Theorem prmuloc2 7762

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7761 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⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 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 7761	. 2 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ Q ∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)))
2		nfv 1574	. . 3 ⊢ Ⅎ𝑥(⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵)
3		nfre1 2573	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈
4		simpr1 1027	. . . . . . . 8 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑥 ∈ 𝐿)
5		simpr3 1029	. . . . . . . . . 10 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))
6		simplrr 536	. . . . . . . . . . 11 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦 ∈ Q)
7		mulidnq 7584	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
8		breq1 4086	. . . . . . . . . . 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⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 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⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦 <Q (𝑥 ·Q 𝐵))
11		simplll 533	. . . . . . . . . 10 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → ⟨𝐿, 𝑈⟩ ∈ P)
12		simpr2 1028	. . . . . . . . . 10 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦 ∈ 𝑈)
13		prcunqu 7680	. . . . . . . . . 10 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑦 ∈ 𝑈) → (𝑦 <Q (𝑥 ·Q 𝐵) → (𝑥 ·Q 𝐵) ∈ 𝑈))
14	11, 12, 13	syl2anc 411	. . . . . . . . 9 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → (𝑦 <Q (𝑥 ·Q 𝐵) → (𝑥 ·Q 𝐵) ∈ 𝑈))
15	10, 14	mpd 13	. . . . . . . 8 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → (𝑥 ·Q 𝐵) ∈ 𝑈)
16		rspe 2579	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐿 ∧ (𝑥 ·Q 𝐵) ∈ 𝑈) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
17	4, 15, 16	syl2anc 411	. . . . . . 7 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
18	17	ex 115	. . . . . 6 ⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → ((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
19	18	anassrs 400	. . . . 5 ⊢ ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ 𝑥 ∈ Q) ∧ 𝑦 ∈ Q) → ((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
20	19	rexlimdva 2648	. . . 4 ⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ 𝑥 ∈ Q) → (∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
21	20	ex 115	. . 3 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → (𝑥 ∈ Q → (∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)))
22	2, 3, 21	rexlimd 2645	. 2 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → (∃𝑥 ∈ Q ∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
23	1, 22	mpd 13	1 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  ⟨cop 3669   class class class wbr 4083  (class class class)co 6007  Qcnq 7475  1_Qc1q 7476   ·_Q cmq 7478   <_Q cltq 7480  Pcnp 7486

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-lti 7502  df-plpq 7539  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-mqqs 7545  df-1nqqs 7546  df-rq 7547  df-ltnqqs 7548  df-enq0 7619  df-nq0 7620  df-0nq0 7621  df-plq0 7622  df-mq0 7623  df-inp 7661

This theorem is referenced by:  recexprlem1ssl  7828  recexprlem1ssu  7829