Theorem ltprordil 7530

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 7446	. . . 4 |- <P C_ ( P. X. P. )
2	1	brel 4656	. . 3 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3		ltdfpr 7447	. . . 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 7416	. . . . . . . 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 7428	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
14	12, 13	syl3an1 1261	. . . . . 6 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
15	6, 7, 9, 14	syl3anc 1228	. . . . 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 7416	. . . . . . . 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 7425	. . . . . . 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 3148	. 2 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> ( 1st ` A ) C_ ( 1st ` B ) )
27	5, 26	rexlimddv 2588	1 |- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   E.wrex 2445   C_ wss 3116   <.cop 3579   class class class wbr 3982   ` cfv 5188   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   <Q cltq 7226   P.cnp 7232   <P cltp 7236

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-mi 7247  df-lti 7248  df-enq 7288  df-nqqs 7289  df-ltnqqs 7294  df-inp 7407  df-iltp 7411

This theorem is referenced by:  ltexprlemrl  7551