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

Theorem xrmaxiflemval 11047
Description: Lemma for xrmaxif 11048. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
Hypothesis
Ref Expression
xrmaxiflemval.m  |-  M  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )
Assertion
Ref Expression
xrmaxiflemval  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( M  e.  RR*  /\  A. x  e.  { A ,  B }  -.  M  <  x  /\  A. x  e.  RR*  ( x  < 
M  ->  E. z  e.  { A ,  B } x  <  z ) ) )
Distinct variable groups:    x, A, z   
x, B, z
Allowed substitution hints:    M( x, z)

Proof of Theorem xrmaxiflemval
StepHypRef Expression
1 xrmaxiflemval.m . . 3  |-  M  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )
2 xrmaxiflemcl 11042 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR* )
31, 2eqeltrid 2227 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  M  e.  RR* )
4 vex 2690 . . . . 5  |-  x  e. 
_V
54elpr 3549 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
6 xrmaxifle 11043 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
76, 1breqtrrdi 3974 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  M )
8 xrlenlt 7849 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  M  e.  RR* )  ->  ( A  <_  M  <->  -.  M  <  A ) )
93, 8syldan 280 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  M  <->  -.  M  <  A ) )
107, 9mpbid 146 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  M  <  A )
11 breq2 3937 . . . . . . 7  |-  ( x  =  A  ->  ( M  <  x  <->  M  <  A ) )
1211notbid 657 . . . . . 6  |-  ( x  =  A  ->  ( -.  M  <  x  <->  -.  M  <  A ) )
1310, 12syl5ibrcom 156 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  =  A  ->  -.  M  <  x ) )
14 xrmaxifle 11043 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  B  <_  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
1514ancoms 266 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
16 xrmaxiflemcom 11046 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) ) )
171, 16syl5eq 2185 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  M  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
1815, 17breqtrrd 3960 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  M )
19 simpr 109 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
20 xrlenlt 7849 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  M  e.  RR* )  ->  ( B  <_  M  <->  -.  M  <  B ) )
2119, 3, 20syl2anc 409 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  M  <->  -.  M  <  B ) )
2218, 21mpbid 146 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  M  <  B )
23 breq2 3937 . . . . . . 7  |-  ( x  =  B  ->  ( M  <  x  <->  M  <  B ) )
2423notbid 657 . . . . . 6  |-  ( x  =  B  ->  ( -.  M  <  x  <->  -.  M  <  B ) )
2522, 24syl5ibrcom 156 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  =  B  ->  -.  M  <  x ) )
2613, 25jaod 707 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( x  =  A  \/  x  =  B )  ->  -.  M  <  x ) )
275, 26syl5bi 151 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  { A ,  B }  ->  -.  M  <  x ) )
2827ralrimiv 2505 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A. x  e.  { A ,  B }  -.  M  <  x
)
29 prid1g 3631 . . . . . . 7  |-  ( A  e.  RR*  ->  A  e. 
{ A ,  B } )
3029ad4antr 486 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  A )  ->  A  e.  { A ,  B } )
31 breq2 3937 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3231rspcev 2790 . . . . . 6  |-  ( ( A  e.  { A ,  B }  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
3330, 32sylancom 417 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
34 prid2g 3632 . . . . . . 7  |-  ( B  e.  RR*  ->  B  e. 
{ A ,  B } )
3534ad4antlr 487 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  B )  ->  B  e.  { A ,  B } )
36 breq2 3937 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3736rspcev 2790 . . . . . 6  |-  ( ( B  e.  { A ,  B }  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
3835, 37sylancom 417 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
39 simplll 523 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  A  e.  RR* )
40 simpllr 524 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  B  e.  RR* )
41 simplr 520 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  x  e.  RR* )
421breq2i 3941 . . . . . . . 8  |-  ( x  <  M  <->  x  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
4342biimpi 119 . . . . . . 7  |-  ( x  <  M  ->  x  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
4443adantl 275 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  x  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
4539, 40, 41, 44xrmaxiflemlub 11045 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  (
x  <  A  \/  x  <  B ) )
4633, 38, 45mpjaodan 788 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  E. z  e.  { A ,  B } x  <  z )
4746ex 114 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( x  <  M  ->  E. z  e.  { A ,  B }
x  <  z )
)
4847ralrimiva 2506 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A. x  e.  RR*  ( x  < 
M  ->  E. z  e.  { A ,  B } x  <  z ) )
493, 28, 483jca 1162 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( M  e.  RR*  /\  A. x  e.  { A ,  B }  -.  M  <  x  /\  A. x  e.  RR*  ( x  < 
M  ->  E. z  e.  { A ,  B } x  <  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   ifcif 3475   {cpr 3529   class class class wbr 3933   supcsup 6873   RRcr 7639   +oocpnf 7817   -oocmnf 7818   RR*cxr 7819    < clt 7820    <_ cle 7821
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-mulrcl 7739  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-mulass 7743  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-1rid 7747  ax-0id 7748  ax-rnegex 7749  ax-precex 7750  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-apti 7755  ax-pre-ltadd 7756  ax-pre-mulgt0 7757  ax-pre-mulext 7758  ax-arch 7759  ax-caucvg 7760
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-ilim 4295  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-frec 6292  df-sup 6875  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-reap 8357  df-ap 8364  df-div 8453  df-inn 8741  df-2 8799  df-3 8800  df-4 8801  df-n0 8998  df-z 9075  df-uz 9347  df-rp 9467  df-seqfrec 10246  df-exp 10320  df-cj 10642  df-re 10643  df-im 10644  df-rsqrt 10798  df-abs 10799
This theorem is referenced by:  xrmaxif  11048
  Copyright terms: Public domain W3C validator