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Theorem genprndu 6505
Description: The upper 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)
genprndu.ord ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
genprndu.com ((x Q y Q) → (x𝐺y) = (y𝐺x))
genprndu.upper ((((A P g (2ndA)) (B P (2ndB))) x Q) → ((g𝐺) <Q xx (2nd ‘(A𝐹B))))
Assertion
Ref Expression
genprndu ((A P B P) → 𝑟 Q (𝑟 (2nd ‘(A𝐹B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(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 genprndu
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, 2genpelvu 6496 . . . . . . . . 9 ((A P B P) → (𝑟 (2nd ‘(A𝐹B)) ↔ 𝑎 (2ndA)𝑏 (2ndB)𝑟 = (𝑎𝐺𝑏)))
4 r2ex 2338 . . . . . . . . 9 (𝑎 (2ndA)𝑏 (2ndB)𝑟 = (𝑎𝐺𝑏) ↔ 𝑎𝑏((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏)))
53, 4syl6bb 185 . . . . . . . 8 ((A P B P) → (𝑟 (2nd ‘(A𝐹B)) ↔ 𝑎𝑏((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏))))
65biimpa 280 . . . . . . 7 (((A P B P) 𝑟 (2nd ‘(A𝐹B))) → 𝑎𝑏((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏)))
76adantrl 447 . . . . . 6 (((A P B P) (𝑟 Q 𝑟 (2nd ‘(A𝐹B)))) → 𝑎𝑏((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏)))
8 prop 6458 . . . . . . . . . . . . . . . 16 (A P → ⟨(1stA), (2ndA)⟩ P)
9 prnminu 6472 . . . . . . . . . . . . . . . 16 ((⟨(1stA), (2ndA)⟩ P 𝑎 (2ndA)) → 𝑐 (2ndA)𝑐 <Q 𝑎)
108, 9sylan 267 . . . . . . . . . . . . . . 15 ((A P 𝑎 (2ndA)) → 𝑐 (2ndA)𝑐 <Q 𝑎)
11 prop 6458 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
12 prnminu 6472 . . . . . . . . . . . . . . . 16 ((⟨(1stB), (2ndB)⟩ P 𝑏 (2ndB)) → 𝑑 (2ndB)𝑑 <Q 𝑏)
1311, 12sylan 267 . . . . . . . . . . . . . . 15 ((B P 𝑏 (2ndB)) → 𝑑 (2ndB)𝑑 <Q 𝑏)
1410, 13anim12i 321 . . . . . . . . . . . . . 14 (((A P 𝑎 (2ndA)) (B P 𝑏 (2ndB))) → (𝑐 (2ndA)𝑐 <Q 𝑎 𝑑 (2ndB)𝑑 <Q 𝑏))
1514an4s 522 . . . . . . . . . . . . 13 (((A P B P) (𝑎 (2ndA) 𝑏 (2ndB))) → (𝑐 (2ndA)𝑐 <Q 𝑎 𝑑 (2ndB)𝑑 <Q 𝑏))
16 reeanv 2473 . . . . . . . . . . . . 13 (𝑐 (2ndA)𝑑 (2ndB)(𝑐 <Q 𝑎 𝑑 <Q 𝑏) ↔ (𝑐 (2ndA)𝑐 <Q 𝑎 𝑑 (2ndB)𝑑 <Q 𝑏))
1715, 16sylibr 137 . . . . . . . . . . . 12 (((A P B P) (𝑎 (2ndA) 𝑏 (2ndB))) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐 <Q 𝑎 𝑑 <Q 𝑏))
18 genprndu.ord . . . . . . . . . . . . . . 15 ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
19 genprndu.com . . . . . . . . . . . . . . 15 ((x Q y Q) → (x𝐺y) = (y𝐺x))
2018, 19genplt2i 6493 . . . . . . . . . . . . . 14 ((𝑐 <Q 𝑎 𝑑 <Q 𝑏) → (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2120reximi 2410 . . . . . . . . . . . . 13 (𝑑 (2ndB)(𝑐 <Q 𝑎 𝑑 <Q 𝑏) → 𝑑 (2ndB)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2221reximi 2410 . . . . . . . . . . . 12 (𝑐 (2ndA)𝑑 (2ndB)(𝑐 <Q 𝑎 𝑑 <Q 𝑏) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2317, 22syl 14 . . . . . . . . . . 11 (((A P B P) (𝑎 (2ndA) 𝑏 (2ndB))) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2423adantrr 448 . . . . . . . . . 10 (((A P B P) ((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏))) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
