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Theorem paddss2 30007
Description: Subset law for projective subspace sum. (unss2 3346 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.
StepHypRef Expression
1 ssel 3174 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim2d 813 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  Z  \/  p  e.  X
)  ->  ( p  e.  Z  \/  p  e.  Y ) ) )
3 ssrexv 3238 . . . . . . . 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 ) ) )
43reximdv 2654 . . . . . . 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 ) ) )
54anim2d 548 . . . . . 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 ) ) ) )
62, 5orim12d 811 . . . . 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 ) ) ) ) )
76adantl 452 . . . 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 958 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
9 simpl3 960 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
10 sstr 3187 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
11103ad2antr2 1121 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
1211ancoms 439 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
13 eqid 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
14 eqid 2283 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
15 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
16 padd0.p . . . . . 6  |-  .+  =  ( + P `  K
)
1713, 14, 15, 16elpadd 29988 . . . . 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 ) ) ) ) )
188, 9, 12, 17syl3anc 1182 . . . 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 959 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Y  C_  A )
2013, 14, 15, 16elpadd 29988 . . . . 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 ) ) ) ) )
218, 9, 19, 20syl3anc 1182 . . . 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 ) ) ) ) )
227, 18, 213imtr4d 259 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Z  .+  X )  ->  p  e.  ( Z  .+  Y ) ) )
2322ssrdv 3185 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( Z  .+  X
)  C_  ( Z  .+  Y ) )
2423ex 423 1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( Z  .+  X )  C_  ( Z  .+  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   + Pcpadd 29984
This theorem is referenced by:  paddss12  30008  pmod1i  30037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-padd 29985
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