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Theorem addcmpblnr 9343
Description: Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addcmpblnr  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R
) )  ->  <. ( A  +P.  F ) ,  ( B  +P.  G
) >.  ~R  <. ( C  +P.  R ) ,  ( D  +P.  S
) >. ) )

Proof of Theorem addcmpblnr
StepHypRef Expression
1 oveq12 6202 . 2  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( A  +P.  D
)  +P.  ( F  +P.  S ) )  =  ( ( B  +P.  C )  +P.  ( G  +P.  R ) ) )
2 addclpr 9291 . . . . . . . 8  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  +P.  F
)  e.  P. )
3 addclpr 9291 . . . . . . . 8  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  +P.  G
)  e.  P. )
42, 3anim12i 566 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. )
)  ->  ( ( A  +P.  F )  e. 
P.  /\  ( B  +P.  G )  e.  P. ) )
54an4s 822 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( ( A  +P.  F )  e. 
P.  /\  ( B  +P.  G )  e.  P. ) )
6 addclpr 9291 . . . . . . . 8  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  +P.  R
)  e.  P. )
7 addclpr 9291 . . . . . . . 8  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  +P.  S
)  e.  P. )
86, 7anim12i 566 . . . . . . 7  |-  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  +P.  R )  e. 
P.  /\  ( D  +P.  S )  e.  P. ) )
98an4s 822 . . . . . 6  |-  ( ( ( C  e.  P.  /\  D  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  +P.  R )  e. 
P.  /\  ( D  +P.  S )  e.  P. ) )
105, 9anim12i 566 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. ) )  /\  ( ( C  e. 
P.  /\  D  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( ( ( A  +P.  F )  e.  P.  /\  ( B  +P.  G )  e. 
P. )  /\  (
( C  +P.  R
)  e.  P.  /\  ( D  +P.  S )  e.  P. ) ) )
1110an4s 822 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( ( ( A  +P.  F )  e.  P.  /\  ( B  +P.  G )  e. 
P. )  /\  (
( C  +P.  R
)  e.  P.  /\  ( D  +P.  S )  e.  P. ) ) )
12 enrbreq 9338 . . . 4  |-  ( ( ( ( A  +P.  F )  e.  P.  /\  ( B  +P.  G )  e.  P. )  /\  ( ( C  +P.  R )  e.  P.  /\  ( D  +P.  S )  e.  P. ) )  ->  ( <. ( A  +P.  F ) ,  ( B  +P.  G
) >.  ~R  <. ( C  +P.  R ) ,  ( D  +P.  S
) >. 
<->  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( ( B  +P.  G )  +P.  ( C  +P.  R
) ) ) )
1311, 12syl 16 . . 3  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( <. ( A  +P.  F ) ,  ( B  +P.  G
) >.  ~R  <. ( C  +P.  R ) ,  ( D  +P.  S
) >. 
<->  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( ( B  +P.  G )  +P.  ( C  +P.  R
) ) ) )
14 addcompr 9294 . . . . . . . 8  |-  ( F  +P.  D )  =  ( D  +P.  F
)
1514oveq1i 6203 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( ( D  +P.  F )  +P.  S )
16 addasspr 9295 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( F  +P.  ( D  +P.  S ) )
17 addasspr 9295 . . . . . . 7  |-  ( ( D  +P.  F )  +P.  S )  =  ( D  +P.  ( F  +P.  S ) )
1815, 16, 173eqtr3i 2488 . . . . . 6  |-  ( F  +P.  ( D  +P.  S ) )  =  ( D  +P.  ( F  +P.  S ) )
1918oveq2i 6204 . . . . 5  |-  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
20 addasspr 9295 . . . . 5  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )
21 addasspr 9295 . . . . 5  |-  ( ( A  +P.  D )  +P.  ( F  +P.  S ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
2219, 20, 213eqtr4i 2490 . . . 4  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( ( A  +P.  D
)  +P.  ( F  +P.  S ) )
23 addcompr 9294 . . . . . . . 8  |-  ( G  +P.  C )  =  ( C  +P.  G
)
2423oveq1i 6203 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( ( C  +P.  G )  +P.  R )
25 addasspr 9295 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( G  +P.  ( C  +P.  R ) )
26 addasspr 9295 . . . . . . 7  |-  ( ( C  +P.  G )  +P.  R )  =  ( C  +P.  ( G  +P.  R ) )
2724, 25, 263eqtr3i 2488 . . . . . 6  |-  ( G  +P.  ( C  +P.  R ) )  =  ( C  +P.  ( G  +P.  R ) )
2827oveq2i 6204 . . . . 5  |-  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
29 addasspr 9295 . . . . 5  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )
30 addasspr 9295 . . . . 5  |-  ( ( B  +P.  C )  +P.  ( G  +P.  R ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
3128, 29, 303eqtr4i 2490 . . . 4  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( ( B  +P.  C
)  +P.  ( G  +P.  R ) )
3222, 31eqeq12i 2471 . . 3  |-  ( ( ( A  +P.  F
)  +P.  ( D  +P.  S ) )  =  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  <-> 
( ( A  +P.  D )  +P.  ( F  +P.  S ) )  =  ( ( B  +P.  C )  +P.  ( G  +P.  R
) ) )
3313, 32syl6bb 261 . 2  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( <. ( A  +P.  F ) ,  ( B  +P.  G
) >.  ~R  <. ( C  +P.  R ) ,  ( D  +P.  S
) >. 
<->  ( ( A  +P.  D )  +P.  ( F  +P.  S ) )  =  ( ( B  +P.  C )  +P.  ( G  +P.  R
) ) ) )
341, 33syl5ibr 221 1  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R
) )  ->  <. ( A  +P.  F ) ,  ( B  +P.  G
) >.  ~R  <. ( C  +P.  R ) ,  ( D  +P.  S
) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3984   class class class wbr 4393  (class class class)co 6193   P.cnp 9130    +P. cpp 9132    ~R cer 9137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-omul 7028  df-er 7204  df-ni 9145  df-pli 9146  df-mi 9147  df-lti 9148  df-plpq 9181  df-mpq 9182  df-ltpq 9183  df-enq 9184  df-nq 9185  df-erq 9186  df-plq 9187  df-mq 9188  df-1nq 9189  df-rq 9190  df-ltnq 9191  df-np 9254  df-plp 9256  df-enr 9330
This theorem is referenced by:  addsrpr  9346
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