Theorem infglb 8396

Description: An infimum is the greatest lower bound. See also infcl 8394 and inflb 8395. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infcl.1	|- ( ph -> R Or A )
infcl.2	|- ( 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
infglb	|- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) -> E. z e. B z R C ) )

Distinct variable groups:   x, A, y, z   x, B, y, z   x, R, y, z   z, C   ph, z

Allowed substitution hints:   ph( x, y)   C( x, y)

Proof of Theorem infglb

Step	Hyp	Ref	Expression
1		df-inf 8349	. . . . 5 |- inf ( B , A , R ) = sup ( B , A , `' R )
2	1	breq1i 4660	. . . 4 |- (inf ( B , A , R ) R C <-> sup ( B , A , `' R ) R C )
3		simpr 477	. . . . 5 |- ( ( ph /\ C e. A ) -> C e. A )
4		infcl.1	. . . . . . . 8 |- ( ph -> R Or A )
5		cnvso 5674	. . . . . . . 8 |- ( R Or A <-> `' R Or A )
6	4, 5	sylib 208	. . . . . . 7 |- ( ph -> `' R Or A )
7		infcl.2	. . . . . . . 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 ) ) )
8	4, 7	infcllem 8393	. . . . . . 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 ) ) )
9	6, 8	supcl 8364	. . . . . 6 |- ( ph -> sup ( B , A , `' R ) e. A )
10	9	adantr 481	. . . . 5 |- ( ( ph /\ C e. A ) -> sup ( B , A , `' R ) e. A )
11		brcnvg 5303	. . . . . 6 |- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( C `' R sup ( B , A , `' R ) <-> sup ( B , A , `' R ) R C ) )
12	11	bicomd 213	. . . . 5 |- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) )
13	3, 10, 12	syl2anc 693	. . . 4 |- ( ( ph /\ C e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) )
14	2, 13	syl5bb 272	. . 3 |- ( ( ph /\ C e. A ) -> (inf ( B , A , R ) R C <-> C `' R sup ( B , A , `' R ) ) )
15	6, 8	suplub 8366	. . . . 5 |- ( ph -> ( ( C e. A /\ C `' R sup ( B , A , `' R ) ) -> E. z e. B C `' R z ) )
16	15	expdimp 453	. . . 4 |- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) -> E. z e. B C `' R z ) )
17		vex 3203	. . . . . 6 |- z e. _V
18		brcnvg 5303	. . . . . 6 |- ( ( C e. A /\ z e. _V ) -> ( C `' R z <-> z R C ) )
19	3, 17, 18	sylancl 694	. . . . 5 |- ( ( ph /\ C e. A ) -> ( C `' R z <-> z R C ) )
20	19	rexbidv 3052	. . . 4 |- ( ( ph /\ C e. A ) -> ( E. z e. B C `' R z <-> E. z e. B z R C ) )
21	16, 20	sylibd 229	. . 3 |- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) -> E. z e. B z R C ) )
22	14, 21	sylbid 230	. 2 |- ( ( ph /\ C e. A ) -> (inf ( B , A , R ) R C -> E. z e. B z R C ) )
23	22	expimpd 629	1 |- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) -> E. z e. B z R C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   Or wor 5034   `'ccnv 5113   supcsup 8346  infcinf 8347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-cnv 5122  df-iota 5851  df-riota 6611  df-sup 8348  df-inf 8349

This theorem is referenced by:  infnlb  8398  omssubaddlem  30361  omssubadd  30362  gtinf  32313  infxrunb2  39584