Theorem ltasr 4003

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 3988	. 2 |- dom +R = (R. X. R.)
3		ltasr.1	. 2 |- A e. V
4		ltrelsr 3974	. 2 |- <R (_ (R. X. R.)
5		0nsr 3982	. 2 |- -. (/) e. R.
6		df-nr 3961	. . . 4 |- R. = ((P. X. P.)/. ~R )
7		opreq1 3006	. . . . . 6 |- ([<.v, u>.] ~R = C -> ([<.v, u>.] ~R +R [<.x, y>.] ~R ) = (C +R [<.x, y>.] ~R ))
8		opreq1 3006	. . . . . 6 |- ([<.v, u>.] ~R = C -> ([<.v, u>.] ~R +R [<.z, w>.] ~R ) = (C +R [<.z, w>.] ~R ))
9	7, 8	breq12d 2073	. . . . 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 470	. . . 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 2065	. . . . 5 |- ([<.x, y>.] ~R = A -> ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> A <R [<.z, w>.] ~R ))
12		opreq2 3007	. . . . . 6 |- ([<.x, y>.] ~R = A -> (C +R [<.x, y>.] ~R ) = (C +R A))
13	12	breq1d 2071	. . . . 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 477	. . . 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 2066	. . . . 5 |- ([<.z, w>.] ~R = B -> (A <R [<.z, w>.] ~R <-> A <R B))
16		opreq2 3007	. . . . . 6 |- ([<.z, w>.] ~R = B -> (C +R [<.z, w>.] ~R ) = (C +R B))
17	16	breq2d 2072	. . . . 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 477	. . . 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 3914	. . . . . . . 8 |- ((v e. P. /\ u e. P.) -> (v +P. u) e. P.)
20	19	adantr 306	. . . . . . 7 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.)) -> (v +P. u) e. P.)
21	20	3adant3 599	. . . . . 6 |- (((v e. P. /\ u e. P.) /\ (x e. P. /\ y e. P.) /\ (z e. P. /\ w e. P.)) -> (v +P. u) e. P.)
22		oprex 3018	. . . . . . . 8 |- (x +P. w) e. V
23		oprex 3018	. . . . . . . 8 |- (y +P. z) e. V
24	22, 23	ltapr 3945	. . . . . . 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))))
25		visset 1350	. . . . . . . 8 |- x e. V
26		visset 1350	. . . . . . . 8 |- y e. V
27		visset 1350	. . . . . . . 8 |- z e. V
28		visset 1350	. . . . . . . 8 |- w e. V
29	25, 26, 27, 28	ltsrpr 3980	. . . . . . 7 |- ([<.x, y>.] ~R <R [<.z, w>.] ~R <-> (x +P. w) <P (y +P. z))
30		oprex 3018	. . . . . . . . 9 |- (v +P. x) e. V
31		oprex 3018	. . . . . . . . 9 |- (u +P. y) e. V
32		oprex 3018	. . . . . . . . 9 |- (v +P. z) e. V
33		oprex 3018	. . . . . . . . 9 |- (u +P. w) e. V
34	30, 31, 32, 33	ltsrpr 3980	. . . . . . . 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)))
35		visset 1350	. . . . . . . . . 10 |- v e. V
36		visset 1350	. . . . . . . . . 10 |- u e. V
37	26, 27	addcompr 3917	. . . . . . . . . 10 |- (y +P. z) = (z +P. y)
38		visset 1350	. . . . . . . . . . 11 |- f e. V
39	27, 38	addasspr 3918	. . . . . . . . . 10 |- ((y +P. z) +P. f) = (y +P. (z +P. f))
40	35, 25, 36, 37, 39, 28	caopr4 3078	. . . . . . . . 9 |- ((v +P. x) +P. (u +P. w)) = ((v +P. u) +P. (x +P. w))
41	31, 32	addcompr 3917	. . . . . . . . . 10 |- ((u +P. y) +P. (v +P. z)) = ((v +P. z) +P. (u +P. y))
42	25, 28	addcompr 3917	. . . . . . . . . . 11 |- (x +P. w) = (w +P. x)
43	28, 38	addasspr 3918	. . . . . . . . . . 11 |- ((x +P. w) +P. f) = (x +P. (w +P. f))
44	35, 27, 36, 42, 43, 26	caopr42 3080	. . . . . . . . . 10 |- ((v +P. z) +P. (u +P. y)) = ((v +P. u) +P. (y +P. z))
45	41, 44	eqtr 1119	. . . . . . . . 9 |- ((u +P. y) +P. (v +P. z)) = ((v +P. u) +P. (y +P. z))
46	40, 45	breq12i 2070	. . . . . . . 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)))
47	34, 46	bitr 151	. . . . . . 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)))
48	24, 29, 47	3bitr4g 428	. . . . . 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 ))
49	21, 48	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 ))
50		addsrpr 3978	. . . . . . 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 )
51	50	3adant3 599	. . . . . 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 )
52		addsrpr 3978	. . . . . . 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 )
53	52	3adant2 598	. . . . . 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 )
54	51, 53	breq12d 2073	. . . . 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 ))
55	49, 54	bitr4d 409	. . . 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 )))
56	6, 10, 14, 18, 55	3ecoptocl 3241	. . 3 |- ((C e. R. /\ A e. R. /\ B e. R.) -> (A <R B <-> (C +R A) <R (C +R B)))
57	56	3coml 617	. 2 |- ((A e. R. /\ B e. R. /\ C e. R.