25 breq2 3759 . . . . . . . . . . . . . 14 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏)))
2625biimprd 147 . . . . . . . . . . . . 13 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → (𝑐𝐺𝑑) <Q 𝑟))
2726reximdv 2414 . . . . . . . . . . . 12 (𝑟 = (𝑎𝐺𝑏) → (𝑑 (2ndB)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → 𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟))
2827reximdv 2414 . . . . . . . . . . 11 (𝑟 = (𝑎𝐺𝑏) → (𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟))
2928ad2antll 460 . . . . . . . . . 10 (((A P B P) ((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏))) → (𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟))
3024, 29mpd 13 . . . . . . . . 9 (((A P B P) ((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏))) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟)
3130ex 108 . . . . . . . 8 ((A P B P) → (((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏)) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟))
3231exlimdvv 1774 . . . . . . 7 ((A P B P) → (𝑎𝑏((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏)) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟))
3332adantr 261 . . . . . 6 (((A P B P) (𝑟 Q 𝑟 (2nd ‘(A𝐹B)))) → (𝑎𝑏((𝑎 (2ndA) 𝑏 (2ndB)) 𝑟 = (𝑎𝐺𝑏)) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟))
347, 33mpd 13 . . . . 5 (((A P B P) (𝑟 Q 𝑟 (2nd ‘(A𝐹B)))) → 𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟)
351, 2genppreclu 6498 . . . . . . . . 9 ((A P B P) → ((𝑐 (2ndA) 𝑑 (2ndB)) → (𝑐𝐺𝑑) (2nd ‘(A𝐹B))))
3635imp 115 . . . . . . . 8 (((A P B P) (𝑐 (2ndA) 𝑑 (2ndB))) → (𝑐𝐺𝑑) (2nd ‘(A𝐹B)))
37 elprnqu 6465 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P 𝑐 (2ndA)) → 𝑐 Q)
388, 37sylan 267 . . . . . . . . . . . 12 ((A P 𝑐 (2ndA)) → 𝑐 Q)
39 elprnqu 6465 . . . . . . . . . . . . 13 ((⟨(1stB), (2ndB)⟩ P 𝑑 (2ndB)) → 𝑑 Q)
4011, 39sylan 267 . . . . . . . . . . . 12 ((B P 𝑑 (2ndB)) → 𝑑 Q)
4138, 40anim12i 321 . . . . . . . . . . 11 (((A P 𝑐 (2ndA)) (B P 𝑑 (2ndB))) → (𝑐 Q 𝑑 Q))
4241an4s 522 . . . . . . . . . 10 (((A P B P) (𝑐 (2ndA) 𝑑 (2ndB))) → (𝑐 Q 𝑑 Q))
432caovcl 5597 . . . . . . . . . 10 ((𝑐 Q 𝑑 Q) → (𝑐𝐺𝑑) Q)
4442, 43syl 14 . . . . . . . . 9 (((A P B P) (𝑐 (2ndA) 𝑑 (2ndB))) → (𝑐𝐺𝑑) Q)
45 breq1 3758 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q 𝑟))
46 eleq1 2097 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 (2nd ‘(A𝐹B)) ↔ (𝑐𝐺𝑑) (2nd ‘(A𝐹B))))
4745, 46anbi12d 442 . . . . . . . . . 10 (𝑞 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 (𝑐𝐺𝑑) (2nd ‘(A𝐹B)))))
4847adantl 262 . . . . . . . . 9 ((((A P B P) (𝑐 (2ndA) 𝑑 (2ndB))) 𝑞 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 (𝑐𝐺𝑑) (2nd ‘(A𝐹B)))))
4944, 48rspcedv 2654 . . . . . . . 8 (((A P B P) (𝑐 (2ndA) 𝑑 (2ndB))) → (((𝑐𝐺𝑑) <Q 𝑟 (𝑐𝐺𝑑) (2nd ‘(A𝐹B))) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))
