Theorem ltprordil 7339

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 7255	. . . 4 |- <P C_ ( P. X. P. )
2	1	brel 4549	. . 3 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3		ltdfpr 7256	. . . 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 501	. . . . . 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 511	. . . . . . 7 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 2nd ` A ) )
9	8	adantr 272	. . . . . 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 7225	. . . . . . . 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 7237	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
14	12, 13	syl3an1 1230	. . . . . 6 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
15	6, 7, 9, 14	syl3anc 1197	. . . . 5 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> y <Q x )
16		simprrr 512	. . . . . . 7 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 1st ` B ) )
17	16	adantr 272	. . . . . 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 7225	. . . . . . . 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 7234	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> ( y <Q x -> y e. ( 1st ` B ) ) )
22	20, 21	sylan 279	. . . . . 6 |- ( ( A <P B /\ x e. ( 1st ` B ) ) -> ( y <Q x -> y e. ( 1st ` B ) ) )
23	6, 17, 22	syl2anc 406	. . . . 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 3067	. 2 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> ( 1st ` A ) C_ ( 1st ` B ) )
27	5, 26	rexlimddv 2526	1 |- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1461   E.wrex 2389   C_ wss 3035   <.cop 3494   class class class wbr 3893   ` cfv 5079   1stc1st 5988   2ndc2nd 5989   Q.cnq 7030   <Q cltq 7035   P.cnp 7041   <P cltp 7045

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-mi 7056  df-lti 7057  df-enq 7097  df-nqqs 7098  df-ltnqqs 7103  df-inp 7216  df-iltp 7220

This theorem is referenced by:  ltexprlemrl  7360