Theorem pmapjlln1 35141

Description: The projective map of the join of a lattice element and a lattice line (expressed as the join Q .\/ R of two atoms). (Contributed by NM, 16-Sep-2012.)

Hypotheses

Ref	Expression
pmapjat.b	|- B = ( Base ` K )
pmapjat.j	|- .\/ = ( join ` K )
pmapjat.a	|- A = ( Atoms ` K )
pmapjat.m	|- M = ( pmap ` K )
pmapjat.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
pmapjlln1	|- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ ( Q .\/ R ) ) ) = ( ( M ` X ) .+ ( M ` ( Q .\/ R ) ) ) )

Proof of Theorem pmapjlln1

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> K e. HL )
2		pmapjat.b	. . . . 5 |- B = ( Base ` K )
3		pmapjat.a	. . . . 5 |- A = ( Atoms ` K )
4		pmapjat.m	. . . . 5 |- M = ( pmap ` K )
5	2, 3, 4	pmapssat 35045	. . . 4 |- ( ( K e. HL /\ X e. B ) -> ( M ` X ) C_ A )
6	5	3ad2antr1 1226	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` X ) C_ A )
7		simpr2 1068	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> Q e. A )
8	2, 3	atbase 34576	. . . . 5 |- ( Q e. A -> Q e. B )
9	7, 8	syl 17	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> Q e. B )
10	2, 3, 4	pmapssat 35045	. . . 4 |- ( ( K e. HL /\ Q e. B ) -> ( M ` Q ) C_ A )
11	9, 10	syldan 487	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` Q ) C_ A )
12		simpr3 1069	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> R e. A )
13	2, 3	atbase 34576	. . . . 5 |- ( R e. A -> R e. B )
14	12, 13	syl 17	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> R e. B )
15	2, 3, 4	pmapssat 35045	. . . 4 |- ( ( K e. HL /\ R e. B ) -> ( M ` R ) C_ A )
16	14, 15	syldan 487	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` R ) C_ A )
17		pmapjat.p	. . . 4 |- .+ = ( +P ` K )
18	3, 17	paddass 35124	. . 3 |- ( ( K e. HL /\ ( ( M ` X ) C_ A /\ ( M ` Q ) C_ A /\ ( M ` R ) C_ A ) ) -> ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) = ( ( M ` X ) .+ ( ( M ` Q ) .+ ( M ` R ) ) ) )
19	1, 6, 11, 16, 18	syl13anc 1328	. 2 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) = ( ( M ` X ) .+ ( ( M ` Q ) .+ ( M ` R ) ) ) )
20		hllat 34650	. . . . . 6 |- ( K e. HL -> K e. Lat )
21	20	adantr 481	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> K e. Lat )
22		simpr1 1067	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> X e. B )
23		pmapjat.j	. . . . . 6 |- .\/ = ( join ` K )
24	2, 23	latjcl 17051	. . . . 5 |- ( ( K e. Lat /\ X e. B /\ Q e. B ) -> ( X .\/ Q ) e. B )
25	21, 22, 9, 24	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( X .\/ Q ) e. B )
26	2, 23, 3, 4, 17	pmapjat1 35139	. . . 4 |- ( ( K e. HL /\ ( X .\/ Q ) e. B /\ R e. A ) -> ( M ` ( ( X .\/ Q ) .\/ R ) ) = ( ( M ` ( X .\/ Q ) ) .+ ( M ` R ) ) )
27	1, 25, 12, 26	syl3anc 1326	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( ( X .\/ Q ) .\/ R ) ) = ( ( M ` ( X .\/ Q ) ) .+ ( M ` R ) ) )
28	2, 23	latjass 17095	. . . . 5 |- ( ( K e. Lat /\ ( X e. B /\ Q e. B /\ R e. B ) ) -> ( ( X .\/ Q ) .\/ R ) = ( X .\/ ( Q .\/ R ) ) )
29	21, 22, 9, 14, 28	syl13anc 1328	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( X .\/ Q ) .\/ R ) = ( X .\/ ( Q .\/ R ) ) )
30	29	fveq2d 6195	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( ( X .\/ Q ) .\/ R ) ) = ( M ` ( X .\/ ( Q .\/ R ) ) ) )
31	2, 23, 3, 4, 17	pmapjat1 35139	. . . . 5 |- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( M ` ( X .\/ Q ) ) = ( ( M ` X ) .+ ( M ` Q ) ) )
32	31	3adant3r3 1276	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ Q ) ) = ( ( M ` X ) .+ ( M ` Q ) ) )
33	32	oveq1d 6665	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( M ` ( X .\/ Q ) ) .+ ( M ` R ) ) = ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) )
34	27, 30, 33	3eqtr3d 2664	. 2 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ ( Q .\/ R ) ) ) = ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) )
35	2, 23, 3, 4, 17	pmapjat1 35139	. . . 4 |- ( ( K e. HL /\ Q e. B /\ R e. A ) -> ( M ` ( Q .\/ R ) ) = ( ( M ` Q ) .+ ( M ` R ) ) )
36	1, 9, 12, 35	syl3anc 1326	. . 3 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( Q .\/ R ) ) = ( ( M ` Q ) .+ ( M ` R ) ) )
37	36	oveq2d 6666	. 2 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( M ` X ) .+ ( M ` ( Q .\/ R ) ) ) = ( ( M ` X ) .+ ( ( M ` Q ) .+ ( M ` R ) ) ) )
38	19, 34, 37	3eqtr4d 2666	1 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ ( Q .\/ R ) ) ) = ( ( M ` X ) .+ ( M ` ( Q .\/ R ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   +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-pmap 34790  df-padd 35082

This theorem is referenced by:  llnmod1i2  35146