Theorem lttrsr 8135

Description: Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)

Assertion

Ref	Expression
lttrsr	|- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( ( f <R g /\ g <R h ) -> f <R h ) )

Distinct variable group:   f, g, h

Proof of Theorem lttrsr

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

Step	Hyp	Ref	Expression
1		df-nr 8100	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4134	. . . 4 |- ( [ <. x , y >. ] ~R = f -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> f <R [ <. z , w >. ] ~R ) )
3	2	anbi1d 469	. . 3 |- ( [ <. x , y >. ] ~R = f -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) ) )
4		breq1 4134	. . 3 |- ( [ <. x , y >. ] ~R = f -> ( [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R <-> f <R [ <. v , u >. ] ~R ) )
5	3, 4	imbi12d 234	. 2 |- ( [ <. x , y >. ] ~R = f -> ( ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) ) )
6		breq2 4135	. . . 4 |- ( [ <. z , w >. ] ~R = g -> ( f <R [ <. z , w >. ] ~R <-> f <R g ) )
7		breq1 4134	. . . 4 |- ( [ <. z , w >. ] ~R = g -> ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> g <R [ <. v , u >. ] ~R ) )
8	6, 7	anbi12d 477	. . 3 |- ( [ <. z , w >. ] ~R = g -> ( ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( f <R g /\ g <R [ <. v , u >. ] ~R ) ) )
9	8	imbi1d 231	. 2 |- ( [ <. z , w >. ] ~R = g -> ( ( ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) <-> ( ( f <R g /\ g <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) ) )
10		breq2 4135	. . . 4 |- ( [ <. v , u >. ] ~R = h -> ( g <R [ <. v , u >. ] ~R <-> g <R h ) )
11	10	anbi2d 468	. . 3 |- ( [ <. v , u >. ] ~R = h -> ( ( f <R g /\ g <R [ <. v , u >. ] ~R ) <-> ( f <R g /\ g <R h ) ) )
12		breq2 4135	. . 3 |- ( [ <. v , u >. ] ~R = h -> ( f <R [ <. v , u >. ] ~R <-> f <R h ) )
13	11, 12	imbi12d 234	. 2 |- ( [ <. v , u >. ] ~R = h -> ( ( ( f <R g /\ g <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) <-> ( ( f <R g /\ g <R h ) -> f <R h ) ) )
14		ltsrprg 8120	. . . . . 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 ) ) )
15	14	3adant3 1048	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. w ) <P ( y +P. z ) ) )
16		ltaprg 7992	. . . . . . . 8 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( r <P s <-> ( t +P. r ) <P ( t +P. s ) ) )
17	16	adantl 277	. . . . . . 7 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( r e. P. /\ s e. P. /\ t e. P. ) ) -> ( r <P s <-> ( t +P. r ) <P ( t +P. s ) ) )
18		simp1l 1052	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
19		simp2r 1055	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
20		addclpr 7910	. . . . . . . 8 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
21	18, 19, 20	syl2anc 415	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x +P. w ) e. P. )
22		simp1r 1053	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
23		simp2l 1054	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
24		addclpr 7910	. . . . . . . 8 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
25	22, 23, 24	syl2anc 415	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y +P. z ) e. P. )
26		simp3r 1057	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
27		addcomprg 7951	. . . . . . . 8 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) = ( s +P. r ) )
28	27	adantl 277	. . . . . . 7 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) = ( s +P. r ) )
29	17, 21, 25, 26, 28	caovord2d 6262	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x +P. w ) <P ( y +P. z ) <-> ( ( x +P. w ) +P. u ) <P ( ( y +P. z ) +P. u ) ) )
30		addassprg 7952	. . . . . . . 8 |- ( ( x e. P. /\ w e. P. /\ u e. P. ) -> ( ( x +P. w ) +P. u ) = ( x +P. ( w +P. u ) ) )
31	18, 19, 26, 30	syl3anc 1278	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x +P. w ) +P. u ) = ( x +P. ( w +P. u ) ) )
32		addassprg 7952	. . . . . . . 8 |- ( ( y e. P. /\ z e. P. /\ u e. P. ) -> ( ( y +P. z ) +P. u ) = ( y +P. ( z +P. u ) ) )
33	22, 23, 26, 32	syl3anc 1278	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( y +P. z ) +P. u ) = ( y +P. ( z +P. u ) ) )
34	31, 33	breq12d 4144	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( x +P. w ) +P. u ) <P ( ( y +P. z ) +P. u ) <-> ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) ) )
