Theorem ltpopr 7952

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 7953. (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 7832	. . . . . . . 8 |- ( s e. P. -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
2		prdisj 7849	. . . . . . . 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 687	. . . . . 6 |- ( ( s e. P. /\ q e. Q. ) -> -. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
6	5	nrexdv 2643	. . . . 5 |- ( s e. P. -> -. E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) )
7		ltdfpr 7863	. . . . . 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 401	. . . . 5 |- ( s e. P. -> ( s <P s <-> E. q e. Q. ( q e. ( 2nd ` s ) /\ q e. ( 1st ` s ) ) ) )
9	6, 8	mtbird 684	. . . 4 |- ( s e. P. -> -. s <P s )
10	9	adantl 277	. . 3 |- ( ( T. /\ s e. P. ) -> -. s <P s )
11		ltdfpr 7863	. . . . . . . . . . 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 1048	. . . . . . . . . 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 7863	. . . . . . . . . . 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 1046	. . . . . . . . . 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 477	. . . . . . . . 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 2721	. . . . . . . . 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 543	. . . . . . . . . . 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 7832	. . . . . . . . . . . . . . . . . 18 |- ( t e. P. -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. )
21		prltlu 7844	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
22	20, 21	syl3an1 1311	. . . . . . . . . . . . . . . . 17 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ r e. ( 2nd ` t ) ) -> q <Q r )
23	22	3adant3r 1266	. . . . . . . . . . . . . . . 16 |- ( ( t e. P. /\ q e. ( 1st ` t ) /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> q <Q r )
24	23	3adant2l 1263	. . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 7832	. . . . . . . . . . . . . . . . 17 |- ( u e. P. -> <. ( 1st ` u ) , ( 2nd ` u ) >. e. P. )
29		prcdnql 7841	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 |- ( ( u e. P. /\ ( r e. ( 2nd ` t ) /\ r e. ( 1st ` u ) ) ) -> ( q <Q r -> q e. ( 1st ` u ) ) )
32	31	adantrl 482	. . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 2672	. . . . . . . 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 2651	. . . . . . 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 7863	. . . . . . . . 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 1047	. . . . . . . 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 4444	. 2 |- ( T. -> <P Po P. )
49	48	mptru 1411	1 |- <P Po P.

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   T. wtru 1403   e. wcel 2209   E.wrex 2529   <.cop 3708   class class class wbr 4125   Po wpo 4434   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   <Q cltq 7642   P.cnp 7648   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-mi 7663  df-lti 7664  df-enq 7704  df-nqqs 7705  df-ltnqqs 7710  df-inp 7823  df-iltp 7827

This theorem is referenced by:  ltsopr  7953