Theorem ltasr 5181

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 5166	. 2 |- dom +R = (R. X. R.)
3		ltasr.1	. 2 |- A e. V
4		ltrelsr 5152	. 2 |- <R (_ (R. X. R.)
5		0nsr 5160	. 2 |- -. (/) e. R.
6		df-nr 5139	. . . 4 |- R. = ((P. X. P.)/. ~R )
7		opreq1 3953	. . . . . 6 |- ([<.v, u>.] ~R = C -> ([<.v, u>.] ~R +R [<.x, y>.] ~R ) = (C +R [<.x, y>.] ~R ))
8		opreq1 3953	. . . . . 6 |- ([<.v, u>.] ~R = C -> ([<.v, u>.] ~R +R [<.z, w>.] ~R ) = (C +R [<.z, w>.] ~R ))
9	7, 8	breq12d 2621	. . . . 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 616	. . . 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 2612	. . . . 5 |- ([<.x, y>.] ~R = A -> ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> A <R [<.z, w>.] ~R ))
12		opreq2 3954	. . . . . 6 |- ([<.x, y>.] ~R = A -> (C +R [<.x, y>.] ~R ) = (C +R A))
13	12	breq1d 2619	. . . . 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 627	. . . 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 2613	. . . . 5 |- ([<.z, w>.] ~R = B -> (A <R [<.z, w>.] ~R <-> A <R B))
16		opreq2 3954	. . . . . 6 |- ([<.z, w>.] ~R = B -> (C +R [<.z, w>.] ~R ) = (C +R B))
17	16	breq2d 2620	. . . . 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 627	. . . 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 5092	. . . . . . 7 |- ((v e. P. /\ u e. P.) -> (v +P. u) e. P.)
20	19	3ad2ant1 798	. . . . . 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 3968	. . . . . . . 8 |- (x +P. w) e. V
22		oprex 3968	. . . . . . . 8 |- (y +P. z) e. V
23	21, 22	ltapr 5123	. . . . . . 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 1804	. . . . . . . 8 |- x e. V
25		visset 1804	. . . . . . . 8 |- y e. V
26		visset 1804	. . . . . . . 8 |- z e. V
27		visset 1804	. . . . . . . 8 |- w e. V
28	24, 25, 26, 27	ltsrpr 5158	. . . . . . 7 |- ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> (x +P. w) <P (y +P. z))
29		oprex 3968	. . . . . . . . 9 |- (v +P. x) e. V
30		oprex 3968	. . . . . . . . 9 |- (u +P. y) e. V
31		oprex 3968	. . . . . . . . 9 |- (v +P. z) e. V
32		oprex 3968	. . . . . . . . 9 |- (u +P. w) e. V
33	29, 30, 31, 32	ltsrpr 5158	. . . . . . . 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 1804	. . . . . . . . . 10 |- v e. V
35		visset 1804	. . . . . . . . . 10 |- u e. V
36	25, 26	addcompr 5095	. . . . . . . . . 10 |- (y +P. z) = (z +P. y)
37		visset 1804	. . . . . . . . . . 11 |- f e. V
38	26, 37	addasspr 5096	. . . . . . . . . 10 |- ((y +P. z) +P. f) = (y +P. (z +P. f))
39	34, 24, 35, 36, 38, 27	caopr4 4050	. . . . . . . . 9 |- ((v +P. x) +P. (u +P. w)) = ((v +P. u) +P. (x +P. w))
40	30, 31	addcompr 5095	. . . . . . . . . 10 |- ((u +P. y) +P. (v +P. z)) = ((v +P. z) +P. (u +P. y))
41	24, 27	addcompr 5095	. . . . . . . . . . 11 |- (x +P. w) = (w +P. x)
42	27, 37	addasspr 5096	. . . . . . . . . . 11 |- ((x +P. w) +P. f) = (x +P. (w +P. f))
43	34, 26, 35, 41, 42, 25	caopr42 4052	. . . . . . . . . 10 |- ((v +P. z) +P. (u +P. y)) = ((v +P. u) +P. (y +P. z))
44	40, 43	eqtr 1487	. . . . . . . . 9 |- ((u +P. y) +P. (v +P. z)) = ((v +P. u) +P. (y +P. z))
45	39, 44	breq12i 2618	. . . . . . . 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	bitr 173	. . . . . . 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 553	. . . . . 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 10	. . . . 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 5156	. . . . . . 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 797	. . . . . 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 5156	. . . . . . 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 796	. . . . . 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 2621	. . . . 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 529	. . . 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 4289	. . 3 |- ((C e. R. /\ A e. R. /\ B e. R.) -> (A <R B <-> (C +R A