Theorem pmod1i 35134

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 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
2		eqid 2622	. . . . 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 35133	. . . 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 1265	. . 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 3833	. . . . 5 |- ( Y i^i Z ) C_ Y
9		simpll 790	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> K e. HL )
10		simplr2 1104	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Y C_ A )
11		simplr1 1103	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> X C_ A )
12	3, 5	paddss2 35104	. . . . . 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 1326	. . . . 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 20	. . . 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 473	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> K e. HL )
16	3, 4	psubssat 35040	. . . . . . . 8 |- ( ( K e. HL /\ Z e. S ) -> Z C_ A )
17	16	3ad2antr3 1228	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Z C_ A )
18		simpr2 1068	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Y C_ A )
19		ssinss1 3841	. . . . . . . 8 |- ( Y C_ A -> ( Y i^i Z ) C_ A )
20	18, 19	syl 17	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> ( Y i^i Z ) C_ A )
21	3, 5	paddss1 35103	. . . . . . 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 1326	. . . . . 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 445	. . . . 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 1105	. . . . . . . 8 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Z e. S )
25	9, 24, 16	syl2anc 693	. . . . . . 7 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Z C_ A )
26		inss2 3834	. . . . . . . 8 |- ( Y i^i Z ) C_ Z
27	3, 5	paddss2 35104	. . . . . . . 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 20	. . . . . . 7 |- ( ( K e. HL /\ Z C_ A /\ Z C_ A ) -> ( Z .+ ( Y i^i Z ) ) C_ ( Z .+ Z ) )
29	9, 25, 25, 28	syl3anc 1326	. . . . . 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 35127	. . . . . . 7 |- ( ( K e. HL /\ Z e. S ) -> ( Z .+ Z ) = Z )
31	9, 24, 30	syl2anc 693	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( Z .+ Z ) = Z )
32	29, 31	sseqtrd 3641	. . . . 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 3613	. . . 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 3837	. . 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 3620	. 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 450	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637   PSubSpcpsubsp 34782   +Pcpadd 35081

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  ax-un 6949

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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-padd 35082

This theorem is referenced by:  pmod2iN  35135  pmodN  35136  pmodl42N  35137  hlmod1i  35142