Theorem pslem 8643

Description: Lemma for psref 8645 and others.

Assertion

Ref	Expression
pslem	|- (R e. Poset -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))

Proof of Theorem pslem

Step	Hyp	Ref	Expression
1		isps 8641	. . 3 |- (R e. Poset -> (R e. Poset <-> (Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R))))
2	1	ibi 594	. 2 |- (R e. Poset -> (Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R)))
3		breq12 2629	. . . . . . . . . 10 |- ((x = A /\ y = B) -> (xRy <-> ARB))
4	3	3adant3 801	. . . . . . . . 9 |- ((x = A /\ y = B /\ z = C) -> (xRy <-> ARB))
5		breq12 2629	. . . . . . . . . 10 |- ((y = B /\ z = C) -> (yRz <-> BRC))
6	5	3adant1 799	. . . . . . . . 9 |- ((x = A /\ y = B /\ z = C) -> (yRz <-> BRC))
7	4, 6	anbi12d 630	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> ((xRy /\ yRz) <-> (ARB /\ BRC)))
8		breq12 2629	. . . . . . . . 9 |- ((x = A /\ z = C) -> (xRz <-> ARC))
9	8	3adant2 800	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> (xRz <-> ARC))
10	7, 9	imbi12d 628	. . . . . . 7 |- ((x = A /\ y = B /\ z = C) -> (((xRy /\ yRz) -> xRz) <-> ((ARB /\ BRC) -> ARC)))
11	10	cla43gv 1870	. . . . . 6 |- ((A e. S /\ B e. T /\ C e. U) -> (A.xA.yA.z((xRy /\ yRz) -> xRz) -> ((ARB /\ BRC) -> ARC)))
12		breq12 2629	. . . . . . . . . . 11 |- ((x = A /\ x = A) -> (xRx <-> ARA))
13	12	anidms 436	. . . . . . . . . 10 |- (x = A -> (xRx <-> ARA))
14	13	rcla4cv 1877	. . . . . . . . 9 |- (A.x e. U.U.RxRx -> (A e. U.U.R -> ARA))
15	14	a1i 8	. . . . . . . 8 |- ((A e. S /\ B e. T) -> (A.x e. U.U.RxRx -> (A e. U.U.R -> ARA)))
16		breq12 2629	. . . . . . . . . . . 12 |- ((y = B /\ x = A) -> (yRx <-> BRA))
17	16	ancoms 438	. . . . . . . . . . 11 |- ((x = A /\ y = B) -> (yRx <-> BRA))
18	3, 17	anbi12d 630	. . . . . . . . . 10 |- ((x = A /\ y = B) -> ((xRy /\ yRx) <-> (ARB /\ BRA)))
19		eqeq12 1490	. . . . . . . . . 10 |- ((x = A /\ y = B) -> (x = y <-> A = B))
20	18, 19	imbi12d 628	. . . . . . . . 9 |- ((x = A /\ y = B) -> (((xRy /\ yRx) -> x = y) <-> ((ARB /\ BRA) -> A = B)))
21	20	cla42gv 1868	. . . . . . . 8 |- ((A e. S /\ B e. T) -> (A.xA.y((xRy /\ yRx) -> x = y) -> ((ARB /\ BRA) -> A = B)))
22	15, 21	anim12d 560	. . . . . . 7 |- ((A e. S /\ B e. T) -> ((A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)) -> ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B))))
23	22	3adant3 801	. . . . . 6 |- ((A e. S /\ B e. T /\ C e. U) -> ((A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)) -> ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B))))
24	11, 23	anim12d 560	. . . . 5 |- ((A e. S /\ B e. T /\ C e. U) -> ((A.xA.yA.z((xRy /\ yRz) -> xRz) /\ (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y))) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
25	24	com12 11	. . . 4 |- ((A.xA.yA.z((xRy /\ yRz) -> xRz) /\ (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y))) -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
26		cotr 3442	. . . 4 |- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
27		asymref2 3446	. . . 4 |- ((R i^i `'R) = (I |` U.U.R) <-> (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)))
28	25, 26, 27	syl2anb 457	. . 3 |- (((R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R)) -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
29	28	3adant1 799	. 2 |- ((Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R)) -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
30	2, 29	syl 10	1 |- (R e. Poset -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777  A.wal 956   = wceq 958   e. wcel 960  A.wral 1648   i^i cin 2049   (_ wss 2050  U.cuni 2507   class class class wbr 2624  Icid 2837  `'ccnv 3175   |` cres 3178   o. ccom 3180  Rel wrel 3181  Posetcps 8629

This theorem is referenced by:  psdmrn 8644  psref 8645  psasym 8647  pstr 8648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ps 8635

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