Theorem prmuloc2 7934

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7933 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 7933	. 2 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ Q ∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)))
2		nfv 1581	. . 3 ⊢ Ⅎ𝑥(⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵)
3		nfre1 2593	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈
4		simpr1 1034	. . . . . . . 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 1036	. . . . . . . . . 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 542	. . . . . . . . . . 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 7756	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
8		breq1 4133	. . . . . . . . . . 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 539	. . . . . . . . . 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 1035	. . . . . . . . . 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 7852	. . . . . . . . . 10 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑦 ∈ 𝑈) → (𝑦 <Q (𝑥 ·Q 𝐵) → (𝑥 ·Q 𝐵) ∈ 𝑈))
14	11, 12, 13	syl2anc 415	. . . . . . . . 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 2599	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐿 ∧ (𝑥 ·Q 𝐵) ∈ 𝑈) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
17	4, 15, 16	syl2anc 415	. . . . . . 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 404	. . . . 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 2668	. . . 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 2665	. 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ⟨cop 3712   class class class wbr 4130  (class class class)co 6085  Qcnq 7647  1_Qc1q 7648   ·_Q cmq 7650   <_Q cltq 7652  Pcnp 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833

This theorem is used by:  recexprlem1ssl  8000  recexprlem1ssu  8001