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Theorem paddss1 30628
Description: Subset law for projective subspace sum. (unss1 3357 analog.) (Contributed by NM, 7-Mar-2012.)
Hypotheses
Ref Expression
padd0.a  |-  A  =  ( Atoms `  K )
padd0.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddss1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )

Proof of Theorem paddss1
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3187 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim1d 812 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  X  \/  p  e.  Z
)  ->  ( p  e.  Y  \/  p  e.  Z ) ) )
3 ssrexv 3251 . . . . . . 7  |-  ( X 
C_  Y  ->  ( E. q  e.  X  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r )  ->  E. q  e.  Y  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) )
43anim2d 548 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) )  ->  ( p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) ) )
52, 4orim12d 811 . . . . 5  |-  ( X 
C_  Y  ->  (
( ( p  e.  X  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) )  ->  ( ( p  e.  Y  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
65adantl 452 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( ( ( p  e.  X  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) )  ->  ( ( p  e.  Y  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
7 simpl1 958 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
8 sstr 3200 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983ad2antr2 1121 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
109ancoms 439 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
11 simpl3 960 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
12 eqid 2296 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2296 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
14 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
15 padd0.p . . . . . 6  |-  .+  =  ( + P `  K
)
1612, 13, 14, 15elpadd 30610 . . . . 5  |-  ( ( K  e.  B  /\  X  C_  A  /\  Z  C_  A )  ->  (
p  e.  ( X 
.+  Z )  <->  ( (
p  e.  X  \/  p  e.  Z )  \/  ( p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) ) ) )
177, 10, 11, 16syl3anc 1182 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( X  .+  Z )  <-> 
( ( p  e.  X  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
1812, 13, 14, 15elpadd 30610 . . . . 5  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  (
p  e.  ( Y 
.+  Z )  <->  ( (
p  e.  Y  \/  p  e.  Z )  \/  ( p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) ) ) )
1918adantr 451 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Y  .+  Z )  <-> 
( ( p  e.  Y  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
206, 17, 193imtr4d 259 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( X  .+  Z )  ->  p  e.  ( Y  .+  Z ) ) )
2120ssrdv 3198 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( X  .+  Z
)  C_  ( Y  .+  Z ) )
2221ex 423 1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   + Pcpadd 30606
This theorem is referenced by:  paddss12  30630  paddasslem12  30642  pmod1i  30659  pl42lem3N  30792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-padd 30607
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