Theorem prmuloc2 7792

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7791 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 7791	. 2 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ Q ∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)))
2		nfv 1576	. . 3 ⊢ Ⅎ𝑥(⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵)
3		nfre1 2574	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈
4		simpr1 1029	. . . . . . . 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 1031	. . . . . . . . . 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 7614	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
8		breq1 4092	. . . . . . . . . . 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 1030	. . . . . . . . . 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 7710	. . . . . . . . . 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 2580	. . . . . . . 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 2649	. . . 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 2646	. 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 1004   = wceq 1397   ∈ wcel 2201  ∃wrex 2510  ⟨cop 3673   class class class wbr 4089  (class class class)co 6023  Qcnq 7505  1_Qc1q 7506   ·_Q cmq 7508   <_Q cltq 7510  Pcnp 7516

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-eprel 4388  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-1o 6587  df-2o 6588  df-oadd 6591  df-omul 6592  df-er 6707  df-ec 6709  df-qs 6713  df-ni 7529  df-pli 7530  df-mi 7531  df-lti 7532  df-plpq 7569  df-mpq 7570  df-enq 7572  df-nqqs 7573  df-plqqs 7574  df-mqqs 7575  df-1nqqs 7576  df-rq 7577  df-ltnqqs 7578  df-enq0 7649  df-nq0 7650  df-0nq0 7651  df-plq0 7652  df-mq0 7653  df-inp 7691

This theorem is referenced by:  recexprlem1ssl  7858  recexprlem1ssu  7859