Theorem infglbti 7084

Description: An infimum is the greatest lower bound. See also infclti 7082 and inflbti 7083. (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 7044	. . . . 5 |- inf ( B , A , R ) = sup ( B , A , `' R )
2	1	breq1i 4036	. . . 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 7078	. . . . . . 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 7077	. . . . . . 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 7057	. . . . . 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 4843	. . . . . 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 7059	. . . . 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 2763	. . . . . 6 |- z e. _V
17		brcnvg 4843	. . . . . 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 2495	. . . 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 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   class class class wbr 4029   `'ccnv 4658   supcsup 7041  infcinf 7042

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-cnv 4667  df-iota 5215  df-riota 5873  df-sup 7043  df-inf 7044

This theorem is referenced by:  infnlbti  7085  zssinfcl  12085