Theorem ltasrg 8050

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 8007	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6035	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = ( C +R [ <. x , y >. ] ~R ) )
3		oveq1 6035	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = ( C +R [ <. z , w >. ] ~R ) )
4	2, 3	breq12d 4106	. . . 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 4096	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
7		oveq2 6036	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( C +R [ <. x , y >. ] ~R ) = ( C +R A ) )
8	7	breq1d 4103	. . . 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 4097	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		oveq2 6036	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( C +R [ <. z , w >. ] ~R ) = ( C +R B ) )
12	11	breq2d 4105	. . . 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 1050	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> x e. P. )
15		simp3r 1053	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> w e. P. )
16		addclpr 7817	. . . . . . 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 1051	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> y e. P. )
19		simp3l 1052	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> z e. P. )
20		addclpr 7817	. . . . . . 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 7817	. . . . . . 7 |- ( ( v e. P. /\ u e. P. ) -> ( v +P. u ) e. P. )
23	22	3ad2ant1 1045	. . . . . 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 7899	. . . . . 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 1274	. . . . 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 8027	. . . . . 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 1042	. . . . 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 1048	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> v e. P. )
29		addclpr 7817	. . . . . . . 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 1049	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> u e. P. )
32		addclpr 7817	. . . . . . . 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 7817	. . . . . . . 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 7817	. . . . . . . 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 8027	. . . . . . 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 1275	. . . . . 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 7858	. . . . . . . . 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 7859	. . . . . . . . 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 7817	. . . . . . . . 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 6217	. . . . . . 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 6189	. . . . . . . 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 6219	. . . . . . . 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 4106	. . . . . 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 8025	. . . . . 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 1044	. . . . 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 8025	. . . . . 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 1043	. . . . 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 4106	. . . 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 6836	. 2 |- ( ( C e. R. /\ A e. R. /\ B e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )
60	59	3coml 1237	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 1005   = wceq 1398   e. wcel 2202   <.cop 3676   class class class wbr 4093  (class class class)co 6028   [cec 6743   P.cnp 7571   +P. cpp 7573   <P cltp 7575   ~R cer 7576   R.cnr 7577   +R cplr 7581   <R cltr 7583

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-iplp 7748  df-iltp 7750  df-enr 8006  df-nr 8007  df-plr 8008  df-ltr 8010

This theorem is referenced by:  addgt0sr  8055  ltadd1sr  8056  caucvgsrlemoffcau  8078  caucvgsrlemoffgt1  8079  caucvgsrlemoffres  8080  caucvgsr  8082  ltpsrprg  8083  mappsrprg  8084  map2psrprg  8085  suplocsrlempr  8087  axpre-ltadd  8166