Theorem addcmpblnr 9890

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

Step	Hyp	Ref	Expression
1		oveq12 6659	. 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 9840	. . . . . . . 8 |- ( ( A e. P. /\ F e. P. ) -> ( A +P. F ) e. P. )
3		addclpr 9840	. . . . . . . 8 |- ( ( B e. P. /\ G e. P. ) -> ( B +P. G ) e. P. )
4	2, 3	anim12i 590	. . . . . . 7 |- ( ( ( A e. P. /\ F e. P. ) /\ ( B e. P. /\ G e. P. ) ) -> ( ( A +P. F ) e. P. /\ ( B +P. G ) e. P. ) )
5	4	an4s 869	. . . . . 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 9840	. . . . . . . 8 |- ( ( C e. P. /\ R e. P. ) -> ( C +P. R ) e. P. )
7		addclpr 9840	. . . . . . . 8 |- ( ( D e. P. /\ S e. P. ) -> ( D +P. S ) e. P. )
8	6, 7	anim12i 590	. . . . . . 7 |- ( ( ( C e. P. /\ R e. P. ) /\ ( D e. P. /\ S e. P. ) ) -> ( ( C +P. R ) e. P. /\ ( D +P. S ) e. P. ) )
9	8	an4s 869	. . . . . 6 |- ( ( ( C e. P. /\ D e. P. ) /\ ( R e. P. /\ S e. P. ) ) -> ( ( C +P. R ) e. P. /\ ( D +P. S ) e. P. ) )
10	5, 9	anim12i 590	. . . . 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. ) ) )
11	10	an4s 869	. . . 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 9885	. . . 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 ) ) ) )
13	11, 12	syl 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 9843	. . . . . . . 8 |- ( F +P. D ) = ( D +P. F )
15	14	oveq1i 6660	. . . . . . 7 |- ( ( F +P. D ) +P. S ) = ( ( D +P. F ) +P. S )
16		addasspr 9844	. . . . . . 7 |- ( ( F +P. D ) +P. S ) = ( F +P. ( D +P. S ) )
17		addasspr 9844	. . . . . . 7 |- ( ( D +P. F ) +P. S ) = ( D +P. ( F +P. S ) )
18	15, 16, 17	3eqtr3i 2652	. . . . . 6 |- ( F +P. ( D +P. S ) ) = ( D +P. ( F +P. S ) )
19	18	oveq2i 6661	. . . . 5 |- ( A +P. ( F +P. ( D +P. S ) ) ) = ( A +P. ( D +P. ( F +P. S ) ) )
20		addasspr 9844	. . . . 5 |- ( ( A +P. F ) +P. ( D +P. S ) ) = ( A +P. ( F +P. ( D +P. S ) ) )
21		addasspr 9844	. . . . 5 |- ( ( A +P. D ) +P. ( F +P. S ) ) = ( A +P. ( D +P. ( F +P. S ) ) )
22	19, 20, 21	3eqtr4i 2654	. . . 4 |- ( ( A +P. F ) +P. ( D +P. S ) ) = ( ( A +P. D ) +P. ( F +P. S ) )
23		addcompr 9843	. . . . . . . 8 |- ( G +P. C ) = ( C +P. G )
24	23	oveq1i 6660	. . . . . . 7 |- ( ( G +P. C ) +P. R ) = ( ( C +P. G ) +P. R )
25		addasspr 9844	. . . . . . 7 |- ( ( G +P. C ) +P. R ) = ( G +P. ( C +P. R ) )
26		addasspr 9844	. . . . . . 7 |- ( ( C +P. G ) +P. R ) = ( C +P. ( G +P. R ) )
27	24, 25, 26	3eqtr3i 2652	. . . . . 6 |- ( G +P. ( C +P. R ) ) = ( C +P. ( G +P. R ) )
28	27	oveq2i 6661	. . . . 5 |- ( B +P. ( G +P. ( C +P. R ) ) ) = ( B +P. ( C +P. ( G +P. R ) ) )
29		addasspr 9844	. . . . 5 |- ( ( B +P. G ) +P. ( C +P. R ) ) = ( B +P. ( G +P. ( C +P. R ) ) )
30		addasspr 9844	. . . . 5 |- ( ( B +P. C ) +P. ( G +P. R ) ) = ( B +P. ( C +P. ( G +P. R ) ) )
31	28, 29, 30	3eqtr4i 2654	. . . 4 |- ( ( B +P. G ) +P. ( C +P. R ) ) = ( ( B +P. C ) +P. ( G +P. R ) )
32	22, 31	eqeq12i 2636	. . 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 ) ) )
33	13, 32	syl6bb 276	. 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 ) ) ) )
34	1, 33	syl5ibr 236	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 ) >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653  (class class class)co 6650   P.cnp 9681   +P. cpp 9683   ~R cer 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-enr 9877

This theorem is referenced by:  addsrmo  9894