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Theorem addcmpblnr 8710
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 5883 . 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 8658 . . . . . . . 8  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  +P.  F
)  e.  P. )
3 addclpr 8658 . . . . . . . 8  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  +P.  G
)  e.  P. )
42, 3anim12i 549 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. )
)  ->  ( ( A  +P.  F )  e. 
P.  /\  ( B  +P.  G )  e.  P. ) )
54an4s 799 . . . . . 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 8658 . . . . . . . 8  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  +P.  R
)  e.  P. )
7 addclpr 8658 . . . . . . . 8  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  +P.  S
)  e.  P. )
86, 7anim12i 549 . . . . . . 7  |-  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  +P.  R )  e. 
P.  /\  ( D  +P.  S )  e.  P. ) )
98an4s 799 . . . . . 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 549 . . . . 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 799 . . . 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 8705 . . . 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 15 . . 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 8661 . . . . . . . 8  |-  ( F  +P.  D )  =  ( D  +P.  F
)
1514oveq1i 5884 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( ( D  +P.  F )  +P.  S )
16 addasspr 8662 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( F  +P.  ( D  +P.  S ) )
17 addasspr 8662 . . . . . . 7  |-  ( ( D  +P.  F )  +P.  S )  =  ( D  +P.  ( F  +P.  S ) )
1815, 16, 173eqtr3i 2324 . . . . . 6  |-  ( F  +P.  ( D  +P.  S ) )  =  ( D  +P.  ( F  +P.  S ) )
1918oveq2i 5885 . . . . 5  |-  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
20 addasspr 8662 . . . . 5  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )
21 addasspr 8662 . . . . 5  |-  ( ( A  +P.  D )  +P.  ( F  +P.  S ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
2219, 20, 213eqtr4i 2326 . . . 4  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( ( A  +P.  D
)  +P.  ( F  +P.  S ) )
23 addcompr 8661 . . . . . . . 8  |-  ( G  +P.  C )  =  ( C  +P.  G
)
2423oveq1i 5884 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( ( C  +P.  G )  +P.  R )
25 addasspr 8662 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( G  +P.  ( C  +P.  R ) )
26 addasspr 8662 . . . . . . 7  |-  ( ( C  +P.  G )  +P.  R )  =  ( C  +P.  ( G  +P.  R ) )
2724, 25, 263eqtr3i 2324 . . . . . 6  |-  ( G  +P.  ( C  +P.  R ) )  =  ( C  +P.  ( G  +P.  R ) )
2827oveq2i 5885 . . . . 5  |-  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
29 addasspr 8662 . . . . 5  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )
30 addasspr 8662 . . . . 5  |-  ( ( B  +P.  C )  +P.  ( G  +P.  R ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
3128, 29, 303eqtr4i 2326 . . . 4  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( ( B  +P.  C
)  +P.  ( G  +P.  R ) )
3222, 31eqeq12i 2309 . . 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 252 . 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 212 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039  (class class class)co 5874   P.cnp 8497    +P. cpp 8499    ~R cer 8504
This theorem is referenced by:  addsrpr  8713
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623  df-enr 8697
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