Theorem ltprordil 7906

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 7822	. . . 4 |- <P C_ ( P. X. P. )
2	1	brel 4804	. . 3 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3		ltdfpr 7823	. . . 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 144	. . 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 527	. . . . . 6 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> A <P B )
7		simpr 110	. . . . . 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 541	. . . . . . 7 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 2nd ` A ) )
9	8	adantr 276	. . . . . 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 112	. . . . . . . 8 |- ( A <P B -> A e. P. )
11		prop 7792	. . . . . . . 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 7804	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
14	12, 13	syl3an1 1307	. . . . . 6 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
15	6, 7, 9, 14	syl3anc 1274	. . . . 5 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> y <Q x )
16		simprrr 542	. . . . . . 7 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 1st ` B ) )
17	16	adantr 276	. . . . . 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 114	. . . . . . . 8 |- ( A <P B -> B e. P. )
19		prop 7792	. . . . . . . 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 7801	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> ( y <Q x -> y e. ( 1st ` B ) ) )
22	20, 21	sylan 283	. . . . . 6 |- ( ( A <P B /\ x e. ( 1st ` B ) ) -> ( y <Q x -> y e. ( 1st ` B ) ) )
23	6, 17, 22	syl2anc 411	. . . . 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 115	. . 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 3246	. 2 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> ( 1st ` A ) C_ ( 1st ` B ) )
27	5, 26	rexlimddv 2667	1 |- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   E.wrex 2523   C_ wss 3213   <.cop 3694   class class class wbr 4111   ` cfv 5354   1stc1st 6334   2ndc2nd 6335   Q.cnq 7597   <Q cltq 7602   P.cnp 7608   <P cltp 7612

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-mi 7623  df-lti 7624  df-enq 7664  df-nqqs 7665  df-ltnqqs 7670  df-inp 7783  df-iltp 7787

This theorem is referenced by:  ltexprlemrl  7927