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

Theorem ltexprlemlol 7522
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7533. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemlol  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( E. r  e. 
Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemlol
StepHypRef Expression
1 simplr 520 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  q  e.  Q. )
2 simprrr 530 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) )
32simpld 111 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  y  e.  ( 2nd `  A ) )
4 simprl 521 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  q  <Q  r
)
5 simpll 519 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  A  <P  B )
6 ltrelpr 7425 . . . . . . . . . . . 12  |-  <P  C_  ( P.  X.  P. )
76brel 4638 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
87simpld 111 . . . . . . . . . 10  |-  ( A 
<P  B  ->  A  e. 
P. )
9 prop 7395 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
10 elprnqu 7402 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
119, 10sylan 281 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
128, 11sylan 281 . . . . . . . . 9  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
135, 3, 12syl2anc 409 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  y  e.  Q. )
14 ltanqi 7322 . . . . . . . 8  |-  ( ( q  <Q  r  /\  y  e.  Q. )  ->  ( y  +Q  q
)  <Q  ( y  +Q  r ) )
154, 13, 14syl2anc 409 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  +Q  q )  <Q  (
y  +Q  r ) )
167simprd 113 . . . . . . . . 9  |-  ( A 
<P  B  ->  B  e. 
P. )
175, 16syl 14 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  B  e.  P. )
182simprd 113 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  +Q  r )  e.  ( 1st `  B ) )
19 prop 7395 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
20 prcdnql 7404 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  r
)  e.  ( 1st `  B ) )  -> 
( ( y  +Q  q )  <Q  (
y  +Q  r )  ->  ( y  +Q  q )  e.  ( 1st `  B ) ) )
2119, 20sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  ( y  +Q  r
)  e.  ( 1st `  B ) )  -> 
( ( y  +Q  q )  <Q  (
y  +Q  r )  ->  ( y  +Q  q )  e.  ( 1st `  B ) ) )
2217, 18, 21syl2anc 409 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( ( y  +Q  q )  <Q 
( y  +Q  r
)  ->  ( y  +Q  q )  e.  ( 1st `  B ) ) )
2315, 22mpd 13 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  +Q  q )  e.  ( 1st `  B ) )
241, 3, 23jca32 308 . . . . 5  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
2524eximi 1580 . . . 4  |-  ( E. y ( ( A 
<P  B  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  (
r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  E. y ( q  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) ) ) )
26 ltexprlem.1 . . . . . . . . . 10  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
2726ltexprlemell 7518 . . . . . . . . 9  |-  ( r  e.  ( 1st `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
28 19.42v 1886 . . . . . . . . 9  |-  ( E. y ( r  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) )  <->  ( r  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
2927, 28bitr4i 186 . . . . . . . 8  |-  ( r  e.  ( 1st `  C
)  <->  E. y ( r  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  r )  e.  ( 1st `  B
) ) ) )
3029anbi2i 453 . . . . . . 7  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  C ) )  <->  ( q  <Q  r  /\  E. y
( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
31 19.42v 1886 . . . . . . 7  |-  ( E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )  <->  ( q  <Q  r  /\  E. y
( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
3230, 31bitr4i 186 . . . . . 6  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  C ) )  <->  E. y
( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
3332anbi2i 453 . . . . 5  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  <->  ( ( A 
<P  B  /\  q  e.  Q. )  /\  E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
34 19.42v 1886 . . . . 5  |-  ( E. y ( ( A 
<P  B  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  (
r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  <-> 
( ( A  <P  B  /\  q  e.  Q. )  /\  E. y ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
3533, 34bitr4i 186 . . . 4  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  <->  E. y ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
3626ltexprlemell 7518 . . . . 5  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
37 19.42v 1886 . . . . 5  |-  ( E. y ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  <->  ( q  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
3836, 37bitr4i 186 . . . 4  |-  ( q  e.  ( 1st `  C
)  <->  E. y ( q  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) ) ) )
3925, 35, 383imtr4i 200 . . 3  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  ->  q  e.  ( 1st `  C ) )
4039ex 114 . 2  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
4140rexlimdvw 2578 1  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( E. r  e. 
Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335   E.wex 1472    e. wcel 2128   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200    +Q cplq 7202    <Q cltq 7205   P.cnp 7211    <P cltp 7215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-ltnqqs 7273  df-inp 7386  df-iltp 7390
This theorem is referenced by:  ltexprlemrnd  7525
  Copyright terms: Public domain W3C validator