35	29, 34	bitrd 188	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x +P. w ) <P ( y +P. z ) <-> ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) ) )
36	15, 35	bitrd 188	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) ) )
37		ltsrprg 8120	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> ( z +P. u ) <P ( w +P. v ) ) )
38	37	3adant1 1046	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> ( z +P. u ) <P ( w +P. v ) ) )
39		addclpr 7910	. . . . . . 7 |- ( ( z e. P. /\ u e. P. ) -> ( z +P. u ) e. P. )
40	23, 26, 39	syl2anc 415	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z +P. u ) e. P. )
41		simp3l 1056	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
42		addclpr 7910	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
43	19, 41, 42	syl2anc 415	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
44		ltaprg 7992	. . . . . 6 |- ( ( ( z +P. u ) e. P. /\ ( w +P. v ) e. P. /\ y e. P. ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) )
45	40, 43, 22, 44	syl3anc 1278	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) )
46	38, 45	bitrd 188	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) )
47	36, 46	anbi12d 477	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) /\ ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) ) )
48		ltsopr 7969	. . . . 5 |- <P Or P.
49		ltrelpr 7878	. . . . 5 |- <P C_ ( P. X. P. )
50	48, 49	sotri 5186	. . . 4 |- ( ( ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) /\ ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) -> ( x +P. ( w +P. u ) ) <P ( y +P. ( w +P. v ) ) )
51		addclpr 7910	. . . . . . . 8 |- ( ( x e. P. /\ u e. P. ) -> ( x +P. u ) e. P. )
52	18, 26, 51	syl2anc 415	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x +P. u ) e. P. )
53		addclpr 7910	. . . . . . . 8 |- ( ( y e. P. /\ v e. P. ) -> ( y +P. v ) e. P. )
54	22, 41, 53	syl2anc 415	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y +P. v ) e. P. )
55		ltaprg 7992	. . . . . . 7 |- ( ( ( x +P. u ) e. P. /\ ( y +P. v ) e. P. /\ w e. P. ) -> ( ( x +P. u ) <P ( y +P. v ) <-> ( w +P. ( x +P. u ) ) <P ( w +P. ( y +P. v ) ) ) )
56	52, 54, 19, 55	syl3anc 1278	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x +P. u ) <P ( y +P. v ) <-> ( w +P. ( x +P. u ) ) <P ( w +P. ( y +P. v ) ) ) )
57	56	biimprd 158	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( w +P. ( x +P. u ) ) <P ( w +P. ( y +P. v ) ) -> ( x +P. u ) <P ( y +P. v ) ) )
58		addassprg 7952	. . . . . . . 8 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( ( r +P. s ) +P. t ) = ( r +P. ( s +P. t ) ) )
59	58	adantl 277	. . . . . . 7 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( r e. P. /\ s e. P. /\ t e. P. ) ) -> ( ( r +P. s ) +P. t ) = ( r +P. ( s +P. t ) ) )
60	18, 19, 26, 28, 59	caov12d 6274	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x +P. ( w +P. u ) ) = ( w +P. ( x +P. u ) ) )
61	22, 19, 41, 28, 59	caov12d 6274	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y +P. ( w +P. v ) ) = ( w +P. ( y +P. v ) ) )
62	60, 61	breq12d 4144	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x +P. ( w +P. u ) ) <P ( y +P. ( w +P. v ) ) <-> ( w +P. ( x +P. u ) ) <P ( w +P. ( y +P. v ) ) ) )
63		ltsrprg 8120	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R <-> ( x +P. u ) <P ( y +P. v ) ) )
64	63	3adant2 1047	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R <-> ( x +P. u ) <P ( y +P. v ) ) )
65	57, 62, 64	3imtr4d 203	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x +P. ( w +P. u ) ) <P ( y +P. ( w +P. v ) ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) )
66	50, 65	syl5 32	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) /\ ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) )
67	47, 66	sylbid 150	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) )
68	1, 5, 9, 13, 67	3ecoptocl 6902	1 |- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( ( f <R g /\ g <R h ) -> f <R h ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4131  (class class class)co 6088   [cec 6809   P.cnp 7664   +P. cpp 7666   <P cltp 7668   ~R cer 7669   R.cnr 7670   <R cltr 7676

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-iltp 7843  df-enr 8099  df-nr 8100  df-ltr 8103

This theorem is used by:  ltposr  8136