Theorem pmod1i 33484

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 2471	. . . . 5 |- ( le ` K ) = ( le ` K )
2		eqid 2471	. . . . 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 33483	. . . 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 1231	. . 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 3643	. . . . 5 |- ( Y i^i Z ) C_ Y
9		simpll 768	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> K e. HL )
10		simplr2 1073	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Y C_ A )
11		simplr1 1072	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> X C_ A )
12	3, 5	paddss2 33454	. . . . . 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 1292	. . . . 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 464	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> K e. HL )
16	3, 4	psubssat 33390	. . . . . . . 8 |- ( ( K e. HL /\ Z e. S ) -> Z C_ A )
17	16	3ad2antr3 1197	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Z C_ A )
18		simpr2 1037	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Y C_ A )
19		ssinss1 3651	. . . . . . . 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 33453	. . . . . . 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 1292	. . . . . 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 436	. . . . 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 1074	. . . . . . . 8 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Z e. S )
25	9, 24, 16	syl2anc 673	. . . . . . 7 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Z C_ A )
26		inss2 3644	. . . . . . . 8 |- ( Y i^i Z ) C_ Z
27	3, 5	paddss2 33454	. . . . . . . 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 1292	. . . . . 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 33477	. . . . . . 7 |- ( ( K e. HL /\ Z e. S ) -> ( Z .+ Z ) = Z )
31	9, 24, 30	syl2anc 673	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> ( Z .+ Z ) = Z )
32	29, 31	sseqtrd 3454	. . . . 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 3428	. . . 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 3647	. . 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 3435	. 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 441	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 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   i^i cin 3389   C_ wss 3390   ` cfv 5589  (class class class)co 6308   lecple 15275   joincjn 16267   Atomscatm 32900   HLchlt 32987   PSubSpcpsubsp 33132   +Pcpadd 33431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-lat 16370  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-psubsp 33139  df-padd 33432

This theorem is referenced by:  pmod2iN  33485  pmodN  33486  pmodl42N  33487  hlmod1i  33492