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

Theorem genprndu 6504
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  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
genpelvl.2  Q.  Q.  G  Q.
genprndu.ord  Q.  Q.  Q.  <Q  G  <Q  G
genprndu.com  Q.  Q.  G  G
genprndu.upper  P.  2nd `  P.  h  2nd `  Q.  G h 
<Q  2nd `  F
Assertion
Ref Expression
genprndu  P.  P.  r  Q.  r  2nd `  F  q  Q.  q  <Q 
r  q  2nd `  F
Distinct variable groups:   ,,,, h,,, q,   ,,,,, h,,, q   , G,,,, h,,, q   , F, q   , r, q,,,,,   , r,, h    h, F, r,,,,,    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  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
2 genpelvl.2 . . . . . . . . . 10  Q.  Q.  G  Q.
31, 2genpelvu 6495 . . . . . . . . 9  P.  P.  r  2nd `  F  a  2nd `  b  2nd `  r  a G b
4 r2ex 2338 . . . . . . . . 9  a  2nd `  b  2nd `  r 
a G b  a b a  2nd `  b  2nd `  r 
a G b
53, 4syl6bb 185 . . . . . . . 8  P.  P.  r  2nd `  F  a b a  2nd `  b  2nd `  r 
a G b
65biimpa 280 . . . . . . 7  P.  P.  r  2nd `  F  a b a  2nd `  b  2nd `  r 
a G b
76adantrl 447 . . . . . 6  P.  P.  r  Q.  r  2nd `  F  a b a  2nd `  b  2nd `  r 
a G b
8 prop 6457 . . . . . . . . . . . . . . . 16  P.  <. 1st `  ,  2nd `  >.  P.
9 prnminu 6471 . . . . . . . . . . . . . . . 16 
<. 1st `  ,  2nd `  >.  P.  a  2nd `  c  2nd `  c 
<Q  a
108, 9sylan 267 . . . . . . . . . . . . . . 15  P.  a  2nd `  c  2nd `  c 
<Q  a
11 prop 6457 . . . . . . . . . . . . . . . 16  P.  <. 1st `  ,  2nd `  >.  P.
12 prnminu 6471 . . . . . . . . . . . . . . . 16 
<. 1st `  ,  2nd `  >.  P.  b  2nd `  d  2nd `  d 
<Q  b
1311, 12sylan 267 . . . . . . . . . . . . . . 15  P.  b  2nd `  d  2nd `  d 
<Q  b
1410, 13anim12i 321 . . . . . . . . . . . . . 14  P.  a  2nd `  P.  b  2nd `  c  2nd `  c  <Q  a  d  2nd `  d  <Q 
b
1514an4s 522 . . . . . . . . . . . . 13  P.  P.  a  2nd `  b  2nd `  c  2nd `  c  <Q 
a  d  2nd `  d  <Q  b
16 reeanv 2473 . . . . . . . . . . . . 13  c  2nd `  d  2nd `  c  <Q  a  d  <Q  b  c  2nd `  c  <Q  a  d  2nd `  d  <Q 
b
1715, 16sylibr 137 . . . . . . . . . . . 12  P.  P.  a  2nd `  b  2nd `  c  2nd `  d  2nd `  c  <Q  a  d  <Q  b
18 genprndu.ord . . . . . . . . . . . . . . 15  Q.  Q.  Q.  <Q  G  <Q  G
19 genprndu.com . . . . . . . . . . . . . . 15  Q.  Q.  G  G
2018, 19genplt2i 6492 . . . . . . . . . . . . . 14  c  <Q  a  d  <Q  b  c G d  <Q  a G b
2120reximi 2410 . . . . . . . . . . . . 13  d  2nd `  c 
<Q  a  d  <Q  b  d  2nd `  c G d  <Q  a G b
2221reximi 2410 . . . . . . . . . . . 12  c  2nd `  d  2nd `  c  <Q  a  d  <Q  b  c  2nd `  d  2nd `  c G d 
<Q  a G b
2317, 22syl 14 . . . . . . . . . . 11  P.  P.  a  2nd `  b  2nd `  c  2nd `  d  2nd `  c G d  <Q  a G b
2423adantrr 448 . . . . . . . . . 10  P.  P.  a  2nd `  b  2nd `  r  a G b  c  2nd `  d  2nd `  c G d  <Q  a G b
25 breq2 3759 . . . . . . . . . . . . . 14  r  a G b  c G d  <Q  r  c G d  <Q  a G b
2625biimprd 147 . . . . . . . . . . . . 13  r  a G b  c G d  <Q  a G b  c G d  <Q  r
2726reximdv 2414 . . . . . . . . . . . 12  r  a G b  d  2nd `  c G d 
<Q  a G b  d  2nd `  c G d  <Q  r
2827reximdv 2414 . . . . . . . . . . 11  r  a G b  c  2nd `  d  2nd `  c G d  <Q  a G b  c  2nd `  d  2nd `  c G d 
<Q  r
2928ad2antll 460 . . . . . . . . . 10  P.  P.  a  2nd `  b  2nd `  r  a G b  c  2nd `  d  2nd `  c G d 
<Q  a G b  c  2nd `  d  2nd `  c G d  <Q  r
3024, 29mpd 13 . . . . . . . . 9  P.  P.  a  2nd `  b  2nd `  r  a G b  c  2nd `  d  2nd `  c G d  <Q  r
3130ex 108 . . . . . . . 8  P.  P.  a  2nd `  b  2nd `  r  a G b  c  2nd `  d  2nd `  c G d  <Q  r
3231exlimdvv 1774 . . . . . . 7  P.  P.  a b a  2nd `  b  2nd `  r  a G b  c  2nd `  d  2nd `  c G d  <Q  r
3332adantr 261 . . . . . 6  P.  P.  r  Q.  r  2nd `  F  a b a  2nd `  b  2nd `  r  a G b  c  2nd `  d  2nd `  c G d  <Q  r
347, 33mpd 13 . . . . 5  P.  P.  r  Q.  r  2nd `  F  c  2nd `  d  2nd `  c G d 
<Q  r
351, 2genppreclu 6497 . . . . . . . . 9  P.  P.  c  2nd `  d  2nd `  c G d  2nd `  F
3635imp 115 . . . . . . . 8  P.  P.  c  2nd `  d  2nd `  c G d  2nd `  F
37 elprnqu 6464 . . . . . . . . . . . . 13 
<. 1st `  ,  2nd `  >.  P.  c  2nd `  c  Q.
