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Theorem infglbti 6667
Description: An infimum is the greatest lower bound. See also infclti 6665 and inflbti 6666. (Contributed by Jim Kingdon, 18-Dec-2021.)
Hypotheses
Ref Expression
infclti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
infclti.ex  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
Assertion
Ref Expression
infglbti  |-  ( ph  ->  ( ( C  e.  A  /\ inf ( B ,  A ,  R ) R C )  ->  E. z  e.  B  z R C ) )
Distinct variable groups:    u, A, v, x, y, z    u, B, v, x, y, z   
u, R, v, x, y, z    ph, u, v, x, y, z    z, C
Allowed substitution hints:    C( x, y, v, u)

Proof of Theorem infglbti
StepHypRef Expression
1 df-inf 6627 . . . . 5  |- inf ( B ,  A ,  R
)  =  sup ( B ,  A ,  `' R )
21breq1i 3829 . . . 4  |-  (inf ( B ,  A ,  R ) R C  <->  sup ( B ,  A ,  `' R ) R C )
3 simpr 108 . . . . 5  |-  ( (
ph  /\  C  e.  A )  ->  C  e.  A )
4 infclti.ti . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
54cnvti 6661 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
6 infclti.ex . . . . . . . 8  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
76cnvinfex 6660 . . . . . . 7  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x `' R y  /\  A. y  e.  A  ( y `' R x  ->  E. z  e.  B  y `' R z ) ) )
85, 7supclti 6640 . . . . . 6  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
98adantr 270 . . . . 5  |-  ( (
ph  /\  C  e.  A )  ->  sup ( B ,  A ,  `' R )  e.  A
)
10 brcnvg 4587 . . . . . 6  |-  ( ( C  e.  A  /\  sup ( B ,  A ,  `' R )  e.  A
)  ->  ( C `' R sup ( B ,  A ,  `' R )  <->  sup ( B ,  A ,  `' R ) R C ) )
1110bicomd 139 . . . . 5  |-  ( ( C  e.  A  /\  sup ( B ,  A ,  `' R )  e.  A
)  ->  ( sup ( B ,  A ,  `' R ) R C  <-> 
C `' R sup ( B ,  A ,  `' R ) ) )
123, 9, 11syl2anc 403 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( sup ( B ,  A ,  `' R ) R C  <-> 
C `' R sup ( B ,  A ,  `' R ) ) )
132, 12syl5bb 190 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  <-> 
C `' R sup ( B ,  A ,  `' R ) ) )
145, 7suplubti 6642 . . . . 5  |-  ( ph  ->  ( ( C  e.  A  /\  C `' R sup ( B ,  A ,  `' R
) )  ->  E. z  e.  B  C `' R z ) )
1514expdimp 255 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R sup ( B ,  A ,  `' R )  ->  E. z  e.  B  C `' R z ) )
16 vex 2618 . . . . . 6  |-  z  e. 
_V
17 brcnvg 4587 . . . . . 6  |-  ( ( C  e.  A  /\  z  e.  _V )  ->  ( C `' R
z  <->  z R C ) )
183, 16, 17sylancl 404 . . . . 5  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R z  <->  z R C ) )
1918rexbidv 2377 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C `' R z  <->  E. z  e.  B  z R C ) )
2015, 19sylibd 147 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R sup ( B ,  A ,  `' R )  ->  E. z  e.  B  z R C ) )
2113, 20sylbid 148 . 2  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  ->  E. z  e.  B  z R C ) )
2221expimpd 355 1  |-  ( ph  ->  ( ( C  e.  A  /\ inf ( B ,  A ,  R ) R C )  ->  E. z  e.  B  z R C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1436   A.wral 2355   E.wrex 2356   _Vcvv 2615   class class class wbr 3822   `'ccnv 4412   supcsup 6624  infcinf 6625
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-cnv 4421  df-iota 4948  df-riota 5571  df-sup 6626  df-inf 6627
This theorem is referenced by:  infnlbti  6668  zssinfcl  10850
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