Theorem prarloclemarch 7226

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7225 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 7200	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
2		mulclnq 7184	. . . 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 7225	. . 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 518	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐴 ∈ Q)
7		1pi 7123	. . . . . . . . . . 11 ⊢ 1o ∈ N
8		opelxpi 4571	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ N ∧ 1o ∈ N) → ⟨𝑥, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 421	. . . . . . . . . 10 ⊢ (𝑥 ∈ N → ⟨𝑥, 1o⟩ ∈ (N × N))
10		enqex 7168	. . . . . . . . . . 11 ⊢ ~Q ∈ V
11	10	ecelqsi 6483	. . . . . . . . . 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 7156	. . . . . . . . 9 ⊢ Q = ((N × N) / ~Q )
14	12, 13	eleqtrrdi 2233	. . . . . . . 8 ⊢ (𝑥 ∈ N → [⟨𝑥, 1o⟩] ~Q ∈ Q)
15	14	adantl 275	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → [⟨𝑥, 1o⟩] ~Q ∈ Q)
16		simplr 519	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐵 ∈ Q)
17		mulclnq 7184	. . . . . . 7 ⊢ (([⟨𝑥, 1o⟩] ~Q ∈ Q ∧ 𝐵 ∈ Q) → ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q)
18	15, 16, 17	syl2anc 408	. . . . . 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 7209	. . . . . 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 1216	. . . . 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 7191	. . . . . . 7 ⊢ (((*Q‘𝐵) ∈ Q ∧ 𝐴 ∈ Q) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
23	19, 6, 22	syl2anc 408	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
24		mulcomnqg 7191	. . . . . . . 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 408	. . . . . . 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 7192	. . . . . . . . 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 1216	. . . . . . . 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 7201	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
29	28	oveq2d 5790	. . . . . . . . 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 7197	. . . . . . . . 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 2176	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)) = [⟨𝑥, 1o⟩] ~Q )
34	25, 33	eqtrd 2172	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1o⟩] ~Q )
35	23, 34	breq12d 3942	. . . . 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 2534	. 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 1331   ∈ wcel 1480  ∃wrex 2417  ⟨cop 3530   class class class wbr 3929   × cxp 4537  ‘cfv 5123  (class class class)co 5774  1_oc1o 6306  [cec 6427   / cqs 6428  Ncnpi 7080   ~_Q ceq 7087  Qcnq 7088  1_Qc1q 7089   ·_Q cmq 7091  *_Qcrq 7092   <_Q cltq 7093

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161

This theorem is referenced by:  prarloclemarch2  7227