Theorem infglbti 6438

Description: An infimum is the greatest lower bound. See also infclti 6436 and inflbti 6437. (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

Step	Hyp	Ref	Expression
1		df-inf 6398	. . . . 5 |- inf ( B , A , R ) = sup ( B , A , `' R )
2	1	breq1i 3792	. . . 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 ) ) )
5	4	cnvti 6432	. . . . . . 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 ) ) )
7	6	cnvinfex 6431	. . . . . . 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 ) ) )
8	5, 7	supclti 6411	. . . . . 6 |- ( ph -> sup ( B , A , `' R ) e. A )
9	8	adantr 270	. . . . 5 |- ( ( ph /\ C e. A ) -> sup ( B , A , `' R ) e. A )
10		brcnvg 4534	. . . . . 6 |- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( C `' R sup ( B , A , `' R ) <-> sup ( B , A , `' R ) R C ) )
11	10	bicomd 139	. . . . 5 |- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) )
12	3, 9, 11	syl2anc 403	. . . 4 |- ( ( ph /\ C e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) )
13	2, 12	syl5bb 190	. . 3 |- ( ( ph /\ C e. A ) -> (inf ( B , A , R ) R C <-> C `' R sup ( B , A , `' R ) ) )
14	5, 7	suplubti 6413	. . . . 5 |- ( ph -> ( ( C e. A /\ C `' R sup ( B , A , `' R ) ) -> E. z e. B C `' R z ) )
15	14	expdimp 255	. . . 4 |- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) -> E. z e. B C `' R z ) )
16		vex 2604	. . . . . 6 |- z e. _V
17		brcnvg 4534	. . . . . 6 |- ( ( C e. A /\ z e. _V ) -> ( C `' R z <-> z R C ) )
18	3, 16, 17	sylancl 404	. . . . 5 |- ( ( ph /\ C e. A ) -> ( C `' R z <-> z R C ) )
19	18	rexbidv 2369	. . . 4 |- ( ( ph /\ C e. A ) -> ( E. z e. B C `' R z <-> E. z e. B z R C ) )
20	15, 19	sylibd 147	. . 3 |- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) -> E. z e. B z R C ) )
21	13, 20	sylbid 148	. 2 |- ( ( ph /\ C e. A ) -> (inf ( B , A , R ) R C -> E. z e. B z R C ) )
22	21	expimpd 355	1 |- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) -> E. z e. B z R C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   A.wral 2348   E.wrex 2349   _Vcvv 2601   class class class wbr 3785   `'ccnv 4362   supcsup 6395  infcinf 6396

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-cnv 4371  df-iota 4887  df-riota 5488  df-sup 6397  df-inf 6398

This theorem is referenced by:  infnlbti  6439  zssinfcl  10344