Theorem ltapr 5163

Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123.

Hypotheses

Ref	Expression
ltapr.1	|- A e. V
ltapr.2	|- B e. V

Assertion

Ref	Expression
ltapr	|- (C e. P. -> (A <P B <-> (C +P. A) <P (C +P. B)))

Proof of Theorem ltapr

Step	Hyp	Ref	Expression
1		ltapr.2	. 2 |- B e. V
2		dmplp 5127	. 2 |- dom +P. = (P. X. P.)
3		ltapr.1	. 2 |- A e. V
4		ltrelpr 5113	. 2 |- <P (_ (P. X. P.)
5		0npr 5108	. 2 |- -. (/) e. P.
6	3, 1	ltaprlem 5162	. . . . . 6 |- (C e. P. -> (A <P B -> (C +P. A) <P (C +P. B)))
7	6	adantr 391	. . . . 5 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (A <P B -> (C +P. A) <P (C +P. B)))
8	1, 3	ltaprlem 5162	. . . . . . . . . . . 12 |- (C e. P. -> (B <P A -> (C +P. B) <P (C +P. A)))
9	8	adantr 391	. . . . . . . . . . 11 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (B <P A -> (C +P. B) <P (C +P. A)))
10		ltsopr 5148	. . . . . . . . . . . . 13 |- <P Or P.
11		sotric 2866	. . . . . . . . . . . . 13 |- (( <P Or P. /\ (B e. P. /\ A e. P.)) -> (B <P A <-> -. (B = A \/ A <P B)))
12	10, 11	mpan 697	. . . . . . . . . . . 12 |- ((B e. P. /\ A e. P.) -> (B <P A <-> -. (B = A \/ A <P B)))
13	12	adantl 390	. . . . . . . . . . 11 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (B <P A <-> -. (B = A \/ A <P B)))
14		addclpr 5132	. . . . . . . . . . . . . 14 |- ((C e. P. /\ B e. P.) -> (C +P. B) e. P.)
15		addclpr 5132	. . . . . . . . . . . . . 14 |- ((C e. P. /\ A e. P.) -> (C +P. A) e. P.)
16	14, 15	anim12i 333	. . . . . . . . . . . . 13 |- (((C e. P. /\ B e. P.) /\ (C e. P. /\ A e. P.)) -> ((C +P. B) e. P. /\ (C +P. A) e. P.))
17	16	anandis 514	. . . . . . . . . . . 12 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. B) e. P. /\ (C +P. A) e. P.))
18		sotric 2866	. . . . . . . . . . . . 13 |- (( <P Or P. /\ ((C +P. B) e. P. /\ (C +P. A) e. P.)) -> ((C +P. B) <P (C +P. A) <-> -. ((C +P. B) = (C +P. A) \/ (C +P. A) <P (C +P. B))))
19	10, 18	mpan 697	. . . . . . . . . . . 12 |- (((C +P. B) e. P. /\ (C +P. A) e. P.) -> ((C +P. B) <P (C +P. A) <-> -. ((C +P. B) = (C +P. A) \/ (C +P. A) <P (C +P. B))))
20	17, 19	syl 10	. . . . . . . . . . 11 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. B) <P (C +P. A) <-> -. ((C +P. B) = (C +P. A) \/ (C +P. A) <P (C +P. B))))
21	9, 13, 20	3imtr3d 544	. . . . . . . . . 10 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (-. (B = A \/ A <P B) -> -. ((C +P. B) = (C +P. A) \/ (C +P. A) <P (C +P. B))))
22	21	a3d 75	. . . . . . . . 9 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (((C +P. B) = (C +P. A) \/ (C +P. A) <P (C +P. B)) -> (B = A \/ A <P B)))
23		olc 268	. . . . . . . . 9 |- ((C +P. A) <P (C +P. B) -> ((C +P. B) = (C +P. A) \/ (C +P. A) <P (C +P. B)))
24	22, 23	syl5 21	. . . . . . . 8 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. A) <P (C +P. B) -> (B = A \/ A <P B)))
25		df-or 224	. . . . . . . 8 |- ((B = A \/ A <P B) <-> (-. B = A -> A <P B))
26	24, 25	syl6ib 212	. . . . . . 7 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. A) <P (C +P. B) -> (-. B = A -> A <P B)))
27	26	com23 32	. . . . . 6 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (-. B = A -> ((C +P. A) <P (C +P. B) -> A <P B)))
28		oprex 3989	. . . . . . . . 9 |- (C +P. A) e. V
29	28, 10, 4	soirri 3448	. . . . . . . 8 |- -. (C +P. A) <P (C +P. A)
30		opreq2 3975	. . . . . . . . 9 |- (B = A -> (C +P. B) = (C +P. A))
31	30	breq2d 2635	. . . . . . . 8 |- (B = A -> ((C +P. A) <P (C +P. B) <-> (C +P. A) <P (C +P. A)))
32	29, 31	mtbiri 719	. . . . . . 7 |- (B = A -> -. (C +P. A) <P (C +P. B))
33	32	pm2.21d 78	. . . . . 6 |- (B = A -> ((C +P. A) <P (C +P. B) -> A <P B))
34	27, 33	pm2.61d2 129	. . . . 5 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. A) <P (C +P. B) -> A <P B))
35	7, 34	impbid 518	. . . 4 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (A <P B <-> (C +P. A) <P (C +P. B)))
36	35	3impb 831	. . 3 |- ((C e. P. /\ B e. P. /\ A e. P.) -> (A <P B <-> (C +P. A) <P (C +P. B)))
37	36	3com13 840	. 2 |- ((A e. P. /\ B e. P. /\ C e. P.) -> (A <P B <-> (C +P. A) <P (C +P. B)))
38	1, 2, 3, 4, 5, 37	ndmord 4056	1 |- (C e. P. -> (A <P B <-> (C +P. A) <P (C +P. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   class class class wbr 2624   Or wor 2845  (class class class)co 3969  P.cnp 4997   +P. cpp 4999   <P cltp 5001

This theorem is referenced by:  addcanpr 5164  ltsrpr 5198  gt0srpr 5199  ltsosr 5215  ltasr 5221  ltpsrpr 5231

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-plp 5100  df-ltp 5102

Copyright terms: Public domain