Theorem lttrsr 7874

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 7839	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4046	. . . 4 |- ( [ <. x , y >. ] ~R = f -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> f <R [ <. z , w >. ] ~R ) )
3	2	anbi1d 465	. . 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 4046	. . 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 4047	. . . 4 |- ( [ <. z , w >. ] ~R = g -> ( f <R [ <. z , w >. ] ~R <-> f <R g ) )
7		breq1 4046	. . . 4 |- ( [ <. z , w >. ] ~R = g -> ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> g <R [ <. v , u >. ] ~R ) )
8	6, 7	anbi12d 473	. . 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 4047	. . . 4 |- ( [ <. v , u >. ] ~R = h -> ( g <R [ <. v , u >. ] ~R <-> g <R h ) )
11	10	anbi2d 464	. . 3 |- ( [ <. v , u >. ] ~R = h -> ( ( f <R g /\ g <R [ <. v , u >. ] ~R ) <-> ( f <R g /\ g <R h ) ) )
12		breq2 4047	. . 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 7859	. . . . . 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 1019	. . . . 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 7731	. . . . . . . 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 1023	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
19		simp2r 1026	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
20		addclpr 7649	. . . . . . . 8 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
21	18, 19, 20	syl2anc 411	. . . . . . 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 1024	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
23		simp2l 1025	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
24		addclpr 7649	. . . . . . . 8 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
25	22, 23, 24	syl2anc 411	. . . . . . 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 1028	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
27		addcomprg 7690	. . . . . . . 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 6115	. . . . . 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 7691	. . . . . . . 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 1249	. . . . . . 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 7691	. . . . . . . 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 1249	. . . . . . 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 4056	. . . . . 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 7859	. . . . . 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 1017	. . . . 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 7649	. . . . . . 7 |- ( ( z e. P. /\ u e. P. ) -> ( z +P. u ) e. P. )
40	23, 26, 39	syl2anc 411	. . . . . 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 1027	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
42		addclpr 7649	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
43	19, 41, 42	syl2anc 411	. . . . . 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 7731	. . . . . 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 1249	. . . . 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 473	. . 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 7708	. . . . 5 |- <P Or P.
49		ltrelpr 7617	. . . . 5 |- <P C_ ( P. X. P. )
50	48, 49	sotri 5077	. . . 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 7649	. . . . . . . 8 |- ( ( x e. P. /\ u e. P. ) -> ( x +P. u ) e. P. )
52	18, 26, 51	syl2anc 411	. . . . . . 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 7649	. . . . . . . 8 |- ( ( y e. P. /\ v e. P. ) -> ( y +P. v ) e. P. )
54	22, 41, 53	syl2anc 411	. . . . . . 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 7731	. . . . . . 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 1249	. . . . . 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 7691	. . . . . . . 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 6127	. . . . . 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 6127	. . . . . 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 4056	. . . . 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 7859	. . . . . 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 1018	. . . . 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 6710	1 |- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( ( f <R g /\ g <R h ) -> f <R h ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   <.cop 3635   class class class wbr 4043  (class class class)co 5943   [cec 6617   P.cnp 7403   +P. cpp 7405   <P cltp 7407   ~R cer 7408   R.cnr 7409   <R cltr 7415

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-iplp 7580  df-iltp 7582  df-enr 7838  df-nr 7839  df-ltr 7842

This theorem is referenced by:  ltposr  7875