Theorem addcmpblnr 5161

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 5100	. . . . . . . 8 |- ((A e. P. /\ F e. P.) -> (A +P. F) e. P.)
2		addclpr 5100	. . . . . . . 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 508	. . . . . 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 5100	. . . . . . . 8 |- ((C e. P. /\ R e. P.) -> (C +P. R) e. P.)
6		addclpr 5100	. . . . . . . 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 508	. . . . . 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 508	. . . 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 5154	. . . 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 1809	. . . . . 6 |- x e. V
17		visset 1809	. . . . . 6 |- y e. V
18	16, 17	addcompr 5103	. . . . 5 |- (x +P. y) = (y +P. x)
19		visset 1809	. . . . . 6 |- z e. V
20	17, 19	addasspr 5104	. . . . 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 4056	. . . 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 4056	. . . 4 |- ((B +P. G) +P. (C +P. R)) = ((B +P. C) +P. (G +P. R))
28	22, 27	eqeq12i 1485	. . 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 535	. 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 3961	. 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 954   e. wcel 956  Vcvv 1807  <.cop 2407   class class class wbr 2614  (class class class)co 3954  P.cnp 4965   +P. cpp 4967   ~R cer 4972

This theorem is referenced by:  addsrpr 5164

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-plp 5068  df-enr 5146

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