Theorem ltprordil 7601

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 7517	. . . 4 |- <P C_ ( P. X. P. )
2	1	brel 4690	. . 3 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3		ltdfpr 7518	. . . 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 539	. . . . . . 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 7487	. . . . . . . 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 7499	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
14	12, 13	syl3an1 1281	. . . . . 6 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ x e. ( 2nd ` A ) ) -> y <Q x )
15	6, 7, 9, 14	syl3anc 1248	. . . . 5 |- ( ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) /\ y e. ( 1st ` A ) ) -> y <Q x )
16		simprrr 540	. . . . . . 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 7487	. . . . . . . 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 7496	. . . . . . 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 3173	. 2 |- ( ( A <P B /\ ( x e. Q. /\ ( x e. ( 2nd ` A ) /\ x e. ( 1st ` B ) ) ) ) -> ( 1st ` A ) C_ ( 1st ` B ) )
27	5, 26	rexlimddv 2609	1 |- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2158   E.wrex 2466   C_ wss 3141   <.cop 3607   class class class wbr 4015   ` cfv 5228   1stc1st 6152   2ndc2nd 6153   Q.cnq 7292   <Q cltq 7297   P.cnp 7303   <P cltp 7307

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-mi 7318  df-lti 7319  df-enq 7359  df-nqqs 7360  df-ltnqqs 7365  df-inp 7478  df-iltp 7482

This theorem is referenced by:  ltexprlemrl  7622