Theorem ltasr 6361

Description: Ordering property of addition.

Hypotheses

Ref	Expression
ltasr.1	|- A e. _V
ltasr.2	|- B e. _V

Assertion

Ref	Expression
ltasr	|- (C e. R. -> (A <R B <-> (C +R A) <R (C +R B)))

Proof of Theorem ltasr

Step	Hyp	Ref	Expression
1		ltasr.2	. 2 |- B e. _V
2		dmaddsr 6346	. 2 |- dom +R = (R. X. R.)
3		ltasr.1	. 2 |- A e. _V
4		ltrelsr 6332	. 2 |- <R C_ (R. X. R.)
5		0nsr 6340	. 2 |- -. (/) e. R.
6		df-nr 6319	. . . 4 |- R. = ((P. X. P.)/. ~R )
7		opreq1 4889	. . . . . 6 |- ([<.v, u>.] ~R = C -> ([<.v, u>.] ~R +R [<.x, y>.] ~R ) = (C +R [<.x, y>.] ~R ))
8		opreq1 4889	. . . . . 6 |- ([<.v, u>.] ~R = C -> ([<.v, u>.] ~R +R [<.z, w>.] ~R ) = (C +R [<.z, w>.] ~R ))
9	7, 8	breq12d 3351	. . . . 5 |- ([<.v, u>.] ~R = C -> (([<.v, u>.] ~R +R [<.x, y>.] ~R ) <R ([<.v, u>.] ~R +R [<.z, w>.] ~R ) <-> (C +R [<.x, y>.] ~R ) <R (C +R [<.z, w>.] ~R )))
10	9	bibi2d 680	. . . 4 |- ([<.v, u>.] ~R = C -> (([<.x, y>.] ~R <R [<.z, w>.] ~R <-> ([<.v, u>.] ~R +R [<.x, y>.] ~R ) <R ([<.v, u>.] ~R +R [<.z, w>.] ~R )) <-> ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> (C +R [<.x, y>.] ~R ) <R (C +R [<.z, w>.] ~R ))))
11		breq1 3341	. . . . 5 |- ([<.x, y>.] ~R = A -> ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> A <R [<.z, w>.] ~R ))
12		opreq2 4890	. . . . . 6 |- ([<.x, y>.] ~R = A -> (C +R [<.x, y>.] ~R ) = (C +R A))
13	12	breq1d 3348	. . . . 5 |- ([<.x, y>.] ~R = A -> ((C +R [<.x, y>.] ~R ) <R (C +R [<.z, w>.] ~R ) <-> (C +R A) <R (C +R [<.z, w>.] ~R )))
14	11, 13	bibi12d 691	. . . 4 |- ([<.x, y>.] ~R = A -> (([<.x, y>.] ~R <R [<.z, w>.] ~R <-> (C +R [<.x, y>.] ~R ) <R (C +R [<.z, w>.] ~R )) <-> (A <R [<.z, w>.] ~R <-> (C +R A) <R (C +R [<.z, w>.] ~R ))))
15		breq2 3342	. . . . 5 |- ([<.z, w>.] ~R = B -> (A <R [<.z, w>.] ~R <-> A <R B))
16		opreq2 4890	. . . . . 6 |- ([<.z, w>.] ~R = B -> (C +R [<.z, w>.] ~R ) = (C +R B))
17	16	breq2d 3350	. . . . 5 |- ([<.z, w>.] ~R = B -> ((C +R A) <R (C +R [<.z, w>.] ~R ) <-> (C +R A) <R (C +R B)))
18	15, 17	bibi12d 691	. . . 4 |- ([<.z, w>.] ~R = B -> ((A <R [<.z, w>.] ~R <-> (C +R A) <R (C +R [<.z, w>.] ~R )) <-> (A <R B <-> (C +R A) <R (C +R B))))
19		addclpr 6272	. . . . . . 7 |- ((v e. P. /\ u e. P.) -> (v +P. u) e. P.)
20	19	3ad2ant1 897	. . . . . 6 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.) /\ (z e. P. /\ w e. P.)) -> (v +P. u) e. P.)
21		oprex 4907	. . . . . . . 8 |- (x +P. w) e. _V
22		oprex 4907	. . . . . . . 8 |- (y +P. z) e. _V
23	21, 22	ltapr 6303	. . . . . . 7 |- ((v +P. u) e. P. -> ((x +P. w) <P (y +P. z) <-> ((v +P. u) +P. (x +P. w)) <P ((v +P. u) +P. (y +P. z))))
24		visset 2295	. . . . . . . 8 |- x e. _V
25		visset 2295	. . . . . . . 8 |- y e. _V
26		visset 2295	. . . . . . . 8 |- z e. _V
27		visset 2295	. . . . . . . 8 |- w e. _V
28	24, 25, 26, 27	ltsrpr 6338	. . . . . . 7 |- ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> (x +P. w) <P (y +P. z))
29		oprex 4907	. . . . . . . . 9 |- (v +P. x) e. _V
30		oprex 4907	. . . . . . . . 9 |- (u +P. y) e. _V
31		oprex 4907	. . . . . . . . 9 |- (v +P. z) e. _V
32		oprex 4907	. . . . . . . . 9 |- (u +P. w) e. _V
33	29, 30, 31, 32	ltsrpr 6338	. . . . . . . 8 |- ([<.(v +P. x), (u +P. y)>.] ~R <R [<.(v +P. z), (u +P. w)>.] ~R <-> ((v +P. x) +P. (u +P. w)) <P ((u +P. y) +P. (v +P. z)))
34		visset 2295	. . . . . . . . . 10 |- v e. _V
35		visset 2295	. . . . . . . . . 10 |- u e. _V
36	25, 26	addcompr 6275	. . . . . . . . . 10 |- (y +P. z) = (z +P. y)
