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

Step	Hyp	Ref	Expression
1		df-inf 6998	. . . . 5 |- inf ( B , A , R ) = sup ( B , A , `' R )
2	1	breq1i 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 ) ) )
5	4	cnvti 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 ) ) )
7	6	cnvinfex 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 ) ) )
8	5, 7	supclti 7011	. . . . . 6 |- ( ph -> sup ( B , A , `' R ) e. A )
9	8	adantr 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 ) )
11	10	bicomd 141	. . . . 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 411	. . . 4 |- ( ( ph /\ C e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) )
13	2, 12	bitrid 192	. . 3 |- ( ( ph /\ C e. A ) -> (inf ( B , A , R ) R C <-> C `' R sup ( B , A , `' R ) ) )
14	5, 7	suplubti 7013	. . . . 5 |- ( ph -> ( ( C e. A /\ C `' R sup ( B , A , `' R ) ) -> E. z e. B C `' R z ) )
15	14	expdimp 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 ) )
18	3, 16, 17	sylancl 413	. . . . 5 |- ( ( ph /\ C e. A ) -> ( C `' R z <-> z R C ) )
19	18	rexbidv 2488	. . . 4 |- ( ( ph /\ C e. A ) -> ( E. z e. B C `' R z <-> E. z e. B z R C ) )
20	15, 19	sylibd 149	. . 3 |- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) -> E. z e. B z R C ) )
21	13, 20	sylbid 150	. 2 |- ( ( ph /\ C e. A ) -> (inf ( B , A , R ) R C -> E. z e. B z R C ) )
22	21	expimpd 363	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 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