Theorem ltapr 5123

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 5087	. 2 |- dom +P. = (P. X. P.)
3		ltapr.1	. 2 |- A e. V
4		ltrelpr 5073	. 2 |- <P (_ (P. X. P.)
5		0npr 5068	. 2 |- -. (/) e. P.
6	3, 1	ltaprlem 5122	. . . . . 6 |- (C e. P. -> (A <P B -> (C +P. A) <P (C +P. B)))
7	6	adantr 389	. . . . 5 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (A <P B -> (C +P. A) <P (C +P. B)))
8	1, 3	ltaprlem 5122	. . . . . . . . . . . 12 |- (C e. P. -> (B <P A -> (C +P. B) <P (C +P. A)))
9	8	adantr 389	. . . . . . . . . . 11 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (B <P A -> (C +P. B) <P (C +P. A)))
10		ltsopr 5108	. . . . . . . . . . . . 13 |- <P Or P.
11		sotric 2851	. . . . . . . . . . . . 13 |- (( <P Or P. /\ (B e. P. /\ A e. P.)) -> (B <P A <-> -. (B = A \/ A <P B)))
12	10, 11	mpan 693	. . . . . . . . . . . 12 |- ((B e. P. /\ A e. P.) -> (B <P A <-> -. (B = A \/ A <P B)))
13	12	adantl 388	. . . . . . . . . . 11 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (B <P A <-> -. (B = A \/ A <P B)))
14		addclpr 5092	. . . . . . . . . . . . . 14 |- ((C e. P. /\ B e. P.) -> (C +P. B) e. P.)
15		addclpr 5092	. . . . . . . . . . . . . 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 511	. . . . . . . . . . . 12 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. B) e. P. /\ (C +P. A) e. P.))
18		sotric 2851	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 540	. . . . . . . . . 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 3968	. . . . . . . . 9 |- (C +P. A) e. V
29	28, 10, 4	soirri 3428	. . . . . . . 8 |- -. (C +P. A) <P (C +P. A)
30		opreq2 3954	. . . . . . . . 9 |- (B = A -> (C +P. B) = (C +P. A))
31	30	breq2d 2620	. . . . . . . 8 |- (B = A -> ((C +P. A) <P (C +P. B) <-> (C +P. A) <P (C +P. A)))
32	29, 31	mtbiri 715	. . . . . . 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 514	. . . 4 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (A <P B <-> (C +P. A) <P (C +P. B)))
36	35	3impb 827	. . 3 |- ((C e. P. /\ B e. P. /\ A e. P.) -> (A <P B <-> (C +P. A) <P (C +P. B)))
37	36	3com13 836	. 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 4036	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 953   e. wcel 955  Vcvv 1802   class class class wbr 2609   Or wor 2830  (class class class)co 3948  P.cnp 4957   +P. cpp 4959   <P cltp 4961

This theorem is referenced by:  addcanpr 5124  ltsrpr 5158  gt0srpr 5159  ltsosr 5175  ltasr 5181  ltpsrpr 5191

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-plp 5060  df-ltp 5062

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