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Theorem ltprordil 7148
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 7064 . . . 4  |-  <P  C_  ( P.  X.  P. )
21brel 4490 . . 3  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
3 ltdfpr 7065 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  A
)  /\  x  e.  ( 1st `  B ) ) ) )
43biimpd 142 . . 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 496 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  A  <P  B )
7 simpr 108 . . . . . 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 506 . . . . . . 7  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  ->  x  e.  ( 2nd `  A ) )
98adantr 270 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  x  e.  ( 2nd `  A ) )
102simpld 110 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
11 prop 7034 . . . . . . . 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 7046 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A )  /\  x  e.  ( 2nd `  A
) )  ->  y  <Q  x )
1412, 13syl3an1 1207 . . . . . 6  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A )  /\  x  e.  ( 2nd `  A
) )  ->  y  <Q  x )
156, 7, 9, 14syl3anc 1174 . . . . 5  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  -> 
y  <Q  x )
16 simprrr 507 . . . . . . 7  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  ->  x  e.  ( 1st `  B ) )
1716adantr 270 . . . . . 6  |-  ( ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  /\  y  e.  ( 1st `  A ) )  ->  x  e.  ( 1st `  B ) )
182simprd 112 . . . . . . . 8  |-  ( A 
<P  B  ->  B  e. 
P. )
19 prop 7034 . . . . . . . 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 7043 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 1st `  B ) )  -> 
( y  <Q  x  ->  y  e.  ( 1st `  B ) ) )
2220, 21sylan 277 . . . . . 6  |-  ( ( A  <P  B  /\  x  e.  ( 1st `  B ) )  -> 
( y  <Q  x  ->  y  e.  ( 1st `  B ) ) )
236, 17, 22syl2anc 403 . . . . 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 113 . . 3  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  -> 
( y  e.  ( 1st `  A )  ->  y  e.  ( 1st `  B ) ) )
2625ssrdv 3031 . 2  |-  ( ( A  <P  B  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  A )  /\  x  e.  ( 1st `  B ) ) ) )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
275, 26rexlimddv 2493 1  |-  ( A 
<P  B  ->  ( 1st `  A )  C_  ( 1st `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   E.wrex 2360    C_ wss 2999   <.cop 3449   class class class wbr 3845   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839    <Q cltq 6844   P.cnp 6850    <P cltp 6854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-mi 6865  df-lti 6866  df-enq 6906  df-nqqs 6907  df-ltnqqs 6912  df-inp 7025  df-iltp 7029
This theorem is referenced by:  ltexprlemrl  7169
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