Theorem glbprop 16810

Description: Properties of greatest lower bound of a poset.

Hypotheses

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

Assertion

Ref	Expression
glbprop	|- ((K e. A /\ S C_ B /\ (G` S) e. B) -> (A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz)))

Distinct variable groups:   z,B   y,z,K   y,S,z   y,G,z

Proof of Theorem glbprop

Step	Hyp	Ref	Expression
1		glbval.b	. . . . 5 |- B = (base` K)
2		glbval.l	. . . . 5 |- L = (geNEW` K)
3		glbval.g	. . . . 5 |- G = (glb` K)
4	1, 2, 3	glbval 16809	. . . 4 |- ((K e. A /\ S C_ B) -> (G` S) = (iota_x e. B(A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))))
5	4	3adant3 896	. . 3 |- ((K e. A /\ S C_ B /\ (G` S) e. B) -> (G` S) = (iota_x e. B(A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))))
6	5	eqcomd 1889	. 2 |- ((K e. A /\ S C_ B /\ (G` S) e. B) -> (iota_x e. B(A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))) = (G` S))
7		simp3 878	. . 3 |- ((K e. A /\ S C_ B /\ (G` S) e. B) -> (G` S) e. B)
8	5, 7	eqeltrrd 1972	. . . 4 |- ((K e. A /\ S C_ B /\ (G` S) e. B) -> (iota_x e. B(A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))) e. B)
9		fvex 4689	. . . . . 6 |- (base` K) e. _V
10	1, 9	eqeltri 1967	. . . . 5 |- B e. _V
11	10	riotaclb 5573	. . . 4 |- (E!x e. B (A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz)) <-> (iota_x e. B(A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))) e. B)
12	8, 11	sylibr 217	. . 3 |- ((K e. A /\ S C_ B /\ (G` S) e. B) -> E!x e. B (A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz)))
13		ax-17 1317	. . . 4 |- (w e. (G` S) -> A.x w e. (G` S))
14		ax-17 1317	. . . . 5 |- ((A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz)) -> A.x(A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz)))
15	14	a1i 8	. . . 4 |- ((G` S) e. B -> ((A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz)) -> A.x(A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz))))
16		breq2 3342	. . . . . 6 |- (x = (G` S) -> (yLx <-> yL(G` S)))
17	16	ralbidv 2123	. . . . 5 |- (x = (G` S) -> (A.y e. S yLx <-> A.y e. S yL(G` S)))
18		breq1 3341	. . . . . . 7 |- (x = (G` S) -> (xLz <-> (G` S)Lz))
19	18	imbi2d 674	. . . . . 6 |- (x = (G` S) -> ((A.y e. S yLz -> xLz) <-> (A.y e. S yLz -> (G` S)Lz)))
20	19	ralbidv 2123	. . . . 5 |- (x = (G` S) -> (A.z e. B (A.y e. S yLz -> xLz) <-> A.z e. B (A.y e. S yLz -> (G` S)Lz)))
21	17, 20	anbi12d 690	. . . 4 |- (x = (G` S) -> ((A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz)) <-> (A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz))))
22	13, 15, 21	riota2f 5579	. . 3 |- (((G` S) e. B /\ E!x e. B (A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))) -> ((A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz)) <-> (iota_x e. B(A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))) = (G` S)))
23	7, 12, 22	syl11anc 524	. 2 |- ((K e. A /\ S C_ B /\ (G` S) e. B) -> ((A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz)) <-> (iota_x e. B(A.y e. S yLx /\ A.z e. B (A.y e. S yLz -> xLz))) = (G` S)))
24	6, 23	mpbird 213	1 |- ((K e. A /\ S C_ B /\ (G` S) e. B) -> (A.y e. S yL(G` S) /\ A.z e. B (A.y e. S yLz -> (G` S)Lz)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  E!wreu 2107  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  ` cfv 3998  iota_crio 5555  basecbs 16758  geNEWcpge 16762  glbcglb 16765

This theorem is referenced by:  glbprople 16812

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-rab 2112  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-mpt 5006  df-iota 5089  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-glb 16800

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