Theorem prarloclemarch 7359

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7358 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 7333	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
2		mulclnq 7317	. . . 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 7358	. . 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 519	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐴 ∈ Q)
7		1pi 7256	. . . . . . . . . . 11 ⊢ 1o ∈ N
8		opelxpi 4636	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ N ∧ 1o ∈ N) → ⟨𝑥, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 422	. . . . . . . . . 10 ⊢ (𝑥 ∈ N → ⟨𝑥, 1o⟩ ∈ (N × N))
10		enqex 7301	. . . . . . . . . . 11 ⊢ ~Q ∈ V
11	10	ecelqsi 6555	. . . . . . . . . 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 7289	. . . . . . . . 9 ⊢ Q = ((N × N) / ~Q )
14	12, 13	eleqtrrdi 2260	. . . . . . . 8 ⊢ (𝑥 ∈ N → [⟨𝑥, 1o⟩] ~Q ∈ Q)
15	14	adantl 275	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → [⟨𝑥, 1o⟩] ~Q ∈ Q)
16		simplr 520	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐵 ∈ Q)
17		mulclnq 7317	. . . . . . 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 7342	. . . . . 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 1228	. . . . 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 7324	. . . . . . 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 7324	. . . . . . . 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 7325	. . . . . . . . 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 1228	. . . . . . . 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 7334	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
29	28	oveq2d 5858	. . . . . . . . 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 7330	. . . . . . . . 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 2202	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)) = [⟨𝑥, 1o⟩] ~Q )
34	25, 33	eqtrd 2198	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1o⟩] ~Q )
35	23, 34	breq12d 3995	. . . . 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 2568	. 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 1343   ∈ wcel 2136  ∃wrex 2445  ⟨cop 3579   class class class wbr 3982   × cxp 4602  ‘cfv 5188  (class class class)co 5842  1_oc1o 6377  [cec 6499   / cqs 6500  Ncnpi 7213   ~_Q ceq 7220  Qcnq 7221  1_Qc1q 7222   ·_Q cmq 7224  *_Qcrq 7225   <_Q cltq 7226

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294

This theorem is referenced by:  prarloclemarch2  7360