Theorem prmuloc2 7529

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7528 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 7528	. 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 1521	. . 3 |- F/ x ( <. L , U >. e. P. /\ 1Q <Q B )
3		nfre1 2513	. . 3 |- F/ x E. x e. L ( x .Q B ) e. U
4		simpr1 998	. . . . . . . 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 1000	. . . . . . . . . 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 531	. . . . . . . . . . 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 7351	. . . . . . . . . . 11 |- ( y e. Q. -> ( y .Q 1Q ) = y )
8		breq1 3992	. . . . . . . . . . 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 528	. . . . . . . . . 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 999	. . . . . . . . . 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 7447	. . . . . . . . . 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 2519	. . . . . . . 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 2587	. . . 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 2584	. 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 973   = wceq 1348   e. wcel 2141   E.wrex 2449   <.cop 3586   class class class wbr 3989  (class class class)co 5853   Q.cnq 7242   1Qc1q 7243   .Q cmq 7245   <Q cltq 7247   P.cnp 7253

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428

This theorem is referenced by:  recexprlem1ssl  7595  recexprlem1ssu  7596