Theorem pmod1i 34662

Description: The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)

Hypotheses

Ref	Expression
pmod.a	|- A = ( Atoms ` K )
pmod.s	|- S = ( PSubSp ` K )
pmod.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
pmod1i	|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> ( X C_ Z -> ( ( X .+ Y ) i^i Z ) = ( X .+ ( Y i^i Z ) ) ) )

Proof of Theorem pmod1i

Step	Hyp	Ref	Expression
1		eqid 2467	. . . . 5 |- ( le ` K ) = ( le ` K )
2		eqid 2467	. . . . 5 |- ( join ` K ) = ( join ` K )
3		pmod.a	. . . . 5 |- A = ( Atoms ` K )
4		pmod.s	. . . . 5 |- S = ( PSubSp ` K )
5		pmod.p	. . . . 5 |- .+ = ( +P ` K )
6	1, 2, 3, 4, 5	pmodlem2 34661	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) -> ( ( X .+ Y ) i^i Z ) C_ ( X .+ ( Y i^i Z ) ) )
7	6	3expa 1196	. . 3 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( ( X .+ Y ) i^i Z ) C_ ( X .+ ( Y i^i Z ) ) )
8		inss1 3718	. . . . 5 |- ( Y i^i Z ) C_ Y
9		simpll 753	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> K e. HL )
10		simplr2 1039	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Y C_ A )
11		simplr1 1038	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> X C_ A )
12	3, 5	paddss2 34632	. . . . . 6 |- ( ( K e. HL /\ Y C_ A /\ X C_ A ) -> ( ( Y i^i Z ) C_ Y -> ( X .+ ( Y i^i Z ) ) C_ ( X .+ Y ) ) )
13	9, 10, 11, 12	syl3anc 1228	. . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( ( Y i^i Z ) C_ Y -> ( X .+ ( Y i^i Z ) ) C_ ( X .+ Y ) ) )
14	8, 13	mpi 17	. . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( X .+ ( Y i^i Z ) ) C_ ( X .+ Y ) )
15		simpl 457	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> K e. HL )
16	3, 4	psubssat 34568	. . . . . . . 8 |- ( ( K e. HL /\ Z e. S ) -> Z C_ A )
17	16	3ad2antr3 1163	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Z C_ A )
18		simpr2 1003	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Y C_ A )
19		ssinss1 3726	. . . . . . . 8 |- ( Y C_ A -> ( Y i^i Z ) C_ A )
20	18, 19	syl 16	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> ( Y i^i Z ) C_ A )
21	3, 5	paddss1 34631	. . . . . . 7 |- ( ( K e. HL /\ Z C_ A /\ ( Y i^i Z ) C_ A ) -> ( X C_ Z -> ( X .+ ( Y i^i Z ) ) C_ ( Z .+ ( Y i^i Z ) ) ) )
22	15, 17, 20, 21	syl3anc 1228	. . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> ( X C_ Z -> ( X .+ ( Y i^i Z ) ) C_ ( Z .+ ( Y i^i Z ) ) ) )
23	22	imp 429	. . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( X .+ ( Y i^i Z ) ) C_ ( Z .+ ( Y i^i Z ) ) )
24		simplr3 1040	. . . . . . . 8 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Z e. S )
25	9, 24, 16	syl2anc 661	. . . . . . 7 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Z C_ A )
26		inss2 3719	. . . . . . . 8 |- ( Y i^i Z ) C_ Z
27	3, 5	paddss2 34632	. . . . . . . 8 |- ( ( K e. HL /\ Z C_ A /\ Z C_ A ) -> ( ( Y i^i Z ) C_ Z -> ( Z .+ ( Y i^i Z ) ) C_ ( Z .+ Z ) ) )
28	26, 27	mpi 17	. . . . . . 7 |- ( ( K e. HL /\ Z C_ A /\ Z C_ A ) -> ( Z .+ ( Y i^i Z ) ) C_ ( Z .+ Z ) )
29	9, 25, 25, 28	syl3anc 1228	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( Z .+ ( Y i^i Z ) ) C_ ( Z .+ Z ) )
30	4, 5	paddidm 34655	. . . . . . 7 |- ( ( K e. HL /\ Z e. S ) -> ( Z .+ Z ) = Z )
31	9, 24, 30	syl2anc 661	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( Z .+ Z ) = Z )
32	29, 31	sseqtrd 3540	. . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( Z .+ ( Y i^i Z ) ) C_ Z )
33	23, 32	sstrd 3514	. . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( X .+ ( Y i^i Z ) ) C_ Z )
34	14, 33	ssind 3722	. . 3 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( X .+ ( Y i^i Z ) ) C_ ( ( X .+ Y ) i^i Z ) )
35	7, 34	eqssd 3521	. 2 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( ( X .+ Y ) i^i Z ) = ( X .+ ( Y i^i Z ) ) )
36	35	ex 434	1 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> ( X C_ Z -> ( ( X .+ Y ) i^i Z ) = ( X .+ ( Y i^i Z ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   i^i cin 3475   C_ wss 3476   ` cfv 5588  (class class class)co 6284   lecple 14562   joincjn 15431   Atomscatm 34078   HLchlt 34165   PSubSpcpsubsp 34310   +Pcpadd 34609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-lat 15533  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-psubsp 34317  df-padd 34610

This theorem is referenced by:  pmod2iN  34663  pmodN  34664  pmodl42N  34665  hlmod1i  34670