Theorem glbvalle 16811

Description: Value of greatest lower bound of a poset. Same as glbval 16809 with less-than-or-equal ordering.

Hypotheses

Ref	Expression
glbvalle.b	|- B = (base` K)
glbvalle.l	|- L = (le` K)
glbvalle.g	|- G = (glb` K)

Assertion

Ref	Expression
glbvalle	|- ((K e. A /\ S C_ B) -> (G` S) = (iota_x e. B(A.y e. S xLy /\ A.z e. B (A.y e. S zLy -> zLx))))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,K,y,z   x,L,y,z   x,S,y,z

Proof of Theorem glbvalle

Step	Hyp	Ref	Expression
1		glbvalle.b	. . 3 |- B = (base` K)
2		eqid 1884	. . 3 |- (geNEW` K) = (geNEW` K)
3		glbvalle.g	. . 3 |- G = (glb` K)
4	1, 2, 3	glbval 16809	. 2 |- ((K e. A /\ S C_ B) -> (G` S) = (iota_x e. B(A.y e. S y(geNEW` K)x /\ A.z e. B (A.y e. S y(geNEW` K)z -> x(geNEW` K)z))))
5		visset 2295	. . . . . . 7 |- y e. _V
6		visset 2295	. . . . . . 7 |- x e. _V
7		glbvalle.l	. . . . . . . 8 |- L = (le` K)
8	7, 2	pgeval 16794	. . . . . . 7 |- ((K e. A /\ y e. _V /\ x e. _V) -> (y(geNEW` K)x <-> xLy))
9	5, 6, 8	mp3an23 1183	. . . . . 6 |- (K e. A -> (y(geNEW` K)x <-> xLy))
10	9	ralbidv 2123	. . . . 5 |- (K e. A -> (A.y e. S y(geNEW` K)x <-> A.y e. S xLy))
11		visset 2295	. . . . . . . . 9 |- z e. _V
12	7, 2	pgeval 16794	. . . . . . . . 9 |- ((K e. A /\ y e. _V /\ z e. _V) -> (y(geNEW` K)z <-> zLy))
13	5, 11, 12	mp3an23 1183	. . . . . . . 8 |- (K e. A -> (y(geNEW` K)z <-> zLy))
14	13	ralbidv 2123	. . . . . . 7 |- (K e. A -> (A.y e. S y(geNEW` K)z <-> A.y e. S zLy))
15	7, 2	pgeval 16794	. . . . . . . 8 |- ((K e. A /\ x e. _V /\ z e. _V) -> (x(geNEW` K)z <-> zLx))
16	6, 11, 15	mp3an23 1183	. . . . . . 7 |- (K e. A -> (x(geNEW` K)z <-> zLx))
17	14, 16	imbi12d 688	. . . . . 6 |- (K e. A -> ((A.y e. S y(geNEW` K)z -> x(geNEW` K)z) <-> (A.y e. S zLy -> zLx)))
18	17	ralbidv 2123	. . . . 5 |- (K e. A -> (A.z e. B (A.y e. S y(geNEW` K)z -> x(geNEW` K)z) <-> A.z e. B (A.y e. S zLy -> zLx)))
19	10, 18	anbi12d 690	. . . 4 |- (K e. A -> ((A.y e. S y(geNEW` K)x /\ A.z e. B (A.y e. S y(geNEW` K)z -> x(geNEW` K)z)) <-> (A.y e. S xLy /\ A.z e. B (A.y e. S zLy -> zLx))))
20	19	riotabidv 5562	. . 3 |- (K e. A -> (iota_x e. B(A.y e. S y(geNEW` K)x /\ A.z e. B (A.y e. S y(geNEW` K)z -> x(geNEW` K)z))) = (iota_x e. B(A.y e. S xLy /\ A.z e. B (A.y e. S zLy -> zLx))))
21	20	adantr 425	. 2 |- ((K e. A /\ S C_ B) -> (iota_x e. B(A.y e. S y(geNEW` K)x /\ A.z e. B (A.y e. S y(geNEW` K)z -> x(geNEW` K)z))) = (iota_x e. B(A.y e. S xLy /\ A.z e. B (A.y e. S zLy -> zLx))))
22	4, 21	eqtrd 1925	1 |- ((K e. A /\ S C_ B) -> (G` S) = (iota_x e. B(A.y e. S xLy /\ A.z e. B (A.y e. S zLy -> zLx))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  ` cfv 3998  iota_crio 5555  basecbs 16758  lecple 16759  geNEWcpge 16762  glbcglb 16765

This theorem is referenced by:  glb0 16920  glbcon 17028

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-mpt 5006  df-iota 5089  df-riota 5560  df-pge 16792  df-glb 16800

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