Theorem prmuloc2 7786

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7785 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 7785	. 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 1576	. . 3 |- F/ x ( <. L , U >. e. P. /\ 1Q <Q B )
3		nfre1 2575	. . 3 |- F/ x E. x e. L ( x .Q B ) e. U
4		simpr1 1029	. . . . . . . 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 1031	. . . . . . . . . 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 538	. . . . . . . . . . 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 7608	. . . . . . . . . . 11 |- ( y e. Q. -> ( y .Q 1Q ) = y )
8		breq1 4091	. . . . . . . . . . 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 147	. . . . . . . . 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 535	. . . . . . . . . 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 1030	. . . . . . . . . 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 7704	. . . . . . . . . 10 |- ( ( <. L , U >. e. P. /\ y e. U ) -> ( y <Q ( x .Q B ) -> ( x .Q B ) e. U ) )
14	11, 12, 13	syl2anc 411	. . . . . . . . 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 2581	. . . . . . . 8 |- ( ( x e. L /\ ( x .Q B ) e. U ) -> E. x e. L ( x .Q B ) e. U )
17	4, 15, 16	syl2anc 411	. . . . . . 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 115	. . . . . 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 400	. . . . 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 2650	. . . 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 115	. . 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 2647	. 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 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   <.cop 3672   class class class wbr 4088  (class class class)co 6017   Q.cnq 7499   1Qc1q 7500   .Q cmq 7502   <Q cltq 7504   P.cnp 7510

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685

This theorem is referenced by:  recexprlem1ssl  7852  recexprlem1ssu  7853