Theorem pmod1i 33588

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 2443	. . . . 5 |- ( le ` K ) = ( le ` K )
2		eqid 2443	. . . . 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 33587	. . . 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 1187	. . 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 3591	. . . . 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 1031	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> Y C_ A )
11		simplr1 1030	. . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) /\ X C_ Z ) -> X C_ A )
12	3, 5	paddss2 33558	. . . . . 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 1218	. . . . 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 33494	. . . . . . . 8 |- ( ( K e. HL /\ Z e. S ) -> Z C_ A )
17	16	3ad2antr3 1155	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Z C_ A )
18		simpr2 995	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) ) -> Y C_ A )
19		ssinss1 3599	. . . . . . . 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 33557	. . . . . . 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 1218	. . . . . 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 1032	. . . . . . . 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 3592	. . . . . . . 8 |- ( Y i^i Z ) C_ Z
27	3, 5	paddss2 33558	. . . . . . . 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 1218	. . . . . 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 33581	. . . . . . 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 3413	. . . . 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 3387	. . . 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 3595	. . 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 3394	. 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 965   = wceq 1369   e. wcel 1756   i^i cin 3348   C_ wss 3349   ` cfv 5439  (class class class)co 6112   lecple 14266   joincjn 15135   Atomscatm 33004   HLchlt 33091   PSubSpcpsubsp 33236   +Pcpadd 33535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-lat 15237  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-psubsp 33243  df-padd 33536

This theorem is referenced by:  pmod2iN  33589  pmodN  33590  pmodl42N  33591  hlmod1i  33596