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Theorem infglbti 7002
Description: An infimum is the greatest lower bound. See also infclti 7000 and inflbti 7001. (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 6962 . . . . 5  |- inf ( B ,  A ,  R
)  =  sup ( B ,  A ,  `' R )
21breq1i 3996 . . . 4  |-  (inf ( B ,  A ,  R ) R C  <->  sup ( B ,  A ,  `' R ) R C )
3 simpr 109 . . . . 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 6996 . . . . . . 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 6995 . . . . . . 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 6975 . . . . . 6  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
98adantr 274 . . . . 5  |-  ( (
ph  /\  C  e.  A )  ->  sup ( B ,  A ,  `' R )  e.  A
)
10 brcnvg 4792 . . . . . 6  |-  ( ( C  e.  A  /\  sup ( B ,  A ,  `' R )  e.  A
)  ->  ( C `' R sup ( B ,  A ,  `' R )  <->  sup ( B ,  A ,  `' R ) R C ) )
1110bicomd 140 . . . . 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 409 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( sup ( B ,  A ,  `' R ) R C  <-> 
C `' R sup ( B ,  A ,  `' R ) ) )
132, 12syl5bb 191 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  <-> 
C `' R sup ( B ,  A ,  `' R ) ) )
145, 7suplubti 6977 . . . . 5  |-  ( ph  ->  ( ( C  e.  A  /\  C `' R sup ( B ,  A ,  `' R
) )  ->  E. z  e.  B  C `' R z ) )
1514expdimp 257 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R sup ( B ,  A ,  `' R )  ->  E. z  e.  B  C `' R z ) )
16 vex 2733 . . . . . 6  |-  z  e. 
_V
17 brcnvg 4792 . . . . . 6  |-  ( ( C  e.  A  /\  z  e.  _V )  ->  ( C `' R
z  <->  z R C ) )
183, 16, 17sylancl 411 . . . . 5  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R z  <->  z R C ) )
1918rexbidv 2471 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C `' R z  <->  E. z  e.  B  z R C ) )
2015, 19sylibd 148 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( C `' R sup ( B ,  A ,  `' R )  ->  E. z  e.  B  z R C ) )
2113, 20sylbid 149 . 2  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  ->  E. z  e.  B  z R C ) )
2221expimpd 361 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 103    <-> wb 104    e. wcel 2141   A.wral 2448   E.wrex 2449   _Vcvv 2730   class class class wbr 3989   `'ccnv 4610   supcsup 6959  infcinf 6960
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-cnv 4619  df-iota 5160  df-riota 5809  df-sup 6961  df-inf 6962
This theorem is referenced by:  infnlbti  7003  zssinfcl  11903
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