ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  maxleast Unicode version
Theorem maxleast 11732
Description: The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)
Assertion
Ref Expression
maxleast |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ C /\ B <_ C ) ) -> sup ( { A , B } , RR , < ) <_ C )
Proof of Theorem maxleast
Dummy variables f g x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioran 757 . . . 4 |- ( -. ( C < A \/ C < B ) <-> ( -. C < A /\ -. C < B ) )
2 simp3 1023 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3 lttri3 8234 . . . . . . . . 9 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
43adantl 277 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
5 maxabslemval 11727 . . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) )
6 3anass 1006 . . . . . . . . . . 11 |- ( ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) <-> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) )
75, 6sylib 122 . . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) )
8 breq1 4086 . . . . . . . . . . . . . 14 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( x < y <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
98notbid 671 . . . . . . . . . . . . 13 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( -. x < y <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
109ralbidv 2530 . . . . . . . . . . . 12 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( A. y e. { A , B } -. x < y <-> A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
11 breq2 4087 . . . . . . . . . . . . . 14 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( y < x <-> y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) )
1211imbi1d 231 . . . . . . . . . . . . 13 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( ( y < x -> E. z e. { A , B } y < z ) <-> ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) )
1312ralbidv 2530 . . . . . . . . . . . 12 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( A. y e. RR ( y < x -> E. z e. { A , B } y < z ) <-> A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) )
1410, 13anbi12d 473 . . . . . . . . . . 11 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) <-> ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) )
1514rspcev 2907 . . . . . . . . . 10 |- ( ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) -> E. x e. RR ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) )
167, 15syl 14 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> E. x e. RR ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) )
17163adant3 1041 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) )
184, 17suplubti 7175 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C e. RR /\ C < sup ( { A , B } , RR , < ) ) -> E. z e. { A , B } C < z ) )
192, 18mpand 429 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < sup ( { A , B } , RR , < ) -> E. z e. { A , B } C < z ) )
20 elpri 3689 . . . . . . . . 9 |- ( z e. { A , B } -> ( z = A \/ z = B ) )
2120adantr 276 . . . . . . . 8 |- ( ( z e. { A , B } /\ C < z ) -> ( z = A \/ z = B ) )
22 breq2 4087 . . . . . . . . . . 11 |- ( z = A -> ( C < z <-> C < A ) )
2322biimpcd 159 . . . . . . . . . 10 |- ( C < z -> ( z = A -> C < A ) )
2423adantl 277 . . . . . . . . 9 |- ( ( z e. { A , B } /\ C < z ) -> ( z = A -> C < A ) )
25 breq2 4087 . . . . . . . . . . 11 |- ( z = B -> ( C < z <-> C < B ) )
2625biimpcd 159 . . . . . . . . . 10 |- ( C < z -> ( z = B -> C < B ) )
2726adantl 277 . . . . . . . . 9 |- ( ( z e. { A , B } /\ C < z ) -> ( z = B -> C < B ) )
2824, 27orim12d 791 . . . . . . . 8 |- ( ( z e. { A , B } /\ C < z ) -> ( ( z = A \/ z = B ) -> ( C < A \/ C < B ) ) )
2921, 28mpd 13 . . . . . . 7 |- ( ( z e. { A , B } /\ C < z ) -> ( C < A \/ C < B ) )
3029rexlimiva 2643 . . . . . 6 |- ( E. z e. { A , B } C < z -> ( C < A \/ C < B ) )
3119, 30syl6 33 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < sup ( { A , B } , RR , < ) -> ( C < A \/ C < B ) ) )
3231con3d 634 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. ( C < A \/ C < B ) -> -. C < sup ( { A , B } , RR , < ) ) )
331, 32biimtrrid 153 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( -. C < A /\ -. C < B ) -> -. C < sup ( { A , B } , RR , < ) ) )
34 simp1 1021 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
3534, 2lenltd 8272 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
36 simp2 1022 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
3736, 2lenltd 8272 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
3835, 37anbi12d 473 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ C /\ B <_ C ) <-> ( -. C < A /\ -. C < B ) ) )
394, 17supclti 7173 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
4039, 2lenltd 8272 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) <_ C <-> -. C < sup ( { A , B } , RR , < ) ) )
4133, 38, 403imtr4d 203 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ C /\ B <_ C ) -> sup ( { A , B } , RR , < ) <_ C ) )
4241imp 124 1 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ C /\ B <_ C ) ) -> sup ( { A , B } , RR , < ) <_ C )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {cpr 3667   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   supcsup 7157   RRcr 8006   + caddc 8010   < clt 8189   <_ cle 8190   - cmin 8325   / cdiv 8827   2c2 9169   abscabs 11516
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-sup 7159  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-uz 9731  df-rp 9858  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518
This theorem is referenced by:  maxleastb  11733  dfabsmax  11736
  Copyright terms: Public domain W3C validator