5036, 49mpan2d 404 . . . . . . 7 (((A P B P) (𝑐 (2ndA) 𝑑 (2ndB))) → ((𝑐𝐺𝑑) <Q 𝑟𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))
5150rexlimdvva 2434 . . . . . 6 ((A P B P) → (𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))
5251adantr 261 . . . . 5 (((A P B P) (𝑟 Q 𝑟 (2nd ‘(A𝐹B)))) → (𝑐 (2ndA)𝑑 (2ndB)(𝑐𝐺𝑑) <Q 𝑟𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))
5334, 52mpd 13 . . . 4 (((A P B P) (𝑟 Q 𝑟 (2nd ‘(A𝐹B)))) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B))))
5453expr 357 . . 3 (((A P B P) 𝑟 Q) → (𝑟 (2nd ‘(A𝐹B)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))
55 genprndu.upper . . . . . . . . . . 11 ((((A P g (2ndA)) (B P (2ndB))) x Q) → ((g𝐺) <Q xx (2nd ‘(A𝐹B))))
561, 2, 55genpcuu 6503 . . . . . . . . . 10 ((A P B P) → (𝑞 (2nd ‘(A𝐹B)) → (𝑞 <Q xx (2nd ‘(A𝐹B)))))
5756alrimdv 1753 . . . . . . . . 9 ((A P B P) → (𝑞 (2nd ‘(A𝐹B)) → x(𝑞 <Q xx (2nd ‘(A𝐹B)))))
58 breq2 3759 . . . . . . . . . . 11 (x = 𝑟 → (𝑞 <Q x𝑞 <Q 𝑟))
59 eleq1 2097 . . . . . . . . . . 11 (x = 𝑟 → (x (2nd ‘(A𝐹B)) ↔ 𝑟 (2nd ‘(A𝐹B))))
6058, 59imbi12d 223 . . . . . . . . . 10 (x = 𝑟 → ((𝑞 <Q xx (2nd ‘(A𝐹B))) ↔ (𝑞 <Q 𝑟𝑟 (2nd ‘(A𝐹B)))))
6160cbvalv 1791 . . . . . . . . 9 (x(𝑞 <Q xx (2nd ‘(A𝐹B))) ↔ 𝑟(𝑞 <Q 𝑟𝑟 (2nd ‘(A𝐹B))))
6257, 61syl6ib 150 . . . . . . . 8 ((A P B P) → (𝑞 (2nd ‘(A𝐹B)) → 𝑟(𝑞 <Q 𝑟𝑟 (2nd ‘(A𝐹B)))))
63 sp 1398 . . . . . . . 8 (𝑟(𝑞 <Q 𝑟𝑟 (2nd ‘(A𝐹B))) → (𝑞 <Q 𝑟𝑟 (2nd ‘(A𝐹B))))
6462, 63syl6 29 . . . . . . 7 ((A P B P) → (𝑞 (2nd ‘(A𝐹B)) → (𝑞 <Q 𝑟𝑟 (2nd ‘(A𝐹B)))))
6564impd 242 . . . . . 6 ((A P B P) → ((𝑞 (2nd ‘(A𝐹B)) 𝑞 <Q 𝑟) → 𝑟 (2nd ‘(A𝐹B))))
6665ancomsd 256 . . . . 5 ((A P B P) → ((𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B))) → 𝑟 (2nd ‘(A𝐹B))))
6766ad2antrr 457 . . . 4 ((((A P B P) 𝑟 Q) 𝑞 Q) → ((𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B))) → 𝑟 (2nd ‘(A𝐹B))))
6867rexlimdva 2427 . . 3 (((A P B P) 𝑟 Q) → (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B))) → 𝑟 (2nd ‘(A𝐹B))))
6954, 68impbid 120 . 2 (((A P B P) 𝑟 Q) → (𝑟 (2nd ‘(A𝐹B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))
7069ralrimiva 2386 1 ((A P B P) → 𝑟 Q (𝑟 (2nd ‘(A𝐹B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884  wal 1240   = wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301  {crab 2304  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  cmpt2 5457  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   <Q cltq 6269  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-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  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-nul 3219  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-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  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-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-lti 6291  df-enq 6331  df-nqqs 6332  df-ltnqqs 6337  df-inp 6449
This theorem is referenced by:  addclpr  6520  mulclpr  6553
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