA)𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑) → 𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2317, 22syl 14 . . . . . . . . . . 11 (((A P B P) (𝑎 (1stA) 𝑏 (1stB))) → 𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2423adantrr 451 . . . . . . . . . 10 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))) → 𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
25 breq1 3741 . . . . . . . . . . . . . 14 (𝑞 = (𝑎𝐺𝑏) → (𝑞 <Q (𝑐𝐺𝑑) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2625biimprd 147 . . . . . . . . . . . . 13 (𝑞 = (𝑎𝐺𝑏) → ((𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑞 <Q (𝑐𝐺𝑑)))
2726reximdv 2398 . . . . . . . . . . . 12 (𝑞 = (𝑎𝐺𝑏) → (𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
2827reximdv 2398 . . . . . . . . . . 11 (𝑞 = (𝑎𝐺𝑏) → (𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
2928ad2antll 464 . . . . . . . . . 10 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))) → (𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
3024, 29mpd 13 . . . . . . . . 9 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑))
3130ex 108 . . . . . . . 8 ((A P B P) → (((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
3231exlimdvv 1759 . . . . . . 7 ((A P B P) → (𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
3332adantr 261 . . . . . 6 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → (𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
347, 33mpd 13 . . . . 5 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑))
351, 2genpprecll 6368 . . . . . . . . 9 ((A P B P) → ((𝑐 (1stA) 𝑑 (1stB)) → (𝑐𝐺𝑑) (1st ‘(A𝐹B))))
3635imp 115 . . . . . . . 8 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐𝐺𝑑) (1st ‘(A𝐹B)))
37 elprnql 6335 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P 𝑐 (1stA)) → 𝑐 Q)
388, 37sylan 267 . . . . . . . . . . . 12 ((A P 𝑐 (1stA)) → 𝑐 Q)
39 elprnql 6335 . . . . . . . . . . . . 13 ((⟨(1stB), (2ndB)⟩ P 𝑑 (1stB)) → 𝑑 Q)
4011, 39sylan 267 . . . . . . . . . . . 12 ((B P 𝑑 (1stB)) → 𝑑 Q)
4138, 40anim12i 321 . . . . . . . . . . 11 (((A P 𝑐 (1stA)) (B P 𝑑 (1stB))) → (𝑐 Q 𝑑 Q))
4241an4s 509 . . . . . . . . . 10 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐 Q 𝑑 Q))
432caovcl 5578 . . . . . . . . . 10 ((𝑐 Q 𝑑 Q) → (𝑐𝐺𝑑) Q)
4442, 43syl 14 . . . . . . . . 9 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐𝐺𝑑) Q)
45 breq2 3742 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟𝑞 <Q (𝑐𝐺𝑑)))
46 eleq1 2082 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑟 (1st ‘(A𝐹B)) ↔ (𝑐𝐺𝑑) (1st ‘(A𝐹B))))
4745, 46anbi12d 445 . . . . . . . . . 10 (𝑟 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) ↔ (𝑞 <Q (𝑐𝐺𝑑) (𝑐𝐺𝑑) (1st ‘(A𝐹B)))))
4847adantl 262 . . . . . . . . 9 ((((A P B P) (𝑐 (1stA) 𝑑 (1stB))) 𝑟 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) ↔ (𝑞 <Q (𝑐𝐺𝑑) (𝑐𝐺𝑑) (1st ‘(A𝐹B)))))
4944, 48rspcedv 2637 . . . . . . . 8 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → ((𝑞 <Q (𝑐𝐺𝑑) (𝑐𝐺𝑑) (1st ‘(A𝐹B))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5036, 49mpan2d 406 . . . . . . 7 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑞 <Q (𝑐𝐺𝑑) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5150rexlimdvva 2418 . . . . . 6 ((A P B P) → (𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5251adantr 261 . . . . 5 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → (𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5334, 52mpd 13 . . . 4 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))))
5453expr 357 . . 3 (((A P B P) 𝑞 Q) → (𝑞 (1st ‘(A𝐹B)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
55 genprndl.lower . . . . . . . . . . 11 ((((A P g (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))
561, 2, 55genpcdl 6374 . . . . . . . . . 10 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → (x <Q 𝑟x (1st ‘(A𝐹B)))))
5756alrimdv 1738 . . . . . . . . 9 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → x(x <Q 𝑟x (1st ‘(A𝐹B)))))
58 breq1 3741 . . . . . . . . . . 11 (x = 𝑞 → (x <Q 𝑟𝑞 <Q 𝑟))
59 eleq1 2082 . . . . . . . . . . 11 (x = 𝑞 → (x (1st ‘(A𝐹B)) ↔ 𝑞 (1st ‘(A𝐹B))))
6058, 59imbi12d 223 . . . . . . . . . 10 (x = 𝑞 → ((x <Q 𝑟x (1st ‘(A𝐹B))) ↔ (𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B)))))
6160cbvalv 1776 . . . . . . . . 9 (x(x <Q 𝑟x (1st ‘(A𝐹B))) ↔ 𝑞(𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B))))
6257, 61syl6ib 150 . . . . . . . 8 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → 𝑞(𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B)))))
63 sp 1382 . . . . . . . 8 (𝑞(𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B))) → (𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B))))
6462, 63syl6 29 . . . . . . 7 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → (𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B)))))
6564impd 242 . . . . . 6 ((A P B P) → ((𝑟 (1st ‘(A𝐹B)) 𝑞 <Q 𝑟) → 𝑞 (1st ‘(A𝐹B))))
6665ancomsd 256 . . . . 5 ((A P B P) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) → 𝑞 (1st ‘(A𝐹B))))
6766ad2antrr 460 . . . 4 ((((A P B P) 𝑞 Q) 𝑟 Q) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) → 𝑞 (1st ‘(A𝐹B))))
6867rexlimdva 2411 . . 3 (((A P B P) 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) → 𝑞 (1st ‘(A𝐹B))))
6954, 68impbid 120 . 2 (((A P B P) 𝑞 Q) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
7069ralrimiva 2370 1 ((A P B P) → 𝑞 Q (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 873  wal 1226   = wceq 1228  wex 1362   wcel 1374  wral 2284  wrex 2285  {crab 2288  cop 3353   class class class wbr 3738  cfv 4829  (class class class)co 5436  cmpt2 5438  1st c1st 5688  2nd c2nd 5689  Qcnq 6138   <Q cltq 6143  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-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  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-nul 3202  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-tr 3829  df-eprel 4000  df-id 4004  df-po 4007  df-iso 4008  df-iord 4052  df-on 4054  df-suc 4057  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-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921  df-er 6017  df-ec 6019  df-qs 6023  df-ni 6164  df-mi 6166  df-lti 6167  df-enq 6206  df-nqqs 6207  df-ltnqqs 6212  df-inp 6320
This theorem is referenced by:  addclpr  6392  mulclpr  6416
  Copyright terms: Public domain W3C validator