Theorem ltprordil 7492

Description: If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)

Assertion

Ref	Expression
ltprordil	|- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) )

Proof of Theorem ltprordil

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 7408	. . . 4 |- <P C_ ( P. X. P. )
2	1	brel 4635	. . 3 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3		ltdfpr 7409	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> E. x e. Q. ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) )
4	3	biimpd 143	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B -> E. x e. Q. ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) )
5	2, 4	mpcom 36	. 2 |- ( A <P B -> E. x e. Q. ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) )
6		simpll 519	. . . . . 6 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> A <P B )
7		simpr 109	. . . . . 6 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> y e. ( 1st ` A ) )
8		simprrl 529	. . . . . . 7 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 2nd ` A ) )
9	8	adantr 274	. . . . . 6 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> x e. ( 2nd ` A ) )
10	2	simpld 111	. . . . . . . 8 |- ( A <P B -> A e. P. )
11		prop 7378	. . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12	10, 11	syl 14	. . . . . . 7 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		prltlu 7390	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
14	12, 13	syl3an1 1253	. . . . . 6 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
15	6, 7, 9, 14	syl3anc 1220	. . . . 5 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> y <Q x )
16		simprrr 530	. . . . . . 7 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 1st ` B ) )
17	16	adantr 274	. . . . . 6 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> x e. ( 1st ` B ) )
18	2	simprd 113	. . . . . . . 8 |- ( A <P B -> B e. P. )
19		prop 7378	. . . . . . . 8 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
20	18, 19	syl 14	. . . . . . 7 |- ( A <P B -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
21		prcdnql 7387	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> ( y <Q x -> y e. ( 1st ` B ) ) )
22	20, 21	sylan 281	. . . . . 6 |- ( ( A <P B /\ x e. ( 1st ` B ) ) -> ( y <Q x -> y e. ( 1st ` B ) ) )
23	6, 17, 22	syl2anc 409	. . . . 5 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> ( y <Q x -> y e. ( 1st ` B ) ) )
24	15, 23	mpd 13	. . . 4 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> y e. ( 1st ` B ) )
25	24	ex 114	. . 3 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> ( y e. ( 1st ` A ) -> y e. ( 1st ` B ) ) )
26	25	ssrdv 3134	. 2 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> ( 1st ` A ) C_ ( 1st ` B ) )
27	5, 26	rexlimddv 2579	1 |- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2128   E.wrex 2436   C_ wss 3102   <.cop 3563   class class class wbr 3965   ` cfv 5167   1stc1st 6080   2ndc2nd 6081   Q.cnq 7183   <Q cltq 7188   P.cnp 7194   <P cltp 7198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4248  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-oadd 6361  df-omul 6362  df-er 6473  df-ec 6475  df-qs 6479  df-ni 7207  df-mi 7209  df-lti 7210  df-enq 7250  df-nqqs 7251  df-ltnqqs 7256  df-inp 7369  df-iltp 7373

This theorem is referenced by:  ltexprlemrl  7513