Theorem prarloclemarch 6516

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6515 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 6490	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
2		mulclnq 6474	. . . 4 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐵) ∈ Q) → (𝐴 ·Q (*Q‘𝐵)) ∈ Q)
3	1, 2	sylan2 270	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q (*Q‘𝐵)) ∈ Q)
4		archnqq 6515	. . 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 481	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐴 ∈ Q)
7		1pi 6413	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ N
8		opelxpi 4376	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ N ∧ 1𝑜 ∈ N) → ⟨𝑥, 1𝑜⟩ ∈ (N × N))
9	7, 8	mpan2 401	. . . . . . . . . 10 ⊢ (𝑥 ∈ N → ⟨𝑥, 1𝑜⟩ ∈ (N × N))
10		enqex 6458	. . . . . . . . . . 11 ⊢ ~Q ∈ V
11	10	ecelqsi 6160	. . . . . . . . . 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 6446	. . . . . . . . 9 ⊢ Q = ((N × N) / ~Q )
14	12, 13	syl6eleqr 2131	. . . . . . . 8 ⊢ (𝑥 ∈ N → [⟨𝑥, 1𝑜⟩] ~Q ∈ Q)
15	14	adantl 262	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → [⟨𝑥, 1𝑜⟩] ~Q ∈ Q)
16		simplr 482	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → 𝐵 ∈ Q)
17		mulclnq 6474	. . . . . . 7 ⊢ (([⟨𝑥, 1𝑜⟩] ~Q ∈ Q ∧ 𝐵 ∈ Q) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ∈ Q)
18	15, 16, 17	syl2anc 391	. . . . . 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 6499	. . . . . 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 1135	. . . . 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 6481	. . . . . . 7 ⊢ (((*Q‘𝐵) ∈ Q ∧ 𝐴 ∈ Q) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
23	19, 6, 22	syl2anc 391	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q‘𝐵)))
24		mulcomnqg 6481	. . . . . . . 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 391	. . . . . . 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 6482	. . . . . . . . 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 1135	. . . . . . . 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 6491	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
29	28	oveq2d 5528	. . . . . . . . 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 6487	. . . . . . . . 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 2076	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q‘𝐵)) = [⟨𝑥, 1𝑜⟩] ~Q )
34	25, 33	eqtrd 2072	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑥 ∈ N) → ((*Q‘𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1𝑜⟩] ~Q )
35	23, 34	breq12d 3777	. . . . 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 177	. . . 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 147	. . 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 2421	. 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 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764   × cxp 4343  ‘cfv 4902  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104   / cqs 6105  Ncnpi 6370   ~_Q ceq 6377  Qcnq 6378  1_Qc1q 6379   ·_Q cmq 6381  *_Qcrq 6382   <_Q cltq 6383

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451

This theorem is referenced by:  prarloclemarch2  6517