Theorem ltasrg 8127

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 8084	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6082	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = ( C +R [ <. x , y >. ] ~R ) )
3		oveq1 6082	. . . . 5 |- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = ( C +R [ <. z , w >. ] ~R ) )
4	2, 3	breq12d 4138	. . . 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 4128	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
7		oveq2 6083	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( C +R [ <. x , y >. ] ~R ) = ( C +R A ) )
8	7	breq1d 4135	. . . 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 4129	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		oveq2 6083	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( C +R [ <. z , w >. ] ~R ) = ( C +R B ) )
12	11	breq2d 4137	. . . 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 1054	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> x e. P. )
15		simp3r 1057	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> w e. P. )
16		addclpr 7894	. . . . . . 7 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
17	14, 15, 16	syl2anc 415	. . . . . 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 1055	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> y e. P. )
19		simp3l 1056	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> z e. P. )
20		addclpr 7894	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
21	18, 19, 20	syl2anc 415	. . . . . 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 7894	. . . . . . 7 |- ( ( v e. P. /\ u e. P. ) -> ( v +P. u ) e. P. )
23	22	3ad2ant1 1049	. . . . . 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 7976	. . . . . 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 1278	. . . . 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 8104	. . . . . 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 1046	. . . . 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 1052	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> v e. P. )
29		addclpr 7894	. . . . . . . 8 |- ( ( v e. P. /\ x e. P. ) -> ( v +P. x ) e. P. )
30	28, 14, 29	syl2anc 415	. . . . . . 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 1053	. . . . . . . 8 |- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> u e. P. )
32		addclpr 7894	. . . . . . . 8 |- ( ( u e. P. /\ y e. P. ) -> ( u +P. y ) e. P. )
33	31, 18, 32	syl2anc 415	. . . . . . 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 7894	. . . . . . . 8 |- ( ( v e. P. /\ z e. P. ) -> ( v +P. z ) e. P. )
35	28, 19, 34	syl2anc 415	. . . . . . 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 7894	. . . . . . . 8 |- ( ( u e. P. /\ w e. P. ) -> ( u +P. w ) e. P. )
37	31, 15, 36	syl2anc 415	. . . . . . 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 8104	. . . . . . 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 1279	. . . . . 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 7935	. . . . . . . . 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 7936	. . . . . . . . 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 7894	. . . . . . . . 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 6264	. . . . . . 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 6236	. . . . . . . 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 6266	. . . . . . . 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 2271	. . . . . . 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 4138	. . . . . 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 8102	. . . . . 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 1048	. . . . 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 8102	. . . . . 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 1047	. . . . 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 4138	. . . 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 6888	. 2 |- ( ( C e. R. /\ A e. R. /\ B e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )
60	59	3coml 1241	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 1009   = wceq 1402   e. wcel 2209   <.cop 3708   class class class wbr 4125  (class class class)co 6075   [cec 6795   P.cnp 7648   +P. cpp 7650   <P cltp 7652   ~R cer 7653   R.cnr 7654   +R cplr 7658   <R cltr 7660

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827  df-enr 8083  df-nr 8084  df-plr 8085  df-ltr 8087

This theorem is referenced by:  addgt0sr  8132  ltadd1sr  8133  caucvgsrlemoffcau  8155  caucvgsrlemoffgt1  8156  caucvgsrlemoffres  8157  caucvgsr  8159  ltpsrprg  8160  mappsrprg  8161  map2psrprg  8162  suplocsrlempr  8164  axpre-ltadd  8243