Theorem pclssN 36034

Description: Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclss.a	|- A = ( Atoms ` K )
pclss.c	|- U = ( PCl ` K )

Assertion

Ref	Expression
pclssN	|- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) C_ ( U ` Y ) )

Proof of Theorem pclssN

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3496	. . . . . 6 |- ( X C_ Y -> ( Y C_ y -> X C_ y ) )
2	1	3ad2ant2 1016	. . . . 5 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( Y C_ y -> X C_ y ) )
3	2	adantr 463	. . . 4 |- ( ( ( K e. V /\ X C_ Y /\ Y C_ A ) /\ y e. ( PSubSp ` K ) ) -> ( Y C_ y -> X C_ y ) )
4	3	ss2rabdv 3567	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> { y e. ( PSubSp ` K ) | Y C_ y } C_ { y e. ( PSubSp ` K ) | X C_ y } )
5		intss 4292	. . 3 |- ( { y e. ( PSubSp ` K ) | Y C_ y } C_ { y e. ( PSubSp ` K ) | X C_ y } -> |^| { y e. ( PSubSp ` K ) | X C_ y } C_ |^| { y e. ( PSubSp ` K ) | Y C_ y } )
6	4, 5	syl 16	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> |^| { y e. ( PSubSp ` K ) | X C_ y } C_ |^| { y e. ( PSubSp ` K ) | Y C_ y } )
7		simp1 994	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> K e. V )
8		sstr 3497	. . . 4 |- ( ( X C_ Y /\ Y C_ A ) -> X C_ A )
9	8	3adant1 1012	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> X C_ A )
10		pclss.a	. . . 4 |- A = ( Atoms ` K )
11		eqid 2454	. . . 4 |- ( PSubSp ` K ) = ( PSubSp ` K )
12		pclss.c	. . . 4 |- U = ( PCl ` K )
13	10, 11, 12	pclvalN 36030	. . 3 |- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
14	7, 9, 13	syl2anc 659	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
15	10, 11, 12	pclvalN 36030	. . 3 |- ( ( K e. V /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
16	15	3adant2 1013	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
17	6, 14, 16	3sstr4d 3532	1 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) C_ ( U ` Y ) )

Syntax hints:   -> wi 4   /\ w3a 971   = wceq 1398   e. wcel 1823   {crab 2808   C_ wss 3461   |^|cint 4271   ` cfv 5570   Atomscatm 35404   PSubSpcpsubsp 35636   PClcpclN 36027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-psubsp 35643  df-pclN 36028

This theorem is referenced by:  pclbtwnN  36037  pclunN  36038  pclfinN  36040  pclss2polN  36061  pclfinclN  36090