Theorem ltasrg 7830

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 7787	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5925	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = ( C +R [ <. x , y >. ] ~R ) )
3		oveq1 5925	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = ( C +R [ <. z , w >. ] ~R ) )
4	2, 3	breq12d 4042	. . . 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 4032	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
7		oveq2 5926	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( C +R [ <. x , y >. ] ~R ) = ( C +R A ) )
8	7	breq1d 4039	. . . 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 4033	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		oveq2 5926	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( C +R [ <. z , w >. ] ~R ) = ( C +R B ) )
12	11	breq2d 4041	. . . 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 1025	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> x e. P. )
15		simp3r 1028	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> w e. P. )
16		addclpr 7597	. . . . . . 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 1026	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> y e. P. )
19		simp3l 1027	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> z e. P. )
20		addclpr 7597	. . . . . . 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 7597	. . . . . . 7 |- ( ( v e. P. /\ u e. P. ) -> ( v +P. u ) e. P. )
23	22	3ad2ant1 1020	. . . . . 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 7679	. . . . . 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 1249	. . . . 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 7807	. . . . . 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 1017	. . . . 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 1023	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> v e. P. )
29		addclpr 7597	. . . . . . . 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 1024	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> u e. P. )
32		addclpr 7597	. . . . . . . 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 7597	. . . . . . . 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 7597	. . . . . . . 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 7807	. . . . . . 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 1250	. . . . . 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 7638	. . . . . . . . 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 7639	. . . . . . . . 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 7597	. . . . . . . . 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 6103	. . . . . . 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 6075	. . . . . . . 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 6105	. . . . . . . 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 2226	. . . . . . 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 4042	. . . . . 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 7805	. . . . . 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 1019	. . . . 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 7805	. . . . . 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 1018	. . . . 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 4042	. . . 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 6678	. 2 |- ( ( C e. R. /\ A e. R. /\ B e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )
60	59	3coml 1212	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 980   = wceq 1364   e. wcel 2164   <.cop 3621   class class class wbr 4029  (class class class)co 5918   [cec 6585   P.cnp 7351   +P. cpp 7353   <P cltp 7355   ~R cer 7356   R.cnr 7357   +R cplr 7361   <R cltr 7363

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530  df-enr 7786  df-nr 7787  df-plr 7788  df-ltr 7790

This theorem is referenced by:  addgt0sr  7835  ltadd1sr  7836  caucvgsrlemoffcau  7858  caucvgsrlemoffgt1  7859  caucvgsrlemoffres  7860  caucvgsr  7862  ltpsrprg  7863  mappsrprg  7864  map2psrprg  7865  suplocsrlempr  7867  axpre-ltadd  7946