Theorem ltpopr 7596

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 7597. (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 7476	. . . . . . . 8 |- ( s e. P. -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
2		prdisj 7493	. . . . . . . 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 676	. . . . . 6 |- ( ( s e. P. /\ q e. Q. ) -> -. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
6	5	nrexdv 2570	. . . . 5 |- ( s e. P. -> -. E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
7		ltdfpr 7507	. . . . . 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 673	. . . 4 |- ( s e. P. -> -. s <P s )
10	9	adantl 277	. . 3 |- ( ( T. /\ s e. P. ) -> -. s <P s )
11		ltdfpr 7507	. . . . . . . . . . 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 1017	. . . . . . . . . 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 7507	. . . . . . . . . . 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 1015	. . . . . . . . . 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 2647	. . . . . . . . 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 7476	. . . . . . . . . . . . . . . . . 18 |- ( t e. P. -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. )
21		prltlu 7488	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
22	20, 21	syl3an1 1271	. . . . . . . . . . . . . . . . 17 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
23	22	3adant3r 1235	. . . . . . . . . . . . . . . 16 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> q <Q r )
24	23	3adant2l 1232	. . . . . . . . . . . . . . 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 1204	. . . . . . . . . . . . . 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 1160	. . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . . 17 |- ( u e. P. -> <. ( 1st ` u ) , ( 2nd ` u ) >. e. P. )
29		prcdnql 7485	. . . . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . 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 2598	. . . . . . . 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 2578	. . . . . . 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 7507	. . . . . . . . 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 1016	. . . . . . . 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 4306	. 2 |- ( T. -> <P Po P. )
49	48	mptru 1362	1 |- <P Po P.

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   T. wtru 1354   e. wcel 2148   E.wrex 2456   <.cop 3597   class class class wbr 4005   Po wpo 4296   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   <Q cltq 7286   P.cnp 7292   <P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-mi 7307  df-lti 7308  df-enq 7348  df-nqqs 7349  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  ltsopr  7597