Theorem prmuloc2 7722

Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7721 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 7721	. 2 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ Q ∃𝑦 ∈ Q (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)))
2		nfv 1554	. . 3 ⊢ Ⅎ𝑥(⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵)
3		nfre1 2553	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈
4		simpr1 1008	. . . . . . . 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 1010	. . . . . . . . . 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 7544	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
8		breq1 4065	. . . . . . . . . . 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 1009	. . . . . . . . . 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 7640	. . . . . . . . . 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 2559	. . . . . . . 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 2628	. . . 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 2625	. 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 983   = wceq 1375   ∈ wcel 2180  ∃wrex 2489  ⟨cop 3649   class class class wbr 4062  (class class class)co 5974  Qcnq 7435  1_Qc1q 7436   ·_Q cmq 7438   <_Q cltq 7440  Pcnp 7446

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-eprel 4357  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-1o 6532  df-2o 6533  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-pli 7460  df-mi 7461  df-lti 7462  df-plpq 7499  df-mpq 7500  df-enq 7502  df-nqqs 7503  df-plqqs 7504  df-mqqs 7505  df-1nqqs 7506  df-rq 7507  df-ltnqqs 7508  df-enq0 7579  df-nq0 7580  df-0nq0 7581  df-plq0 7582  df-mq0 7583  df-inp 7621

This theorem is referenced by:  recexprlem1ssl  7788  recexprlem1ssu  7789