Theorem prmuloc2 7878

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7877 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 7877	. 2 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ Q ∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)))
2		nfv 1577	. . 3 ⊢ Ⅎ𝑥(⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵)
3		nfre1 2585	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈
4		simpr1 1030	. . . . . . . 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 1032	. . . . . . . . . 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 538	. . . . . . . . . . 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 7700	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
8		breq1 4111	. . . . . . . . . . 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 535	. . . . . . . . . 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 1031	. . . . . . . . . 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 7796	. . . . . . . . . 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 2591	. . . . . . . 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 2660	. . . 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 2657	. 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 1005   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  ⟨cop 3691   class class class wbr 4108  (class class class)co 6049  Qcnq 7591  1_Qc1q 7592   ·_Q cmq 7594   <_Q cltq 7596  Pcnp 7602

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664  df-enq0 7735  df-nq0 7736  df-0nq0 7737  df-plq0 7738  df-mq0 7739  df-inp 7777

This theorem is referenced by:  recexprlem1ssl  7944  recexprlem1ssu  7945