388, 37sylan 267 . . . . . . . . . . . 12  P.  c  2nd `  c  Q.
39 elprnqu 6464 . . . . . . . . . . . . 13 
<. 1st `  ,  2nd `  >.  P.  d  2nd `  d  Q.
4011, 39sylan 267 . . . . . . . . . . . 12  P.  d  2nd `  d  Q.
4138, 40anim12i 321 . . . . . . . . . . 11  P.  c  2nd `  P.  d  2nd `  c  Q.  d  Q.
4241an4s 522 . . . . . . . . . 10  P.  P.  c  2nd `  d  2nd `  c  Q.  d  Q.
432caovcl 5597 . . . . . . . . . 10  c  Q.  d  Q.  c G d  Q.
4442, 43syl 14 . . . . . . . . 9  P.  P.  c  2nd `  d  2nd `  c G d 
Q.
45 breq1 3758 . . . . . . . . . . 11  q  c G d 
q  <Q  r  c G d  <Q  r
46 eleq1 2097 . . . . . . . . . . 11  q  c G d 
q  2nd `  F  c G d  2nd `  F
4745, 46anbi12d 442 . . . . . . . . . 10  q  c G d  q  <Q  r  q  2nd `  F  c G d 
<Q  r 
c G d  2nd `  F
4847adantl 262 . . . . . . . . 9  P.  P.  c  2nd `  d  2nd `  q  c G d  q  <Q  r  q  2nd `  F  c G d 
<Q  r 
c G d  2nd `  F
4944, 48rspcedv 2654 . . . . . . . 8  P.  P.  c  2nd `  d  2nd `  c G d  <Q  r  c G d  2nd `  F  q  Q.  q  <Q  r  q  2nd `  F
5036, 49mpan2d 404 . . . . . . 7  P.  P.  c  2nd `  d  2nd `  c G d 
<Q  r  q  Q.  q  <Q 
r  q  2nd `  F
5150rexlimdvva 2434 . . . . . 6  P.  P.  c  2nd `  d  2nd `  c G d  <Q  r  q  Q. 
q  <Q  r  q  2nd `  F
5251adantr 261 . . . . 5  P.  P.  r  Q.  r  2nd `  F  c  2nd `  d  2nd `  c G d  <Q  r  q  Q. 
q  <Q  r  q  2nd `  F
5334, 52mpd 13 . . . 4  P.  P.  r  Q.  r  2nd `  F  q  Q.  q  <Q  r  q  2nd `  F
5453expr 357 . . 3  P.  P.  r  Q.  r  2nd `  F  q  Q.  q  <Q  r  q  2nd `  F
55 genprndu.upper . . . . . . . . . . 11  P.  2nd `  P.  h  2nd `  Q.  G h 
<Q  2nd `  F
561, 2, 55genpcuu 6502 . . . . . . . . . 10  P.  P.  q  2nd `  F  q  <Q  2nd `  F
5756alrimdv 1753 . . . . . . . . 9  P.  P.  q  2nd `  F  q 
<Q  2nd `  F
58 breq2 3759 . . . . . . . . . . 11  r 
q  <Q  q  <Q 
r
59 eleq1 2097 . . . . . . . . . . 11  r  2nd `  F  r  2nd `  F
6058, 59imbi12d 223 . . . . . . . . . 10  r  q  <Q  2nd `  F  q 
<Q  r  r  2nd `  F
6160cbvalv 1791 . . . . . . . . 9  q  <Q  2nd `  F  r q  <Q  r  r  2nd `  F
6257, 61syl6ib 150 . . . . . . . 8  P.  P.  q  2nd `  F  r q 
<Q  r  r  2nd `  F
63 sp 1398 . . . . . . . 8  r q  <Q 
r  r  2nd `  F  q 
<Q  r  r  2nd `  F
6462, 63syl6 29 . . . . . . 7  P.  P.  q  2nd `  F  q  <Q  r  r  2nd `  F
6564impd 242 . . . . . 6  P.  P.  q  2nd `  F  q  <Q  r  r  2nd `  F
6665ancomsd 256 . . . . 5  P.  P.  q  <Q 
r  q  2nd `  F  r  2nd `  F
6766ad2antrr 457 . . . 4  P.  P.  r  Q.  q  Q.  q 
<Q  r  q  2nd `  F  r  2nd `  F
6867rexlimdva 2427 . . 3  P.  P.  r  Q.  q  Q.  q  <Q 
r  q  2nd `  F  r  2nd `  F
6954, 68impbid 120 . 2  P.  P.  r  Q.  r  2nd `  F  q  Q.  q  <Q  r  q  2nd `  F
7069ralrimiva 2386 1  P.  P.  r  Q.  r  2nd `  F  q  Q.  q  <Q 
r  q  2nd `  F
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   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264    <Q cltq 6269   P.cnp 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 6448
This theorem is referenced by:  addclpr  6519  mulclpr  6552
  Copyright terms: Public domain W3C validator