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Theorem addcmpblnr 9446
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 6293 . 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 9396 . . . . . . . 8  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  +P.  F
)  e.  P. )
3 addclpr 9396 . . . . . . . 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 824 . . . . . 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 9396 . . . . . . . 8  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  +P.  R
)  e.  P. )
7 addclpr 9396 . . . . . . . 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 824 . . . . . 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 824 . . . 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 9441 . . . 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 9399 . . . . . . . 8  |-  ( F  +P.  D )  =  ( D  +P.  F
)
1514oveq1i 6294 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( ( D  +P.  F )  +P.  S )
16 addasspr 9400 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( F  +P.  ( D  +P.  S ) )
17 addasspr 9400 . . . . . . 7  |-  ( ( D  +P.  F )  +P.  S )  =  ( D  +P.  ( F  +P.  S ) )
1815, 16, 173eqtr3i 2504 . . . . . 6  |-  ( F  +P.  ( D  +P.  S ) )  =  ( D  +P.  ( F  +P.  S ) )
1918oveq2i 6295 . . . . 5  |-  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
20 addasspr 9400 . . . . 5  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )
21 addasspr 9400 . . . . 5  |-  ( ( A  +P.  D )  +P.  ( F  +P.  S ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
2219, 20, 213eqtr4i 2506 . . . 4  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( ( A  +P.  D
)  +P.  ( F  +P.  S ) )
23 addcompr 9399 . . . . . . . 8  |-  ( G  +P.  C )  =  ( C  +P.  G
)
2423oveq1i 6294 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( ( C  +P.  G )  +P.  R )
25 addasspr 9400 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( G  +P.  ( C  +P.  R ) )
26 addasspr 9400 . . . . . . 7  |-  ( ( C  +P.  G )  +P.  R )  =  ( C  +P.  ( G  +P.  R ) )
2724, 25, 263eqtr3i 2504 . . . . . 6  |-  ( G  +P.  ( C  +P.  R ) )  =  ( C  +P.  ( G  +P.  R ) )
2827oveq2i 6295 . . . . 5  |-  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
29 addasspr 9400 . . . . 5  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )
30 addasspr 9400 . . . . 5  |-  ( ( B  +P.  C )  +P.  ( G  +P.  R ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
3128, 29, 303eqtr4i 2506 . . . 4  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( ( B  +P.  C
)  +P.  ( G  +P.  R ) )
3222, 31eqeq12i 2487 . . 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 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447  (class class class)co 6284   P.cnp 9237    +P. cpp 9239    ~R cer 9242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-omul 7135  df-er 7311  df-ni 9250  df-pli 9251  df-mi 9252  df-lti 9253  df-plpq 9286  df-mpq 9287  df-ltpq 9288  df-enq 9289  df-nq 9290  df-erq 9291  df-plq 9292  df-mq 9293  df-1nq 9294  df-rq 9295  df-ltnq 9296  df-np 9359  df-plp 9361  df-enr 9433
This theorem is referenced by:  addsrmo  9450
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