Theorem paddvaln0N 35087

Description: Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
paddfval.l	|- .<_ = ( le ` K )
paddfval.j	|- .\/ = ( join ` K )
paddfval.a	|- A = ( Atoms ` K )
paddfval.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
paddvaln0N	|- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( X .+ Y ) = { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } )

Distinct variable groups:   A, p, q, r   K, p   r, q, K   X, p, q   Y, p, q, r   .\/ , p   .<_ , p   A, q, r   .\/ , q, r   .<_ , q, r   X, r

Allowed substitution hints:   .+ ( r, q, p)

Proof of Theorem paddvaln0N

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		paddfval.l	. . . 4 |- .<_ = ( le ` K )
2		paddfval.j	. . . 4 |- .\/ = ( join ` K )
3		paddfval.a	. . . 4 |- A = ( Atoms ` K )
4		paddfval.p	. . . 4 |- .+ = ( +P ` K )
5	1, 2, 3, 4	elpaddn0 35086	. . 3 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( s e. ( X .+ Y ) <-> ( s e. A /\ E. q e. X E. r e. Y s .<_ ( q .\/ r ) ) ) )
6		breq1 4656	. . . . 5 |- ( p = s -> ( p .<_ ( q .\/ r ) <-> s .<_ ( q .\/ r ) ) )
7	6	2rexbidv 3057	. . . 4 |- ( p = s -> ( E. q e. X E. r e. Y p .<_ ( q .\/ r ) <-> E. q e. X E. r e. Y s .<_ ( q .\/ r ) ) )
8	7	elrab 3363	. . 3 |- ( s e. { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } <-> ( s e. A /\ E. q e. X E. r e. Y s .<_ ( q .\/ r ) ) )
9	5, 8	syl6bbr 278	. 2 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( s e. ( X .+ Y ) <-> s e. { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } ) )
10	9	eqrdv 2620	1 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( X .+ Y ) = { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   +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-lub 16974  df-join 16976  df-lat 17046  df-ats 34554  df-padd 35082

This theorem is referenced by: (None)