Theorem addcmpblnr 5193

Description: Lemma showing compatibility of addition.

Hypotheses

Ref	Expression
cmpblnr.1	|- A e. V
cmpblnr.2	|- B e. V
cmpblnr.3	|- C e. V
cmpblnr.4	|- D e. V
cmpblnr.5	|- F e. V
cmpblnr.6	|- G e. V
cmpblnr.7	|- R e. V
cmpblnr.8	|- S e. V

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		addclpr 5132	. . . . . . . 8 |- ((A e. P. /\ F e. P.) -> (A +P. F) e. P.)
2		addclpr 5132	. . . . . . . 8 |- ((B e. P. /\ G e. P.) -> (B +P. G) e. P.)
3	1, 2	anim12i 333	. . . . . . 7 |- (((A e. P. /\ F e. P.) /\ (B e. P. /\ G e. P.)) -> ((A +P. F) e. P. /\ (B +P. G) e. P.))
4	3	an4s 510	. . . . . 6 |- (((A e. P. /\ B e. P.) /\ (F e. P. /\ G e. P.)) -> ((A +P. F) e. P. /\ (B +P. G) e. P.))
5		addclpr 5132	. . . . . . . 8 |- ((C e. P. /\ R e. P.) -> (C +P. R) e. P.)
6		addclpr 5132	. . . . . . . 8 |- ((D e. P. /\ S e. P.) -> (D +P. S) e. P.)
7	5, 6	anim12i 333	. . . . . . 7 |- (((C e. P. /\ R e. P.) /\ (D e. P. /\ S e. P.)) -> ((C +P. R) e. P. /\ (D +P. S) e. P.))
8	7	an4s 510	. . . . . 6 |- (((C e. P. /\ D e. P.) /\ (R e. P. /\ S e. P.)) -> ((C +P. R) e. P. /\ (D +P. S) e. P.))
9	4, 8	anim12i 333	. . . . 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.)))
10	9	an4s 510	. . . 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.)))
11		enrbreq 5186	. . . 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))))
12	10, 11	syl 10	. . 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))))
13		cmpblnr.1	. . . . 5 |- A e. V
14		cmpblnr.5	. . . . 5 |- F e. V
15		cmpblnr.4	. . . . 5 |- D e. V
16		visset 1816	. . . . . 6 |- x e. V
17		visset 1816	. . . . . 6 |- y e. V
18	16, 17	addcompr 5135	. . . . 5 |- (x +P. y) = (y +P. x)
19		visset 1816	. . . . . 6 |- z e. V
20	17, 19	addasspr 5136	. . . . 5 |- ((x +P. y) +P. z) = (x +P. (y +P. z))
21		cmpblnr.8	. . . . 5 |- S e. V
22	13, 14, 15, 18, 20, 21	caopr4 4070	. . . 4 |- ((A +P. F) +P. (D +P. S)) = ((A +P. D) +P. (F +P. S))
23		cmpblnr.2	. . . . 5 |- B e. V
24		cmpblnr.6	. . . . 5 |- G e. V
25		cmpblnr.3	. . . . 5 |- C e. V
26		cmpblnr.7	. . . . 5 |- R e. V
27	23, 24, 25, 18, 20, 26	caopr4 4070	. . . 4 |- ((B +P. G) +P. (C +P. R)) = ((B +P. C) +P. (G +P. R))
28	22, 27	eqeq12i 1491	. . 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)))
29	12, 28	syl6bb 538	. 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))))
30		opreq12 3976	. 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)))
31	29, 30	syl5bir 210	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 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  <.cop 2415   class class class wbr 2624  (class class class)co 3969  P.cnp 4997   +P. cpp 4999   ~R cer 5004

This theorem is referenced by:  addsrpr 5196

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-plp 5100  df-enr 5178

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