Theorem ltapr 9867

Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)

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		dmplp 9834	. 2 |- dom +P. = ( P. X. P. )
2		ltrelpr 9820	. 2 |- <P C_ ( P. X. P. )
3		0npr 9814	. 2 |- -. (/) e. P.
4		ltaprlem 9866	. . . . . 6 |- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
5	4	adantr 481	. . . . 5 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
6		olc 399	. . . . . . . . 9 |- ( ( C +P. A ) <P ( C +P. B ) -> ( ( C +P. B ) = ( C +P. A ) \/ ( C +P. A ) <P ( C +P. B ) ) )
7		ltaprlem 9866	. . . . . . . . . . . 12 |- ( C e. P. -> ( B <P A -> ( C +P. B ) <P ( C +P. A ) ) )
8	7	adantr 481	. . . . . . . . . . 11 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( B <P A -> ( C +P. B ) <P ( C +P. A ) ) )
9		ltsopr 9854	. . . . . . . . . . . . 13 |- <P Or P.
10		sotric 5061	. . . . . . . . . . . . 13 |- ( ( <P Or P. /\ ( B e. P. /\ A e. P. ) ) -> ( B <P A <-> -. ( B = A \/ A <P B ) ) )
11	9, 10	mpan 706	. . . . . . . . . . . 12 |- ( ( B e. P. /\ A e. P. ) -> ( B <P A <-> -. ( B = A \/ A <P B ) ) )
12	11	adantl 482	. . . . . . . . . . 11 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( B <P A <-> -. ( B = A \/ A <P B ) ) )
13		addclpr 9840	. . . . . . . . . . . . 13 |- ( ( C e. P. /\ B e. P. ) -> ( C +P. B ) e. P. )
14		addclpr 9840	. . . . . . . . . . . . 13 |- ( ( C e. P. /\ A e. P. ) -> ( C +P. A ) e. P. )
15	13, 14	anim12dan 882	. . . . . . . . . . . 12 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. B ) e. P. /\ ( C +P. A ) e. P. ) )
16		sotric 5061	. . . . . . . . . . . 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 ) ) ) )
17	9, 15, 16	sylancr 695	. . . . . . . . . . 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 ) ) ) )
18	8, 12, 17	3imtr3d 282	. . . . . . . . . 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 ) ) ) )
19	18	con4d 114	. . . . . . . . 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 ) ) )
20	6, 19	syl5 34	. . . . . . . 8 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. A ) <P ( C +P. B ) -> ( B = A \/ A <P B ) ) )
21		df-or 385	. . . . . . . 8 |- ( ( B = A \/ A <P B ) <-> ( -. B = A -> A <P B ) )
22	20, 21	syl6ib 241	. . . . . . 7 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. A ) <P ( C +P. B ) -> ( -. B = A -> A <P B ) ) )
23	22	com23 86	. . . . . 6 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( -. B = A -> ( ( C +P. A ) <P ( C +P. B ) -> A <P B ) ) )
24	9, 2	soirri 5522	. . . . . . . 8 |- -. ( C +P. A ) <P ( C +P. A )
25		oveq2 6658	. . . . . . . . 9 |- ( B = A -> ( C +P. B ) = ( C +P. A ) )
26	25	breq2d 4665	. . . . . . . 8 |- ( B = A -> ( ( C +P. A ) <P ( C +P. B ) <-> ( C +P. A ) <P ( C +P. A ) ) )
27	24, 26	mtbiri 317	. . . . . . 7 |- ( B = A -> -. ( C +P. A ) <P ( C +P. B ) )
28	27	pm2.21d 118	. . . . . 6 |- ( B = A -> ( ( C +P. A ) <P ( C +P. B ) -> A <P B ) )
29	23, 28	pm2.61d2 172	. . . . 5 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. A ) <P ( C +P. B ) -> A <P B ) )
30	5, 29	impbid 202	. . . 4 |- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )
31	30	3impb 1260	. . 3 |- ( ( C e. P. /\ B e. P. /\ A e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )
32	31	3com13 1270	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )
33	1, 2, 3, 32	ndmovord 6824	1 |- ( C e. P. -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   Or wor 5034  (class class class)co 6650   P.cnp 9681   +P. cpp 9683   <P cltp 9685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807

This theorem is referenced by:  addcanpr  9868  ltsrpr  9898  gt0srpr  9899  ltsosr  9915  ltasr  9921  ltpsrpr  9930  map2psrpr  9931