Theorem ltpopr 7782

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 7783. (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 7662	. . . . . . . 8 |- ( s e. P. -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
2		prdisj 7679	. . . . . . . 8 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ q e. Q. ) -> -. ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) )
3	1, 2	sylan 283	. . . . . . 7 |- ( ( s e. P. /\ q e. Q. ) -> -. ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) )
4		ancom 266	. . . . . . 7 |- ( ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) <-> ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
5	3, 4	sylnib 680	. . . . . 6 |- ( ( s e. P. /\ q e. Q. ) -> -. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
6	5	nrexdv 2623	. . . . 5 |- ( s e. P. -> -. E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
7		ltdfpr 7693	. . . . . 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 397	. . . . 5 |- ( s e. P. -> ( s <P s <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) ) )
9	6, 8	mtbird 677	. . . 4 |- ( s e. P. -> -. s <P s )
10	9	adantl 277	. . 3 |- ( ( T. /\ s e. P. ) -> -. s <P s )
11		ltdfpr 7693	. . . . . . . . . . 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 1041	. . . . . . . . . 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 7693	. . . . . . . . . . 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 1039	. . . . . . . . . 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 473	. . . . . . . . 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 2701	. . . . . . . . 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	bitr4di 198	. . . . . . . 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 296	. . . . . . 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 537	. . . . . . . . . . 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 7662	. . . . . . . . . . . . . . . . . 18 |- ( t e. P. -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. )
21		prltlu 7674	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
22	20, 21	syl3an1 1304	. . . . . . . . . . . . . . . . 17 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
23	22	3adant3r 1259	. . . . . . . . . . . . . . . 16 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> q <Q r )
24	23	3adant2l 1256	. . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 7662	. . . . . . . . . . . . . . . . 17 |- ( u e. P. -> <. ( 1st ` u ) , ( 2nd ` u ) >. e. P. )
29		prcdnql 7671	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` u ) , ( 2nd ` u ) >. e. P. /\ r e. ( 1st ` u ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
30	28, 29	sylan 283	. . . . . . . . . . . . . . . 16 |- ( ( u e. P. /\ r e. ( 1st ` u ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
31	30	adantrl 478	. . . . . . . . . . . . . . 15 |- ( ( u e. P. /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
32	31	adantrl 478	. . . . . . . . . . . . . 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 1185	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 306	. . . . . . . . . 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 115	. . . . . . . . 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 2652	. . . . . . . 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 2631	. . . . . . 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 7693	. . . . . . . . 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 1040	. . . . . . . 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 158	. . . . . . 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 276	. . . . . 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 115	. . . 4 |- ( ( s e. P. /\ t e. P. /\ u e. P. ) -> ( ( s <P t /\ t <P u ) -> s <P u ) )
47	46	adantl 277	. . 3 |- ( ( T. /\ ( s e. P. /\ t e. P. /\ u e. P. ) ) -> ( ( s <P t /\ t <P u ) -> s <P u ) )
48	10, 47	ispod 4395	. 2 |- ( T. -> <P Po P. )
49	48	mptru 1404	1 |- <P Po P.

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   T. wtru 1396   e. wcel 2200   E.wrex 2509   <.cop 3669   class class class wbr 4083   Po wpo 4385   ` cfv 5318   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   <Q cltq 7472   P.cnp 7478   <P cltp 7482

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-mi 7493  df-lti 7494  df-enq 7534  df-nqqs 7535  df-ltnqqs 7540  df-inp 7653  df-iltp 7657

This theorem is referenced by:  ltsopr  7783