Theorem ltapr 6303

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 6267	. 2 |- dom +P. = (P. X. P.)
3		ltapr.1	. 2 |- A e. _V
4		ltrelpr 6253	. 2 |- <P C_ (P. X. P.)
5		0npr 6248	. 2 |- -. (/) e. P.
6	3, 1	ltaprlem 6302	. . . . . 6 |- (C e. P. -> (A <P B -> (C +P. A) <P (C +P. B)))
7	6	adantr 425	. . . . 5 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (A <P B -> (C +P. A) <P (C +P. B)))
8	1, 3	ltaprlem 6302	. . . . . . . . . . . 12 |- (C e. P. -> (B <P A -> (C +P. B) <P (C +P. A)))
9	8	adantr 425	. . . . . . . . . . 11 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (B <P A -> (C +P. B) <P (C +P. A)))
10		ltsopr 6288	. . . . . . . . . . . . 13 |- <P Or P.
11		sotric 3615	. . . . . . . . . . . . 13 |- (( <P Or P. /\ (B e. P. /\ A e. P.)) -> (B <P A <-> -. (B = A \/ A <P B)))
12	10, 11	mpan 759	. . . . . . . . . . . 12 |- ((B e. P. /\ A e. P.) -> (B <P A <-> -. (B = A \/ A <P B)))
13	12	adantl 424	. . . . . . . . . . 11 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (B <P A <-> -. (B = A \/ A <P B)))
14		sotric 3615	. . . . . . . . . . . 12 |- (( <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))))
15		addclpr 6272	. . . . . . . . . . . . . 14 |- ((C e. P. /\ B e. P.) -> (C +P. B) e. P.)
16		addclpr 6272	. . . . . . . . . . . . . 14 |- ((C e. P. /\ A e. P.) -> (C +P. A) e. P.)
17	15, 16	anim12i 360	. . . . . . . . . . . . 13 |- (((C e. P. /\ B e. P.) /\ (C e. P. /\ A e. P.)) -> ((C +P. B) e. P. /\ (C +P. A) e. P.))
18	17	anandis 570	. . . . . . . . . . . 12 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. B) e. P. /\ (C +P. A) e. P.))
19	14, 10, 18	sylancr 526	. . . . . . . . . . 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))))
20	9, 13, 19	3imtr3d 601	. . . . . . . . . 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))))
21	20	con4d 91	. . . . . . . . 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)))
22		olc 290	. . . . . . . . 9 |- ((C +P. A) <P (C +P. B) -> ((C +P. B) = (C +P. A) \/ (C +P. A) <P (C +P. B)))
23	21, 22	syl5 20	. . . . . . . 8 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. A) <P (C +P. B) -> (B = A \/ A <P B)))
24		df-or 241	. . . . . . . 8 |- ((B = A \/ A <P B) <-> (-. B = A -> A <P B))
25	23, 24	syl6ib 229	. . . . . . 7 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. A) <P (C +P. B) -> (-. B = A -> A <P B)))
26	25	com23 36	. . . . . 6 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (-. B = A -> ((C +P. A) <P (C +P. B) -> A <P B)))
27		oprex 4907	. . . . . . . . 9 |- (C +P. A) e. _V
28	27, 10, 4	soirri 4314	. . . . . . . 8 |- -. (C +P. A) <P (C +P. A)
29		opreq2 4890	. . . . . . . . 9 |- (B = A -> (C +P. B) = (C +P. A))
30	29	breq2d 3350	. . . . . . . 8 |- (B = A -> ((C +P. A) <P (C +P. B) <-> (C +P. A) <P (C +P. A)))
31	28, 30	mtbiri 785	. . . . . . 7 |- (B = A -> -. (C +P. A) <P (C +P. B))
32	31	pm2.21d 94	. . . . . 6 |- (B = A -> ((C +P. A) <P (C +P. B) -> A <P B))
33	26, 32	pm2.61d2 143	. . . . 5 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> ((C +P. A) <P (C +P. B) -> A <P B))
34	7, 33	impbid 574	. . . 4 |- ((C e. P. /\ (B e. P. /\ A e. P.)) -> (A <P B <-> (C +P. A) <P (C +P. B)))
35	34	3impb 1063	. . 3 |- ((C e. P. /\ B e. P. /\ A e. P.) -> (A <P B <-> (C +P. A) <P (C +P. B)))
36	35	3com13 1073	. 2 |- ((A e. P. /\ B e. P. /\ C e. P.) -> (A <P B <-> (C +P. A) <P (C +P. B)))
37	1, 2, 3, 4, 5, 36	ndmord 4983	1 |- (C e. P. -> (A <P B <-> (C +P. A) <P (C +P. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338   Or wor 3590  (class class class)co 4884  P.cnp 6137   +P. cpp 6139   <P cltp 6141

This theorem is referenced by:  addcanpr 6304  ltsrpr 6338  gt0srpr 6339  ltsosr 6355  ltasr 6361  ltpsrpr 6371

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

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