Theorem ltpopr 7215

Description: Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 7216. (Contributed by Jim Kingdon, 15-Dec-2019.)

Assertion

Ref	Expression
ltpopr	|- <P Po P.

Proof of Theorem ltpopr

Dummy variables r q s t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7095	. . . . . . . 8 |- ( s e. P. -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
2		prdisj 7112	. . . . . . . 8 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ q e. Q. ) -> -. ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) )
3	1, 2	sylan 278	. . . . . . 7 |- ( ( s e. P. /\ q e. Q. ) -> -. ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) )
4		ancom 263	. . . . . . 7 |- ( ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) <-> ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
5	3, 4	sylnib 637	. . . . . 6 |- ( ( s e. P. /\ q e. Q. ) -> -. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
6	5	nrexdv 2467	. . . . 5 |- ( s e. P. -> -. E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
7		ltdfpr 7126	. . . . . 6 |- ( ( s e. P. /\ s e. P. ) -> ( s <P s <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) ) )
8	7	anidms 390	. . . . 5 |- ( s e. P. -> ( s <P s <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) ) )
9	6, 8	mtbird 634	. . . 4 |- ( s e. P. -> -. s <P s )
10	9	adantl 272	. . 3 |- ( ( T. /\ s e. P. ) -> -. s <P s )
11		ltdfpr 7126	. . . . . . . . . . 11 |- ( ( s e. P. /\ t e. P. ) -> ( s <P t <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) ) )
12	11	3adant3 964	. . . . . . . . . 10 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( s <P t <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) ) )
13		ltdfpr 7126	. . . . . . . . . . 11 |- ( ( t e. P. /\ u e. P. ) -> ( t <P u <-> E. r e. Q. ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) )
14	13	3adant1 962	. . . . . . . . . 10 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( t <P u <-> E. r e. Q. ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) )
15	12, 14	anbi12d 458	. . . . . . . . 9 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( ( s <P t /\ t <P u ) <-> ( E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ E. r e. Q. ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) )
16		reeanv 2537	. . . . . . . . 9 |- ( E. q e. Q. E. r e. Q. ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) <-> ( E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ E. r e. Q. ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) )
17	15, 16	syl6bbr 197	. . . . . . . 8 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( ( s <P t /\ t <P u ) <-> E. q e. Q. E. r e. Q. ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) )
18	17	biimpa 291	. . . . . . 7 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) -> E. q e. Q. E. r e. Q. ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) )
19		simprll 505	. . . . . . . . . . 11 |- ( ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> q e. ( 2nd ` s ) )
20		prop 7095	. . . . . . . . . . . . . . . . . 18 |- ( t e. P. -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. )
21		prltlu 7107	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
22	20, 21	syl3an1 1208	. . . . . . . . . . . . . . . . 17 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
23	22	3adant3r 1172	. . . . . . . . . . . . . . . 16 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> q <Q r )
24	23	3adant2l 1169	. . . . . . . . . . . . . . 15 |- ( ( t e. P. /\ ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> q <Q r )
25	24	3expb 1145	. . . . . . . . . . . . . 14 |- ( ( t e. P. /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> q <Q r )
26	25	3ad2antl2 1107	. . . . . . . . . . . . 13 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> q <Q r )
27	26	adantlr 462	. . . . . . . . . . . 12 |- ( ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> q <Q r )
28		prop 7095	. . . . . . . . . . . . . . . . 17 |- ( u e. P. -> <. ( 1st ` u ) , ( 2nd ` u ) >. e. P. )
29		prcdnql 7104	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` u ) , ( 2nd ` u ) >. e. P. /\ r e. ( 1st ` u ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
30	28, 29	sylan 278	. . . . . . . . . . . . . . . 16 |- ( ( u e. P. /\ r e. ( 1st ` u ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
31	30	adantrl 463	. . . . . . . . . . . . . . 15 |- ( ( u e. P. /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
32	31	adantrl 463	. . . . . . . . . . . . . 14 |- ( ( u e. P. /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
33	32	3ad2antl3 1108	. . . . . . . . . . . . 13 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
34	33	adantlr 462	. . . . . . . . . . . 12 |- ( ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
35	27, 34	mpd 13	. . . . . . . . . . 11 |- ( ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> q e. ( 1st ` u ) )
36	19, 35	jca 301	. . . . . . . . . 10 |- ( ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) /\ ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) ) -> ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) )
37	36	ex 114	. . . . . . . . 9 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) -> ( ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) ) )
38	37	rexlimdvw 2493	. . . . . . . 8 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) -> ( E. r e. Q. ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) ) )
39	38	reximdv 2475	. . . . . . 7 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) -> ( E. q e. Q. E. r e. Q. ( ( q e. ( 2nd ` s ) /\ q e. ( 1st ` t ) ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) ) )
40	18, 39	mpd 13	. . . . . 6 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) -> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) )
41		ltdfpr 7126	. . . . . . . . 9 |- ( ( s e. P. /\ u e. P. ) -> ( s <P u <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) ) )
42	41	3adant2 963	. . . . . . . 8 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( s <P u <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) ) )
43	42	biimprd 157	. . . . . . 7 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) -> s <P u ) )
44	43	adantr 271	. . . . . 6 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) -> ( E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` u ) ) -> s <P u ) )
45	40, 44	mpd 13	. . . . 5 |- ( ( ( s e. P. /\ t e. P. /\ u e. P. ) /\ ( s <P t /\ t <P u ) ) -> s <P u )
46	45	ex 114	. . . 4 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( ( s <P t /\ t <P u ) -> s <P u ) )
47	46	adantl 272	. . 3 |- ( ( T. /\ ( s e. P. /\ t e. P. /\ u e. P. ) ) -> ( ( s <P t /\ t <P u ) -> s <P u ) )
48	10, 47	ispod 4140	. 2 |- ( T. -> <P Po P. )
49	48	mptru 1299	1 |- <P Po P.

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   T. wtru 1291   e. wcel 1439   E.wrex 2361   <.cop 3453   class class class wbr 3851   Po wpo 4130   ` cfv 5028   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900   <Q cltq 6905   P.cnp 6911   <P cltp 6915

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-mi 6926  df-lti 6927  df-enq 6967  df-nqqs 6968  df-ltnqqs 6973  df-inp 7086  df-iltp 7090

This theorem is referenced by:  ltsopr  7216