Theorem pclssN 33371

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 3414	. . . . . 6 |- ( X C_ Y -> ( Y C_ y -> X C_ y ) )
2	1	3ad2ant2 1027	. . . . 5 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( Y C_ y -> X C_ y ) )
3	2	adantr 466	. . . 4 |- ( ( ( K e. V /\ X C_ Y /\ Y C_ A ) /\ y e. ( PSubSp ` K ) ) -> ( Y C_ y -> X C_ y ) )
4	3	ss2rabdv 3485	. . 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 4219	. . 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 1005	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> K e. V )
8		sstr 3415	. . . 4 |- ( ( X C_ Y /\ Y C_ A ) -> X C_ A )
9	8	3adant1 1023	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> X C_ A )
10		pclss.a	. . . 4 |- A = ( Atoms ` K )
11		eqid 2428	. . . 4 |- ( PSubSp ` K ) = ( PSubSp ` K )
12		pclss.c	. . . 4 |- U = ( PCl ` K )
13	10, 11, 12	pclvalN 33367	. . 3 |- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
14	7, 9, 13	syl2anc 665	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
15	10, 11, 12	pclvalN 33367	. . 3 |- ( ( K e. V /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
16	15	3adant2 1024	. 2 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` Y ) = |^| { y e. ( PSubSp ` K ) | Y C_ y } )
17	6, 14, 16	3sstr4d 3450	1 |- ( ( K e. V /\ X C_ Y /\ Y C_ A ) -> ( U ` X ) C_ ( U ` Y ) )

Syntax hints:   -> wi 4   /\ w3a 982   = wceq 1437   e. wcel 1872   {crab 2718   C_ wss 3379   |^|cint 4198   ` cfv 5544   Atomscatm 32741   PSubSpcpsubsp 32973   PClcpclN 33364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-psubsp 32980  df-pclN 33365

This theorem is referenced by:  pclbtwnN  33374  pclunN  33375  pclfinN  33377  pclss2polN  33398  pclfinclN  33427