Theorem addcmpblnr 6333

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 6272	. . . . . . . 8 |- ((A e. P. /\ F e. P.) -> (A +P. F) e. P.)
2		addclpr 6272	. . . . . . . 8 |- ((B e. P. /\ G e. P.) -> (B +P. G) e. P.)
3	1, 2	anim12i 360	. . . . . . 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 566	. . . . . 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 6272	. . . . . . . 8 |- ((C e. P. /\ R e. P.) -> (C +P. R) e. P.)
6		addclpr 6272	. . . . . . . 8 |- ((D e. P. /\ S e. P.) -> (D +P. S) e. P.)
7	5, 6	anim12i 360	. . . . . . 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 566	. . . . . 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 360	. . . . 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 566	. . . 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 6326	. . . 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 12	. . 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 2295	. . . . . 6 |- x e. _V
17		visset 2295	. . . . . 6 |- y e. _V
18	16, 17	addcompr 6275	. . . . 5 |- (x +P. y) = (y +P. x)
19		visset 2295	. . . . . 6 |- z e. _V
20	17, 19	addasspr 6276	. . . . 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 4997	. . . 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 4997	. . . 4 |- ((B +P. G) +P. (C +P. R)) = ((B +P. C) +P. (G +P. R))
28	22, 27	eqeq12i 1897	. . 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 595	. 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 4891	. 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 227	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046   class class class wbr 3338  (class class class)co 4884  P.cnp 6137   +P. cpp 6139   ~R cer 6144

This theorem is referenced by:  addsrpr 6336

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-plp 6240  df-enr 6318

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