Theorem ltpopr 7910

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 7911. (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 7790	. . . . . . . 8 |- ( s e. P. -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
2		prdisj 7807	. . . . . . . 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 683	. . . . . 6 |- ( ( s e. P. /\ q e. Q. ) -> -. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
6	5	nrexdv 2635	. . . . 5 |- ( s e. P. -> -. E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
7		ltdfpr 7821	. . . . . 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 680	. . . 4 |- ( s e. P. -> -. s <P s )
10	9	adantl 277	. . 3 |- ( ( T. /\ s e. P. ) -> -. s <P s )
11		ltdfpr 7821	. . . . . . . . . . 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 1044	. . . . . . . . . 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 7821	. . . . . . . . . . 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 1042	. . . . . . . . . 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 2713	. . . . . . . . 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 539	. . . . . . . . . . 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 7790	. . . . . . . . . . . . . . . . . 18 |- ( t e. P. -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. )
21		prltlu 7802	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
22	20, 21	syl3an1 1307	. . . . . . . . . . . . . . . . 17 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
23	22	3adant3r 1262	. . . . . . . . . . . . . . . 16 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> q <Q r )
24	23	3adant2l 1259	. . . . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . 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 1187	. . . . . . . . . . . . 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 7790	. . . . . . . . . . . . . . . . 17 |- ( u e. P. -> <. ( 1st ` u ) , ( 2nd ` u ) >. e. P. )
29		prcdnql 7799	. . . . . . . . . . . . . . . . 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 1188	. . . . . . . . . . . . 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 2664	. . . . . . . 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 2643	. . . . . . 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 7821	. . . . . . . . 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 1043	. . . . . . . 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 4425	. 2 |- ( T. -> <P Po P. )
49	48	mptru 1407	1 |- <P Po P.

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   T. wtru 1399   e. wcel 2203   E.wrex 2521   <.cop 3692   class class class wbr 4109   Po wpo 4415   ` cfv 5352   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   <Q cltq 7600   P.cnp 7606   <P cltp 7610

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-mi 7621  df-lti 7622  df-enq 7662  df-nqqs 7663  df-ltnqqs 7668  df-inp 7781  df-iltp 7785

This theorem is referenced by:  ltsopr  7911