Theorem prarloclemarch 7380

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7379 in the sense that we provide an integer which is larger than a given rational 𝐴 even after being multiplied by a second rational 𝐵. (Contributed by Jim Kingdon, 30-Nov-2019.)

Assertion

Ref	Expression
prarloclemarch	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑥 ∈ N 𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem prarloclemarch

Step	Hyp	Ref	Expression
1		recclnq 7354	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
2		mulclnq 7338	. . . 4 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐵) ∈ Q) → (𝐴 ·Q (*Q‘𝐵)) ∈ Q)
3	1, 2	sylan2 284	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q (*Q‘𝐵)) ∈ Q)
4		archnqq 7379	. . 3 ⊢ ((𝐴 ·Q (*Q‘𝐵)) ∈ Q → ∃𝑥 ∈ N (𝐴 ·Q (*Q‘𝐵)) <Q [⟨𝑥, 1o⟩] ~Q )
5	3, 4	syl 14	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑥 ∈ N (𝐴 ·Q (*Q‘𝐵)) <Q [⟨𝑥, 1o⟩] ~Q )
6		simpll 524	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐴 ∈ Q)
7		1pi 7277	. . . . . . . . . . 11 ⊢ 1o ∈ N
8		opelxpi 4643	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ N ∧ 1o ∈ N) → ⟨𝑥, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 423	. . . . . . . . . 10 ⊢ (𝑥 ∈ N → ⟨𝑥, 1o⟩ ∈ (N × N))
10		enqex 7322	. . . . . . . . . . 11 ⊢ ~Q ∈ V
11	10	ecelqsi 6567	. . . . . . . . . 10 ⊢ (⟨𝑥, 1o⟩ ∈ (N × N) → [⟨𝑥, 1o⟩] ~Q ∈ ((N × N) / ~Q ))
12	9, 11	syl 14	. . . . . . . . 9 ⊢ (𝑥 ∈ N → [⟨𝑥, 1o⟩] ~Q ∈ ((N × N) / ~Q ))
13		df-nqqs 7310	. . . . . . . . 9 ⊢ Q = ((N × N) / ~Q )
14	12, 13	eleqtrrdi 2264	. . . . . . . 8 ⊢ (𝑥 ∈ N → [⟨𝑥, 1o⟩] ~Q ∈ Q)
15	14	adantl 275	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → [⟨𝑥, 1o⟩] ~Q ∈ Q)
16		simplr 525	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐵 ∈ Q)
17		mulclnq 7338	. . . . . . 7 ⊢ (([⟨𝑥, 1o⟩] ~Q ∈ Q ∧ 𝐵 ∈ Q) → ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q)
18	15, 16, 17	syl2anc 409	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q)
19	16, 1	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (*Q‘𝐵) ∈ Q)
20		ltmnqg 7363	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q ∧ (*Q‘𝐵) ∈ Q) → (𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ↔ ((*Q‘𝐵) ·Q 𝐴) <Q ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵))))
21	6, 18, 19, 20	syl3anc 1233	. . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ↔ ((*Q‘𝐵) ·Q 𝐴) <Q ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵))))
22		mulcomnqg 7345	. . . . . . 7 ⊢ (((*Q‘𝐵) ∈ Q ∧ 𝐴 ∈ Q) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
23	19, 6, 22	syl2anc 409	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
24		mulcomnqg 7345	. . . . . . . 8 ⊢ (((*Q‘𝐵) ∈ Q ∧ ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q) → ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)))
25	19, 18, 24	syl2anc 409	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)))
26		mulassnqg 7346	. . . . . . . . 9 ⊢ (([⟨𝑥, 1o⟩] ~Q ∈ Q ∧ 𝐵 ∈ Q ∧ (*Q‘𝐵) ∈ Q) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)) = ([⟨𝑥, 1o⟩] ~Q ·Q (𝐵 ·Q (*Q‘𝐵))))
27	15, 16, 19, 26	syl3anc 1233	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)) = ([⟨𝑥, 1o⟩] ~Q ·Q (𝐵 ·Q (*Q‘𝐵))))
28		recidnq 7355	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
29	28	oveq2d 5869	. . . . . . . . 9 ⊢ (𝐵 ∈ Q → ([⟨𝑥, 1o⟩] ~Q ·Q (𝐵 ·Q (*Q‘𝐵))) = ([⟨𝑥, 1o⟩] ~Q ·Q 1Q))
30	16, 29	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ([⟨𝑥, 1o⟩] ~Q ·Q (𝐵 ·Q (*Q‘𝐵))) = ([⟨𝑥, 1o⟩] ~Q ·Q 1Q))
31		mulidnq 7351	. . . . . . . . 9 ⊢ ([⟨𝑥, 1o⟩] ~Q ∈ Q → ([⟨𝑥, 1o⟩] ~Q ·Q 1Q) = [⟨𝑥, 1o⟩] ~Q )
32	15, 31	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ([⟨𝑥, 1o⟩] ~Q ·Q 1Q) = [⟨𝑥, 1o⟩] ~Q )
33	27, 30, 32	3eqtrd 2207	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)) = [⟨𝑥, 1o⟩] ~Q )
34	25, 33	eqtrd 2203	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1o⟩] ~Q )
35	23, 34	breq12d 4002	. . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (((*Q‘𝐵) ·Q 𝐴) <Q ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) ↔ (𝐴 ·Q (*Q‘𝐵)) <Q [⟨𝑥, 1o⟩] ~Q ))
36	21, 35	bitrd 187	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ↔ (𝐴 ·Q (*Q‘𝐵)) <Q [⟨𝑥, 1o⟩] ~Q ))
37	36	biimprd 157	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((𝐴 ·Q (*Q‘𝐵)) <Q [⟨𝑥, 1o⟩] ~Q → 𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)))
38	37	reximdva 2572	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑥 ∈ N (𝐴 ·Q (*Q‘𝐵)) <Q [⟨𝑥, 1o⟩] ~Q → ∃𝑥 ∈ N 𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)))
39	5, 38	mpd 13	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑥 ∈ N 𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∃wrex 2449  ⟨cop 3586   class class class wbr 3989   × cxp 4609  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  [cec 6511   / cqs 6512  Ncnpi 7234   ~_Q ceq 7241  Qcnq 7242  1_Qc1q 7243   ·_Q cmq 7245  *_Qcrq 7246   <_Q cltq 7247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  prarloclemarch2  7381