Theorem ltasrg 7769

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 7726	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5882	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = ( C +R [ <. x , y >. ] ~R ) )
3		oveq1 5882	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = ( C +R [ <. z , w >. ] ~R ) )
4	2, 3	breq12d 4017	. . . 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 4007	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
7		oveq2 5883	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( C +R [ <. x , y >. ] ~R ) = ( C +R A ) )
8	7	breq1d 4014	. . . 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 4008	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		oveq2 5883	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( C +R [ <. z , w >. ] ~R ) = ( C +R B ) )
12	11	breq2d 4016	. . . 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 1023	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> x e. P. )
15		simp3r 1026	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> w e. P. )
16		addclpr 7536	. . . . . . 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 1024	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> y e. P. )
19		simp3l 1025	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> z e. P. )
20		addclpr 7536	. . . . . . 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 7536	. . . . . . 7 |- ( ( v e. P. /\ u e. P. ) -> ( v +P. u ) e. P. )
23	22	3ad2ant1 1018	. . . . . 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 7618	. . . . . 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 1238	. . . . 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 7746	. . . . . 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 1015	. . . . 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 1021	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> v e. P. )
29		addclpr 7536	. . . . . . . 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 1022	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> u e. P. )
32		addclpr 7536	. . . . . . . 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 7536	. . . . . . . 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 7536	. . . . . . . 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 7746	. . . . . . 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 1239	. . . . . 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 7577	. . . . . . . . 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 7578	. . . . . . . . 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 7536	. . . . . . . . 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 6059	. . . . . . 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 6031	. . . . . . . 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 6061	. . . . . . . 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 2210	. . . . . . 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 4017	. . . . . 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 7744	. . . . . 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 1017	. . . . 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 7744	. . . . . 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 1016	. . . . 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 4017	. . . 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 6624	. 2 |- ( ( C e. R. /\ A e. R. /\ B e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )
60	59	3coml 1210	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 978   = wceq 1353   e. wcel 2148   <.cop 3596   class class class wbr 4004  (class class class)co 5875   [cec 6533   P.cnp 7290   +P. cpp 7292   <P cltp 7294   ~R cer 7295   R.cnr 7296   +R cplr 7300   <R cltr 7302

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-iltp 7469  df-enr 7725  df-nr 7726  df-plr 7727  df-ltr 7729

This theorem is referenced by:  addgt0sr  7774  ltadd1sr  7775  caucvgsrlemoffcau  7797  caucvgsrlemoffgt1  7798  caucvgsrlemoffres  7799  caucvgsr  7801  ltpsrprg  7802  mappsrprg  7803  map2psrprg  7804  suplocsrlempr  7806  axpre-ltadd  7885