Theorem joinval2 16816

Description: Value of join for a poset with GLB expanded.

Hypotheses

Ref	Expression
joinval2.b	|- B = (base` K)
joinval2.l	|- L = (le` K)
joinval2.j	|- J = (join` K)

Assertion

Ref	Expression
joinval2	|- ((K e. A /\ X e. B /\ Y e. B) -> (XJY) = (iota_x e. B((XLx /\ YLx) /\ A.z e. B ((XLz /\ YLz) -> xLz))))

Distinct variable groups:   x,z,B   x,J,z   x,K,z   x,L,z   x,X,z   x,Y,z

Proof of Theorem joinval2

Step	Hyp	Ref	Expression
1		joinval2.b	. . 3 |- B = (base` K)
2		eqid 1884	. . 3 |- (lub` K) = (lub` K)
3		joinval2.j	. . 3 |- J = (join` K)
4	1, 2, 3	joinval 16815	. 2 |- ((K e. A /\ X e. B /\ Y e. B) -> (XJY) = ((lub` K)` {X, Y}))
5		joinval2.l	. . . . . 6 |- L = (le` K)
6	1, 5, 2	lubval 16804	. . . . 5 |- ((K e. A /\ {X, Y} C_ B) -> ((lub` K)` {X, Y}) = (iota_x e. B(A.y e. {X, Y}yLx /\ A.z e. B (A.y e. {X, Y}yLz -> xLz))))
7		prssg 3140	. . . . . 6 |- ((X e. B /\ Y e. B) -> ((X e. B /\ Y e. B) <-> {X, Y} C_ B))
8	7	ibi 652	. . . . 5 |- ((X e. B /\ Y e. B) -> {X, Y} C_ B)
9	6, 8	sylan2 500	. . . 4 |- ((K e. A /\ (X e. B /\ Y e. B)) -> ((lub` K)` {X, Y}) = (iota_x e. B(A.y e. {X, Y}yLx /\ A.z e. B (A.y e. {X, Y}yLz -> xLz))))
10		ralprg 3078	. . . . . . . 8 |- ((X e. B /\ Y e. B) -> (A.y e. {X, Y}yLx <-> ([X / y]yLx /\ [Y / y]yLx)))
11		breq1 3341	. . . . . . . . . 10 |- (y = X -> (yLx <-> XLx))
12	11	sbcieg 2484	. . . . . . . . 9 |- (X e. B -> ([X / y]yLx <-> XLx))
13		breq1 3341	. . . . . . . . . 10 |- (y = Y -> (yLx <-> YLx))
14	13	sbcieg 2484	. . . . . . . . 9 |- (Y e. B -> ([Y / y]yLx <-> YLx))
15	12, 14	bi2anan9 694	. . . . . . . 8 |- ((X e. B /\ Y e. B) -> (([X / y]yLx /\ [Y / y]yLx) <-> (XLx /\ YLx)))
16	10, 15	bitrd 587	. . . . . . 7 |- ((X e. B /\ Y e. B) -> (A.y e. {X, Y}yLx <-> (XLx /\ YLx)))
17		ralprg 3078	. . . . . . . . . 10 |- ((X e. B /\ Y e. B) -> (A.y e. {X, Y}yLz <-> ([X / y]yLz /\ [Y / y]yLz)))
18		breq1 3341	. . . . . . . . . . . 12 |- (y = X -> (yLz <-> XLz))
19	18	sbcieg 2484	. . . . . . . . . . 11 |- (X e. B -> ([X / y]yLz <-> XLz))
20		breq1 3341	. . . . . . . . . . . 12 |- (y = Y -> (yLz <-> YLz))
21	20	sbcieg 2484	. . . . . . . . . . 11 |- (Y e. B -> ([Y / y]yLz <-> YLz))
22	19, 21	bi2anan9 694	. . . . . . . . . 10 |- ((X e. B /\ Y e. B) -> (([X / y]yLz /\ [Y / y]yLz) <-> (XLz /\ YLz)))
23	17, 22	bitrd 587	. . . . . . . . 9 |- ((X e. B /\ Y e. B) -> (A.y e. {X, Y}yLz <-> (XLz /\ YLz)))
24	23	imbi1d 675	. . . . . . . 8 |- ((X e. B /\ Y e. B) -> ((A.y e. {X, Y}yLz -> xLz) <-> ((XLz /\ YLz) -> xLz)))
25	24	ralbidv 2123	. . . . . . 7 |- ((X e. B /\ Y e. B) -> (A.z e. B (A.y e. {X, Y}yLz -> xLz) <-> A.z e. B ((XLz /\ YLz) -> xLz)))
26	16, 25	anbi12d 690	. . . . . 6 |- ((X e. B /\ Y e. B) -> ((A.y e. {X, Y}yLx /\ A.z e. B (A.y e. {X, Y}yLz -> xLz)) <-> ((XLx /\ YLx) /\ A.z e. B ((XLz /\ YLz) -> xLz))))
27	26	riotabidv 5562	. . . . 5 |- ((X e. B /\ Y e. B) -> (iota_x e. B(A.y e. {X, Y}yLx /\ A.z e. B (A.y e. {X, Y}yLz -> xLz))) = (iota_x e. B((XLx /\ YLx) /\ A.z e. B ((XLz /\ YLz) -> xLz))))
28	27	adantl 424	. . . 4 |- ((K e. A /\ (X e. B /\ Y e. B)) -> (iota_x e. B(A.y e. {X, Y}yLx /\ A.z e. B (A.y e. {X, Y}yLz -> xLz))) = (iota_x e. B((XLx /\ YLx) /\ A.z e. B ((XLz /\ YLz) -> xLz))))
29	9, 28	eqtrd 1925	. . 3 |- ((K e. A /\ (X e. B /\ Y e. B)) -> ((lub` K)` {X, Y}) = (iota_x e. B((XLx /\ YLx) /\ A.z e. B ((XLz /\ YLz) -> xLz))))
30	29	3impb 1063	. 2 |- ((K e. A /\ X e. B /\ Y e. B) -> ((lub` K)` {X, Y}) = (iota_x e. B((XLx /\ YLx) /\ A.z e. B ((XLz /\ YLz) -> xLz))))
31	4, 30	eqtrd 1925	1 |- ((K e. A /\ X e. B /\ Y e. B) -> (XJY) = (iota_x e. B((XLx /\ YLx) /\ A.z e. B ((XLz /\ YLz) -> xLz))))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105   C_ wss 2593  {cpr 3045   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  iota_crio 5555  basecbs 16758  lecple 16759  lubclub 16764  joincjn 16766

This theorem is referenced by:  joinlem 16817

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-csb 2541  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-opr 4886  df-oprab 4887  df-mpt 5006  df-mpt2 5007  df-iota 5089  df-riota 5560  df-lub 16799  df-join 16801

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