Theorem ltasrg 7990

Description: Ordering property of addition. (Contributed by NM, 10-May-1996.)

Assertion

Ref	Expression
ltasrg	|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )

Proof of Theorem ltasrg

Dummy variables x y z w v u s r t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 7947	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6025	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = ( C +R [ <. x , y >. ] ~R ) )
3		oveq1 6025	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = ( C +R [ <. z , w >. ] ~R ) )
4	2, 3	breq12d 4101	. . . 4 |- ( [ <. 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 ) ) )
5	4	bibi2d 232	. . 3 |- ( [ <. 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 ) ) ) )
6		breq1 4091	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
7		oveq2 6026	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( C +R [ <. x , y >. ] ~R ) = ( C +R A ) )
8	7	breq1d 4098	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( ( C +R [ <. x , y >. ] ~R ) <R ( C +R [ <. z , w >. ] ~R ) <-> ( C +R A ) <R ( C +R [ <. z , w >. ] ~R ) ) )
9	6, 8	bibi12d 235	. . 3 |- ( [ <. 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 ) ) ) )
10		breq2 4092	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		oveq2 6026	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( C +R [ <. z , w >. ] ~R ) = ( C +R B ) )
12	11	breq2d 4100	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( ( C +R A ) <R ( C +R [ <. z , w >. ] ~R ) <-> ( C +R A ) <R ( C +R B ) ) )
13	10, 12	bibi12d 235	. . 3 |- ( [ <. 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 ) ) ) )
14		simp2l 1049	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> x e. P. )
15		simp3r 1052	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> w e. P. )
16		addclpr 7757	. . . . . . 7 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
17	14, 15, 16	syl2anc 411	. . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) e. P. )
18		simp2r 1050	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> y e. P. )
19		simp3l 1051	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> z e. P. )
20		addclpr 7757	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
21	18, 19, 20	syl2anc 411	. . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) e. P. )
22		addclpr 7757	. . . . . . 7 |- ( ( v e. P. /\ u e. P. ) -> ( v +P. u ) e. P. )
23	22	3ad2ant1 1044	. . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( v +P. u ) e. P. )
24		ltaprg 7839	. . . . . 6 |- ( ( ( x +P. w ) e. P. /\ ( y +P. z ) e. P. /\ ( 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	17, 21, 23, 24	syl3anc 1273	. . . . 5 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x +P. w ) <P ( y +P. z ) <-> ( ( v +P. u ) +P. ( x +P. w ) ) <P ( ( v +P. u ) +P. ( y +P. z ) ) ) )
26		ltsrprg 7967	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. w ) <P ( y +P. z ) ) )
27	26	3adant1 1041	. . . . 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 <-> ( x +P. w ) <P ( y +P. z ) ) )
28		simp1l 1047	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> v e. P. )
29		addclpr 7757	. . . . . . . 8 |- ( ( v e. P. /\ x e. P. ) -> ( v +P. x ) e. P. )
30	28, 14, 29	syl2anc 411	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( v +P. x ) e. P. )
31		simp1r 1048	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> u e. P. )
32		addclpr 7757	. . . . . . . 8 |- ( ( u e. P. /\ y e. P. ) -> ( u +P. y ) e. P. )
33	31, 18, 32	syl2anc 411	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( u +P. y ) e. P. )
34		addclpr 7757	. . . . . . . 8 |- ( ( v e. P. /\ z e. P. ) -> ( v +P. z ) e. P. )
35	28, 19, 34	syl2anc 411	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( v +P. z ) e. P. )
36		addclpr 7757	. . . . . . . 8 |- ( ( u e. P. /\ w e. P. ) -> ( u +P. w ) e. P. )
37	31, 15, 36	syl2anc 411	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( u +P. w ) e. P. )
38		ltsrprg 7967	. . . . . . 7 |- ( ( ( ( v +P. x ) e. P. /\ ( u +P. y ) e. P. ) /\ ( ( v +P. z ) e. P. /\ ( u +P. w ) e. P. ) ) -> ( [ <. ( 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 ) ) ) )
39	30, 33, 35, 37, 38	syl22anc 1274	. . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. ( 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 ) ) ) )
40		addcomprg 7798	. . . . . . . . 9 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) = ( s +P. r ) )
41	40	adantl 277	. . . . . . . 8 |- ( ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) = ( s +P. r ) )
42		addassprg 7799	. . . . . . . . 9 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( ( r +P. s ) +P. t ) = ( r +P. ( s +P. t ) ) )
43	42	adantl 277	. . . . . . . 8 |- ( ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( r e. P. /\ s e. P. /\ t e. P. ) ) -> ( ( r +P. s ) +P. t ) = ( r +P. ( s +P. t ) ) )
44		addclpr 7757	. . . . . . . . 9 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) e. P. )
45	44	adantl 277	. . . . . . . 8 |- ( ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) e. P. )
46	28, 14, 31, 41, 43, 15, 45	caov4d 6207	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( v +P. x ) +P. ( u +P. w ) ) = ( ( v +P. u ) +P. ( x +P. w ) ) )
47	41, 33, 35	caovcomd 6179	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( u +P. y ) +P. ( v +P. z ) ) = ( ( v +P. z ) +P. ( u +P. y ) ) )
48	28, 19, 31, 41, 43, 18, 45	caov42d 6209	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( v +P. z ) +P. ( u +P. y ) ) = ( ( v +P. u ) +P. ( y +P. z ) ) )
49	47, 48	eqtrd 2264	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( u +P. y ) +P. ( v +P. z ) ) = ( ( v +P. u ) +P. ( y +P. z ) ) )
50	46, 49	breq12d 4101	. . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( ( 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 ) ) ) )
51	39, 50	bitrd 188	. . . . 5 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. ( 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 ) ) ) )
52	25, 27, 51	3bitr4d 220	. . . 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 +P. x ) , ( u +P. y ) >. ] ~R <R [ <. ( v +P. z ) , ( u +P. w ) >. ] ~R ) )
53		addsrpr 7965	. . . . . 6 |- ( ( ( 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 )
54	53	3adant3 1043	. . . . 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 ) = [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R )
55		addsrpr 7965	. . . . . 6 |- ( ( ( 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 )
56	55	3adant2 1042	. . . . 5 |- ( ( ( 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 )
57	54, 56	breq12d 4101	. . . 4 |- ( ( ( 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 ) )
58	52, 57	bitr4d 191	. . 3 |- ( ( ( 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 ) ) )
59	1, 5, 9, 13, 58	3ecoptocl 6793	. 2 |- ( ( C e. R. /\ A e. R. /\ B e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )
60	59	3coml 1236	1 |- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   <.cop 3672   class class class wbr 4088  (class class class)co 6018   [cec 6700   P.cnp 7511   +P. cpp 7513   <P cltp 7515   ~R cer 7516   R.cnr 7517   +R cplr 7521   <R cltr 7523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-iplp 7688  df-iltp 7690  df-enr 7946  df-nr 7947  df-plr 7948  df-ltr 7950

This theorem is referenced by:  addgt0sr  7995  ltadd1sr  7996  caucvgsrlemoffcau  8018  caucvgsrlemoffgt1  8019  caucvgsrlemoffres  8020  caucvgsr  8022  ltpsrprg  8023  mappsrprg  8024  map2psrprg  8025  suplocsrlempr  8027  axpre-ltadd  8106