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Theorem ltprordil 7339
Description: If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
Assertion
Ref Expression
ltprordil  |-  ( A 
<P  B  ->  ( 1st `  A )  C_  ( 1st `  B ) )

Proof of Theorem ltprordil
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7255 . . . 4  |-  <P  C_  ( P.  X.  P. )
21brel 4549 . . 3  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
3 ltdfpr 7256 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  A
)  /\  x  e.  ( 1st `  B ) ) ) )
43biimpd 143 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )
52, 4mpcom 36 . 2  |-  ( A 
<P  B  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  A
)  /\  x  e.  ( 1st `  B ) ) )
6 simpll 501 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  A  <P  B )
7 simpr 109 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  ( 1st `  A ) )
8 simprrl 511 . . . . . . 7  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  ->  x  e.  ( 2nd `  A ) )
98adantr 272 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  x  e.  ( 2nd `  A ) )
102simpld 111 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
11 prop 7225 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
1210, 11syl 14 . . . . . . 7  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 prltlu 7237 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A )  /\  x  e.  ( 2nd `  A
) )  ->  y  <Q  x )
1412, 13syl3an1 1230 . . . . . 6  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A )  /\  x  e.  ( 2nd `  A
) )  ->  y  <Q  x )
156, 7, 9, 14syl3anc 1197 . . . . 5  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  -> 
y  <Q  x )
16 simprrr 512 . . . . . . 7  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  ->  x  e.  ( 1st `  B ) )
1716adantr 272 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  x  e.  ( 1st `  B ) )
182simprd 113 . . . . . . . 8  |-  ( A 
<P  B  ->  B  e. 
P. )
19 prop 7225 . . . . . . . 8  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2018, 19syl 14 . . . . . . 7  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
21 prcdnql 7234 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 1st `  B ) )  -> 
( y  <Q  x  ->  y  e.  ( 1st `  B ) ) )
2220, 21sylan 279 . . . . . 6  |-  ( ( A  <P  B  /\  x  e.  ( 1st `  B ) )  -> 
( y  <Q  x  ->  y  e.  ( 1st `  B ) ) )
236, 17, 22syl2anc 406 . . . . 5  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  -> 
( y  <Q  x  ->  y  e.  ( 1st `  B ) ) )
2415, 23mpd 13 . . . 4  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  ( 1st `  B ) )
2524ex 114 . . 3  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  -> 
( y  e.  ( 1st `  A )  ->  y  e.  ( 1st `  B ) ) )
2625ssrdv 3067 . 2  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
275, 26rexlimddv 2526 1  |-  ( A 
<P  B  ->  ( 1st `  A )  C_  ( 1st `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1461   E.wrex 2389    C_ wss 3035   <.cop 3494   class class class wbr 3893   ` cfv 5079   1stc1st 5988   2ndc2nd 5989   Q.cnq 7030    <Q cltq 7035   P.cnp 7041    <P cltp 7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-mi 7056  df-lti 7057  df-enq 7097  df-nqqs 7098  df-ltnqqs 7103  df-inp 7216  df-iltp 7220
This theorem is referenced by:  ltexprlemrl  7360
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