37		visset 2295	. . . . . . . . . . 11 |- f e. _V
38	26, 37	addasspr 6276	. . . . . . . . . 10 |- ((y +P. z) +P. f) = (y +P. (z +P. f))
39	34, 24, 35, 36, 38, 27	caopr4 4997	. . . . . . . . 9 |- ((v +P. x) +P. (u +P. w)) = ((v +P. u) +P. (x +P. w))
40	30, 31	addcompr 6275	. . . . . . . . . 10 |- ((u +P. y) +P. (v +P. z)) = ((v +P. z) +P. (u +P. y))
41	24, 27	addcompr 6275	. . . . . . . . . . 11 |- (x +P. w) = (w +P. x)
42	27, 37	addasspr 6276	. . . . . . . . . . 11 |- ((x +P. w) +P. f) = (x +P. (w +P. f))
43	34, 26, 35, 41, 42, 25	caopr42 4999	. . . . . . . . . 10 |- ((v +P. z) +P. (u +P. y)) = ((v +P. u) +P. (y +P. z))
44	40, 43	eqtri 1908	. . . . . . . . 9 |- ((u +P. y) +P. (v +P. z)) = ((v +P. u) +P. (y +P. z))
45	39, 44	breq12i 3347	. . . . . . . 8 |- (((v +P. x) +P. (u +P. w)) <P ((u +P. y) +P. (v +P. z)) <-> ((v +P. u) +P. (x +P. w)) <P ((v +P. u) +P. (y +P. z)))
46	33, 45	bitri 190	. . . . . . 7 |- ([<.(v +P. x), (u +P. y)>.] ~R <R [<.(v +P. z), (u +P. w)>.] ~R <-> ((v +P. u) +P. (x +P. w)) <P ((v +P. u) +P. (y +P. z)))
47	23, 28, 46	3bitr4g 614	. . . . . 6 |- ((v +P. u) e. P. -> ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> [<.(v +P. x), (u +P. y)>.] ~R <R [<.(v +P. z), (u +P. w)>.] ~R ))
48	20, 47	syl 12	. . . . 5 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.) /\ (z e. P. /\ w e. P.)) -> ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> [<.(v +P. x), (u +P. y)>.] ~R <R [<.(v +P. z), (u +P. w)>.] ~R ))
49		addsrpr 6336	. . . . . . 7 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.)) -> ([<.v, u>.] ~R +R [<.x, y>.] ~R ) = [<.(v +P. x), (u +P. y)>.] ~R )
50	49	3adant3 896	. . . . . 6 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.) /\ (z e. P. /\ w e. P.)) -> ([<.v, u>.] ~R +R [<.x, y>.] ~R ) = [<.(v +P. x), (u +P. y)>.] ~R )
51		addsrpr 6336	. . . . . . 7 |- (((v e. P. /\ u e. P.) /\ (z e. P. /\ w e. P.)) -> ([<.v, u>.] ~R +R [<.z, w>.] ~R ) = [<.(v +P. z), (u +P. w)>.] ~R )
52	51	3adant2 895	. . . . . 6 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.) /\ (z e. P. /\ w e. P.)) -> ([<.v, u>.] ~R +R [<.z, w>.] ~R ) = [<.(v +P. z), (u +P. w)>.] ~R )
53	50, 52	breq12d 3351	. . . . 5 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.) /\ (z e. P. /\ w e. P.)) -> (([<.v, u>.] ~R +R [<.x, y>.] ~R ) <R ([<.v, u>.] ~R +R [<.z, w>.] ~R ) <-> [<.(v +P. x), (u +P. y)>.] ~R <R [<.(v +P. z), (u +P. w)>.] ~R ))
54	48, 53	bitr4d 590	. . . 4 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.) /\ (z e. P. /\ w e. P.)) -> ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> ([<.v, u>.] ~R +R [<.x, y>.] ~R ) <R ([<.v, u>.] ~R +R [<.z, w>.] ~R )))
55	6, 10, 14, 18, 54	3ecoptocl 5364	. . 3 |- ((C e. R. /\ A e. R. /\ B e. R.) -> (A <R B <-> (C +R A) <R (C +R B)))
56	55	3coml 1075	. 2 |- ((A e. R. /\ B e. R. /\ C e. R.) -> (A <R B <-> (C +R A) <R (C +R B)))
57	1, 2, 3, 4, 5, 56	ndmord 4983	1 |- (C e. R. -> (A <R B <-> (C +R A) <R (C +R B)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046   class class class wbr 3338  (class class class)co 4884  [cec 5316  P.cnp 6137   +P. cpp 6139   <P cltp 6141   ~R cer 6144  R.cnr 6145   +R cplr 6149   <R cltr 6151

This theorem is referenced by:  addgt0sr 6365  sqgt0sr 6367  supsrlem3 6379  pre-axltadd 6442

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-ltp 6242  df-plpr 6316  df-enr 6318  df-nr 6319  df-plr 6320  df-ltr 6322

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