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Theorem infglbti 7038
Description: An infimum is the greatest lower bound. See also infclti 7036 and inflbti 7037. (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 6998 . . . . 5  |- inf ( B ,  A ,  R
)  =  sup ( B ,  A ,  `' R )
21breq1i 4022 . . . 4  |-  (inf ( B ,  A ,  R ) R C  <->  sup ( B ,  A ,  `' R ) R C )
3 simpr 110 . . . . 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 7032 . . . . . . 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 7031 . . . . . . 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 7011 . . . . . 6  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
98adantr 276 . . . . 5  |-  ( (
ph  /\  C  e.  A )  ->  sup ( B ,  A ,  `' R )  e.  A
)
10 brcnvg 4820 . . . . . 6  |-  ( ( C  e.  A  /\  sup ( B ,  A ,  `' R )  e.  A
)  ->  ( C `' R sup ( B ,  A ,  `' R )  <->  sup ( B ,  A ,  `' R ) R C ) )
1110bicomd 141 . . . . 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 411 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( sup ( B ,  A ,  `' R ) R C  <-> 
C `' R sup ( B ,  A ,  `' R ) ) )
132, 12bitrid 192 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  <-> 
C `' R sup ( B ,  A ,  `' R ) ) )
145, 7suplubti 7013 . . . . 5  |-  ( ph  ->  ( ( C  e.  A  /\  C `' R sup ( B ,  A ,  `' R
) )  ->  E. z  e.  B  C `' R z ) )
1514expdimp 259 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R sup ( B ,  A ,  `' R )  ->  E. z  e.  B  C `' R z ) )
16 vex 2752 . . . . . 6  |-  z  e. 
_V
17 brcnvg 4820 . . . . . 6  |-  ( ( C  e.  A  /\  z  e.  _V )  ->  ( C `' R
z  <->  z R C ) )
183, 16, 17sylancl 413 . . . . 5  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R z  <->  z R C ) )
1918rexbidv 2488 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C `' R z  <->  E. z  e.  B  z R C ) )
2015, 19sylibd 149 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R sup ( B ,  A ,  `' R )  ->  E. z  e.  B  z R C ) )
2113, 20sylbid 150 . 2  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  ->  E. z  e.  B  z R C ) )
2221expimpd 363 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 104    <-> wb 105    e. wcel 2158   A.wral 2465   E.wrex 2466   _Vcvv 2749   class class class wbr 4015   `'ccnv 4637   supcsup 6995  infcinf 6996
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-cnv 4646  df-iota 5190  df-riota 5844  df-sup 6997  df-inf 6998
This theorem is referenced by:  infnlbti  7039  zssinfcl  11963
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