Theorem prmuloc2 7489

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7488 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio B, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)

Assertion

Ref	Expression
prmuloc2	|- ( ( <. L , U >. e. P. /\ 1Q <Q B ) -> E. x e. L ( x .Q B ) e. U )

Distinct variable groups:   x, B   x, L   x, U

Proof of Theorem prmuloc2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmuloc 7488	. 2 |- ( ( <. L , U >. e. P. /\ 1Q <Q B ) -> E. x e. Q. E. y e. Q. ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) )
2		nfv 1508	. . 3 |- F/ x ( <. L , U >. e. P. /\ 1Q <Q B )
3		nfre1 2500	. . 3 |- F/ x E. x e. L ( x .Q B ) e. U
4		simpr1 988	. . . . . . . 8 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> x e. L )
5		simpr3 990	. . . . . . . . . 10 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> ( y .Q 1Q ) <Q ( x .Q B ) )
6		simplrr 526	. . . . . . . . . . 11 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> y e. Q. )
7		mulidnq 7311	. . . . . . . . . . 11 |- ( y e. Q. -> ( y .Q 1Q ) = y )
8		breq1 3970	. . . . . . . . . . 11 |- ( ( y .Q 1Q ) = y -> ( ( y .Q 1Q ) <Q ( x .Q B ) <-> y <Q ( x .Q B ) ) )
9	6, 7, 8	3syl 17	. . . . . . . . . 10 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> ( ( y .Q 1Q ) <Q ( x .Q B ) <-> y <Q ( x .Q B ) ) )
10	5, 9	mpbid 146	. . . . . . . . 9 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> y <Q ( x .Q B ) )
11		simplll 523	. . . . . . . . . 10 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> <. L , U >. e. P. )
12		simpr2 989	. . . . . . . . . 10 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> y e. U )
13		prcunqu 7407	. . . . . . . . . 10 |- ( ( <. L , U >. e. P. /\ y e. U ) -> ( y <Q ( x .Q B ) -> ( x .Q B ) e. U ) )
14	11, 12, 13	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> ( y <Q ( x .Q B ) -> ( x .Q B ) e. U ) )
15	10, 14	mpd 13	. . . . . . . 8 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> ( x .Q B ) e. U )
16		rspe 2506	. . . . . . . 8 |- ( ( x e. L /\ ( x .Q B ) e. U ) -> E. x e. L ( x .Q B ) e. U )
17	4, 15, 16	syl2anc 409	. . . . . . 7 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) /\ ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) ) -> E. x e. L ( x .Q B ) e. U )
18	17	ex 114	. . . . . 6 |- ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ ( x e. Q. /\ y e. Q. ) ) -> ( ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) -> E. x e. L ( x .Q B ) e. U ) )
19	18	anassrs 398	. . . . 5 |- ( ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ x e. Q. ) /\ y e. Q. ) -> ( ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) -> E. x e. L ( x .Q B ) e. U ) )
20	19	rexlimdva 2574	. . . 4 |- ( ( ( <. L , U >. e. P. /\ 1Q <Q B ) /\ x e. Q. ) -> ( E. y e. Q. ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) -> E. x e. L ( x .Q B ) e. U ) )
21	20	ex 114	. . 3 |- ( ( <. L , U >. e. P. /\ 1Q <Q B ) -> ( x e. Q. -> ( E. y e. Q. ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) -> E. x e. L ( x .Q B ) e. U ) ) )
22	2, 3, 21	rexlimd 2571	. 2 |- ( ( <. L , U >. e. P. /\ 1Q <Q B ) -> ( E. x e. Q. E. y e. Q. ( x e. L /\ y e. U /\ ( y .Q 1Q ) <Q ( x .Q B ) ) -> E. x e. L ( x .Q B ) e. U ) )
23	1, 22	mpd 13	1 |- ( ( <. L , U >. e. P. /\ 1Q <Q B ) -> E. x e. L ( x .Q B ) e. U )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1335   e. wcel 2128   E.wrex 2436   <.cop 3564   class class class wbr 3967  (class class class)co 5826   Q.cnq 7202   1Qc1q 7203   .Q cmq 7205   <Q cltq 7207   P.cnp 7213

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 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549

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 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-plpq 7266  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-plqqs 7271  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275  df-enq0 7346  df-nq0 7347  df-0nq0 7348  df-plq0 7349  df-mq0 7350  df-inp 7388

This theorem is referenced by:  recexprlem1ssl  7555  recexprlem1ssu  7556