Theorem paddass 35124

Description: Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (Contributed by NM, 29-Dec-2011.)

Hypotheses

Ref	Expression
paddass.a	|- A = ( Atoms ` K )
paddass.p	|- .+ = ( +P ` K )

Assertion

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

Proof of Theorem paddass

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> K e. HL )
2		simpr3 1069	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> Z C_ A )
3		simpr2 1068	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> Y C_ A )
4		simpr1 1067	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> X C_ A )
5		paddass.a	. . . . 5 |- A = ( Atoms ` K )
6		paddass.p	. . . . 5 |- .+ = ( +P ` K )
7	5, 6	paddasslem18 35123	. . . 4 |- ( ( K e. HL /\ ( Z C_ A /\ Y C_ A /\ X C_ A ) ) -> ( Z .+ ( Y .+ X ) ) C_ ( ( Z .+ Y ) .+ X ) )
8	1, 2, 3, 4, 7	syl13anc 1328	. . 3 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Z .+ ( Y .+ X ) ) C_ ( ( Z .+ Y ) .+ X ) )
9		hllat 34650	. . . . . . 7 |- ( K e. HL -> K e. Lat )
10	5, 6	paddcom 35099	. . . . . . 7 |- ( ( K e. Lat /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) = ( Y .+ X ) )
11	9, 10	syl3an1 1359	. . . . . 6 |- ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) = ( Y .+ X ) )
12	11	3adant3r3 1276	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ Y ) = ( Y .+ X ) )
13	12	oveq1d 6665	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( X .+ Y ) .+ Z ) = ( ( Y .+ X ) .+ Z ) )
14	5, 6	paddssat 35100	. . . . . 6 |- ( ( K e. HL /\ Y C_ A /\ X C_ A ) -> ( Y .+ X ) C_ A )
15	1, 3, 4, 14	syl3anc 1326	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Y .+ X ) C_ A )
16	5, 6	paddcom 35099	. . . . . 6 |- ( ( K e. Lat /\ ( Y .+ X ) C_ A /\ Z C_ A ) -> ( ( Y .+ X ) .+ Z ) = ( Z .+ ( Y .+ X ) ) )
17	9, 16	syl3an1 1359	. . . . 5 |- ( ( K e. HL /\ ( Y .+ X ) C_ A /\ Z C_ A ) -> ( ( Y .+ X ) .+ Z ) = ( Z .+ ( Y .+ X ) ) )
18	1, 15, 2, 17	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( Y .+ X ) .+ Z ) = ( Z .+ ( Y .+ X ) ) )
19	13, 18	eqtrd 2656	. . 3 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( X .+ Y ) .+ Z ) = ( Z .+ ( Y .+ X ) ) )
20	5, 6	paddcom 35099	. . . . . . 7 |- ( ( K e. Lat /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) = ( Z .+ Y ) )
21	9, 20	syl3an1 1359	. . . . . 6 |- ( ( K e. HL /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) = ( Z .+ Y ) )
22	21	3adant3r1 1274	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Y .+ Z ) = ( Z .+ Y ) )
23	22	oveq2d 6666	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( Y .+ Z ) ) = ( X .+ ( Z .+ Y ) ) )
24	5, 6	paddssat 35100	. . . . . 6 |- ( ( K e. HL /\ Z C_ A /\ Y C_ A ) -> ( Z .+ Y ) C_ A )
25	1, 2, 3, 24	syl3anc 1326	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Z .+ Y ) C_ A )
26	5, 6	paddcom 35099	. . . . . 6 |- ( ( K e. Lat /\ X C_ A /\ ( Z .+ Y ) C_ A ) -> ( X .+ ( Z .+ Y ) ) = ( ( Z .+ Y ) .+ X ) )
27	9, 26	syl3an1 1359	. . . . 5 |- ( ( K e. HL /\ X C_ A /\ ( Z .+ Y ) C_ A ) -> ( X .+ ( Z .+ Y ) ) = ( ( Z .+ Y ) .+ X ) )
28	1, 4, 25, 27	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( Z .+ Y ) ) = ( ( Z .+ Y ) .+ X ) )
29	23, 28	eqtrd 2656	. . 3 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( Y .+ Z ) ) = ( ( Z .+ Y ) .+ X ) )
30	8, 19, 29	3sstr4d 3648	. 2 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( X .+ Y ) .+ Z ) C_ ( X .+ ( Y .+ Z ) ) )
31	5, 6	paddasslem18 35123	. 2 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )
32	30, 31	eqssd 3620	1 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Latclat 17045   Atomscatm 34550   HLchlt 34637   +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-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-padd 35082

This theorem is referenced by:  padd12N  35125  padd4N  35126  pmodl42N  35137  pmapjlln1  35141