Theorem prarloclemarch 7775

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7774 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 7749	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
2		mulclnq 7733	. . . 4 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐵) ∈ Q) → (𝐴 ·Q (*Q‘𝐵)) ∈ Q)
3	1, 2	sylan2 286	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q (*Q‘𝐵)) ∈ Q)
4		archnqq 7774	. . 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 531	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐴 ∈ Q)
7		1pi 7672	. . . . . . . . . . 11 ⊢ 1o ∈ N
8		opelxpi 4801	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ N ∧ 1o ∈ N) → ⟨𝑥, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 429	. . . . . . . . . 10 ⊢ (𝑥 ∈ N → ⟨𝑥, 1o⟩ ∈ (N × N))
10		enqex 7717	. . . . . . . . . . 11 ⊢ ~Q ∈ V
11	10	ecelqsi 6853	. . . . . . . . . 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 7705	. . . . . . . . 9 ⊢ Q = ((N × N) / ~Q )
14	12, 13	eleqtrrdi 2332	. . . . . . . 8 ⊢ (𝑥 ∈ N → [⟨𝑥, 1o⟩] ~Q ∈ Q)
15	14	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → [⟨𝑥, 1o⟩] ~Q ∈ Q)
16		simplr 533	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐵 ∈ Q)
17		mulclnq 7733	. . . . . . 7 ⊢ (([⟨𝑥, 1o⟩] ~Q ∈ Q ∧ 𝐵 ∈ Q) → ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q)
18	15, 16, 17	syl2anc 415	. . . . . 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 7758	. . . . . 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 1278	. . . . 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 7740	. . . . . . 7 ⊢ (((*Q‘𝐵) ∈ Q ∧ 𝐴 ∈ Q) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
23	19, 6, 22	syl2anc 415	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
24		mulcomnqg 7740	. . . . . . . 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 415	. . . . . . 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 7741	. . . . . . . . 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 1278	. . . . . . . 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 7750	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
29	28	oveq2d 6091	. . . . . . . . 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 7746	. . . . . . . . 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 2275	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)) = [⟨𝑥, 1o⟩] ~Q )
34	25, 33	eqtrd 2271	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1o⟩] ~Q )
35	23, 34	breq12d 4138	. . . . 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 188	. . . 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 158	. . 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 2652	. 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 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ⟨cop 3708   class class class wbr 4125   × cxp 4767  ‘cfv 5372  (class class class)co 6075  1_oc1o 6670  [cec 6795   / cqs 6796  Ncnpi 7629   ~_Q ceq 7636  Qcnq 7637  1_Qc1q 7638   ·_Q cmq 7640  *_Qcrq 7641   <_Q cltq 7642

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710

This theorem is referenced by:  prarloclemarch2  7776