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Theorem ltprordil 7607
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 7523 . . . 4  |-  <P  C_  ( P.  X.  P. )
21brel 4693 . . 3  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
3 ltdfpr 7524 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  A
)  /\  x  e.  ( 1st `  B ) ) ) )
43biimpd 144 . . 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 527 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  A  <P  B )
7 simpr 110 . . . . . 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 539 . . . . . . 7  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  ->  x  e.  ( 2nd `  A ) )
98adantr 276 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  x  e.  ( 2nd `  A ) )
102simpld 112 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
11 prop 7493 . . . . . . . 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 7505 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A )  /\  x  e.  ( 2nd `  A
) )  ->  y  <Q  x )
1412, 13syl3an1 1282 . . . . . 6  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A )  /\  x  e.  ( 2nd `  A
) )  ->  y  <Q  x )
156, 7, 9, 14syl3anc 1249 . . . . 5  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  -> 
y  <Q  x )
16 simprrr 540 . . . . . . 7  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  ->  x  e.  ( 1st `  B ) )
1716adantr 276 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  x  e.  ( 1st `  B ) )
182simprd 114 . . . . . . . 8  |-  ( A 
<P  B  ->  B  e. 
P. )
19 prop 7493 . . . . . . . 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 7502 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 1st `  B ) )  -> 
( y  <Q  x  ->  y  e.  ( 1st `  B ) ) )
2220, 21sylan 283 . . . . . 6  |-  ( ( A  <P  B  /\  x  e.  ( 1st `  B ) )  -> 
( y  <Q  x  ->  y  e.  ( 1st `  B ) ) )
236, 17, 22syl2anc 411 . . . . 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 115 . . 3  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  -> 
( y  e.  ( 1st `  A )  ->  y  e.  ( 1st `  B ) ) )
2625ssrdv 3176 . 2  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
275, 26rexlimddv 2612 1  |-  ( A 
<P  B  ->  ( 1st `  A )  C_  ( 1st `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   E.wrex 2469    C_ wss 3144   <.cop 3610   class class class wbr 4018   ` cfv 5231   1stc1st 6157   2ndc2nd 6158   Q.cnq 7298    <Q cltq 7303   P.cnp 7309    <P cltp 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7322  df-mi 7324  df-lti 7325  df-enq 7365  df-nqqs 7366  df-ltnqqs 7371  df-inp 7484  df-iltp 7488
This theorem is referenced by:  ltexprlemrl  7628
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