Theorem paddss2 35104

Description: Subset law for projective subspace sum. (unss2 3784 analog.) (Contributed by NM, 7-Mar-2012.)

Hypotheses

Ref	Expression
padd0.a	|- A = ( Atoms ` K )
padd0.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
paddss2	|- ( ( K e. B /\ Y C_ A /\ Z C_ A ) -> ( X C_ Y -> ( Z .+ X ) C_ ( Z .+ Y ) ) )

Proof of Theorem paddss2

Dummy variables q p r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . . . 7 |- ( X C_ Y -> ( p e. X -> p e. Y ) )
2	1	orim2d 885	. . . . . 6 |- ( X C_ Y -> ( ( p e. Z \/ p e. X ) -> ( p e. Z \/ p e. Y ) ) )
3		ssrexv 3667	. . . . . . . 8 |- ( X C_ Y -> ( E. r e. X p ( le ` K ) ( q ( join ` K ) r ) -> E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) ) )
4	3	reximdv 3016	. . . . . . 7 |- ( X C_ Y -> ( E. q e. Z E. r e. X p ( le ` K ) ( q ( join ` K ) r ) -> E. q e. Z E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) ) )
5	4	anim2d 589	. . . . . 6 |- ( X C_ Y -> ( ( p e. A /\ E. q e. Z E. r e. X p ( le ` K ) ( q ( join ` K ) r ) ) -> ( p e. A /\ E. q e. Z E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) ) ) )
6	2, 5	orim12d 883	. . . . 5 |- ( X C_ Y -> ( ( ( p e. Z \/ p e. X ) \/ ( p e. A /\ E. q e. Z E. r e. X p ( le ` K ) ( q ( join ` K ) r ) ) ) -> ( ( p e. Z \/ p e. Y ) \/ ( p e. A /\ E. q e. Z E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) ) ) ) )
7	6	adantl 482	. . . 4 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> ( ( ( p e. Z \/ p e. X ) \/ ( p e. A /\ E. q e. Z E. r e. X p ( le ` K ) ( q ( join ` K ) r ) ) ) -> ( ( p e. Z \/ p e. Y ) \/ ( p e. A /\ E. q e. Z E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) ) ) ) )
8		simpl1 1064	. . . . 5 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> K e. B )
9		simpl3 1066	. . . . 5 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> Z C_ A )
10		sstr 3611	. . . . . . 7 |- ( ( X C_ Y /\ Y C_ A ) -> X C_ A )
11	10	3ad2antr2 1227	. . . . . 6 |- ( ( X C_ Y /\ ( K e. B /\ Y C_ A /\ Z C_ A ) ) -> X C_ A )
12	11	ancoms 469	. . . . 5 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> X C_ A )
13		eqid 2622	. . . . . 6 |- ( le ` K ) = ( le ` K )
14		eqid 2622	. . . . . 6 |- ( join ` K ) = ( join ` K )
15		padd0.a	. . . . . 6 |- A = ( Atoms ` K )
16		padd0.p	. . . . . 6 |- .+ = ( +P ` K )
17	13, 14, 15, 16	elpadd 35085	. . . . 5 |- ( ( K e. B /\ Z C_ A /\ X C_ A ) -> ( p e. ( Z .+ X ) <-> ( ( p e. Z \/ p e. X ) \/ ( p e. A /\ E. q e. Z E. r e. X p ( le ` K ) ( q ( join ` K ) r ) ) ) ) )
18	8, 9, 12, 17	syl3anc 1326	. . . 4 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> ( p e. ( Z .+ X ) <-> ( ( p e. Z \/ p e. X ) \/ ( p e. A /\ E. q e. Z E. r e. X p ( le ` K ) ( q ( join ` K ) r ) ) ) ) )
19		simpl2 1065	. . . . 5 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> Y C_ A )
20	13, 14, 15, 16	elpadd 35085	. . . . 5 |- ( ( K e. B /\ Z C_ A /\ Y C_ A ) -> ( p e. ( Z .+ Y ) <-> ( ( p e. Z \/ p e. Y ) \/ ( p e. A /\ E. q e. Z E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) ) ) ) )
21	8, 9, 19, 20	syl3anc 1326	. . . 4 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> ( p e. ( Z .+ Y ) <-> ( ( p e. Z \/ p e. Y ) \/ ( p e. A /\ E. q e. Z E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) ) ) ) )
22	7, 18, 21	3imtr4d 283	. . 3 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> ( p e. ( Z .+ X ) -> p e. ( Z .+ Y ) ) )
23	22	ssrdv 3609	. 2 |- ( ( ( K e. B /\ Y C_ A /\ Z C_ A ) /\ X C_ Y ) -> ( Z .+ X ) C_ ( Z .+ Y ) )
24	23	ex 450	1 |- ( ( K e. B /\ Y C_ A /\ Z C_ A ) -> ( X C_ Y -> ( Z .+ X ) C_ ( Z .+ Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-padd 35082

This theorem is referenced by:  paddss12  35105  pmod1i  35134