Theorem pclssN 35180

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 3610	. . . . . 6 |- ( X C_ Y -> ( Y C_ y -> X C_ y ) )
2	1	3ad2ant2 1083	. . . . 5 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( Y C_ y -> X C_ y ) )
3	2	adantr 481	. . . 4 |- ( ( ( K e. V /\ X C_ Y /\ Y C_ A ) /\ y e. ( PSubSp ` K ) ) -> ( Y C_ y -> X C_ y ) )
4	3	ss2rabdv 3683	. . 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 4498	. . 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 17	. 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 1061	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> K e. V )
8		sstr 3611	. . . 4 |- ( ( X C_ Y /\ Y C_ A ) -> X C_ A )
9	8	3adant1 1079	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> X C_ A )
10		pclss.a	. . . 4 |- A = ( Atoms ` K )
11		eqid 2622	. . . 4 |- ( PSubSp ` K ) = ( PSubSp ` K )
12		pclss.c	. . . 4 |- U = ( PCl ` K )
13	10, 11, 12	pclvalN 35176	. . 3 |- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
14	7, 9, 13	syl2anc 693	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
15	10, 11, 12	pclvalN 35176	. . 3 |- ( ( K e. V /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
16	15	3adant2 1080	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
17	6, 14, 16	3sstr4d 3648	1 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) C_ ( U ` Y ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   |^|cint 4475   ` cfv 5888   Atomscatm 34550   PSubSpcpsubsp 34782   PClcpclN 35173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-psubsp 34789  df-pclN 35174

This theorem is referenced by:  pclbtwnN  35183  pclunN  35184  pclfinN  35186  pclss2polN  35207  pclfinclN  35236