Theorem pclssN 33841

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 3458	. . . . . 6 |- ( X C_ Y -> ( Y C_ y -> X C_ y ) )
2	1	3ad2ant2 1010	. . . . 5 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( Y C_ y -> X C_ y ) )
3	2	adantr 465	. . . 4 |- ( ( ( K e. V /\ X C_ Y /\ Y C_ A ) /\ y e. ( PSubSp ` K ) ) -> ( Y C_ y -> X C_ y ) )
4	3	ss2rabdv 3528	. . 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 4244	. . 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 988	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> K e. V )
8		sstr 3459	. . . 4 |- ( ( X C_ Y /\ Y C_ A ) -> X C_ A )
9	8	3adant1 1006	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> X C_ A )
10		pclss.a	. . . 4 |- A = ( Atoms ` K )
11		eqid 2451	. . . 4 |- ( PSubSp ` K ) = ( PSubSp ` K )
12		pclss.c	. . . 4 |- U = ( PCl ` K )
13	10, 11, 12	pclvalN 33837	. . 3 |- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
14	7, 9, 13	syl2anc 661	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
15	10, 11, 12	pclvalN 33837	. . 3 |- ( ( K e. V /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
16	15	3adant2 1007	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
17	6, 14, 16	3sstr4d 3494	1 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) C_ ( U ` Y ) )

Syntax hints:   -> wi 4   /\ w3a 965   = wceq 1370   e. wcel 1758   {crab 2797   C_ wss 3423   |^|cint 4223   ` cfv 5513   Atomscatm 33211   PSubSpcpsubsp 33443   PClcpclN 33834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-psubsp 33450  df-pclN 33835

This theorem is referenced by:  pclbtwnN  33844  pclunN  33845  pclfinN  33847  pclss2polN  33868  pclfinclN  33897