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Theorem addcmpblnr 9492
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 6314 . 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 9442 . . . . . . . 8  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  +P.  F
)  e.  P. )
3 addclpr 9442 . . . . . . . 8  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  +P.  G
)  e.  P. )
42, 3anim12i 568 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. )
)  ->  ( ( A  +P.  F )  e. 
P.  /\  ( B  +P.  G )  e.  P. ) )
54an4s 833 . . . . . 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 9442 . . . . . . . 8  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  +P.  R
)  e.  P. )
7 addclpr 9442 . . . . . . . 8  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  +P.  S
)  e.  P. )
86, 7anim12i 568 . . . . . . 7  |-  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  +P.  R )  e. 
P.  /\  ( D  +P.  S )  e.  P. ) )
98an4s 833 . . . . . 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 568 . . . . 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 833 . . . 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 9487 . . . 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 17 . . 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 9445 . . . . . . . 8  |-  ( F  +P.  D )  =  ( D  +P.  F
)
1514oveq1i 6315 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( ( D  +P.  F )  +P.  S )
16 addasspr 9446 . . . . . . 7  |-  ( ( F  +P.  D )  +P.  S )  =  ( F  +P.  ( D  +P.  S ) )
17 addasspr 9446 . . . . . . 7  |-  ( ( D  +P.  F )  +P.  S )  =  ( D  +P.  ( F  +P.  S ) )
1815, 16, 173eqtr3i 2466 . . . . . 6  |-  ( F  +P.  ( D  +P.  S ) )  =  ( D  +P.  ( F  +P.  S ) )
1918oveq2i 6316 . . . . 5  |-  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
20 addasspr 9446 . . . . 5  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( A  +P.  ( F  +P.  ( D  +P.  S ) ) )
21 addasspr 9446 . . . . 5  |-  ( ( A  +P.  D )  +P.  ( F  +P.  S ) )  =  ( A  +P.  ( D  +P.  ( F  +P.  S ) ) )
2219, 20, 213eqtr4i 2468 . . . 4  |-  ( ( A  +P.  F )  +P.  ( D  +P.  S ) )  =  ( ( A  +P.  D
)  +P.  ( F  +P.  S ) )
23 addcompr 9445 . . . . . . . 8  |-  ( G  +P.  C )  =  ( C  +P.  G
)
2423oveq1i 6315 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( ( C  +P.  G )  +P.  R )
25 addasspr 9446 . . . . . . 7  |-  ( ( G  +P.  C )  +P.  R )  =  ( G  +P.  ( C  +P.  R ) )
26 addasspr 9446 . . . . . . 7  |-  ( ( C  +P.  G )  +P.  R )  =  ( C  +P.  ( G  +P.  R ) )
2724, 25, 263eqtr3i 2466 . . . . . 6  |-  ( G  +P.  ( C  +P.  R ) )  =  ( C  +P.  ( G  +P.  R ) )
2827oveq2i 6316 . . . . 5  |-  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
29 addasspr 9446 . . . . 5  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( B  +P.  ( G  +P.  ( C  +P.  R ) ) )
30 addasspr 9446 . . . . 5  |-  ( ( B  +P.  C )  +P.  ( G  +P.  R ) )  =  ( B  +P.  ( C  +P.  ( G  +P.  R ) ) )
3128, 29, 303eqtr4i 2468 . . . 4  |-  ( ( B  +P.  G )  +P.  ( C  +P.  R ) )  =  ( ( B  +P.  C
)  +P.  ( G  +P.  R ) )
3222, 31eqeq12i 2449 . . 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 264 . 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 224 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   <.cop 4008   class class class wbr 4426  (class class class)co 6305   P.cnp 9283    +P. cpp 9285    ~R cer 9288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-omul 7195  df-er 7371  df-ni 9296  df-pli 9297  df-mi 9298  df-lti 9299  df-plpq 9332  df-mpq 9333  df-ltpq 9334  df-enq 9335  df-nq 9336  df-erq 9337  df-plq 9338  df-mq 9339  df-1nq 9340  df-rq 9341  df-ltnq 9342  df-np 9405  df-plp 9407  df-enr 9479
This theorem is referenced by:  addsrmo  9496
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