Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  prarloclemarch Structured version   GIF version

Theorem prarloclemarch 6275
 Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6274 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
StepHypRef Expression
1 recclnq 6251 . . . 4 (B Q → (*QB) Q)
2 mulclnq 6235 . . . 4 ((A Q (*QB) Q) → (A ·Q (*QB)) Q)
31, 2sylan2 270 . . 3 ((A Q B Q) → (A ·Q (*QB)) Q)
4 archnqq 6274 . . 3 ((A ·Q (*QB)) Qx N (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q )
53, 4syl 14 . 2 ((A Q B Q) → x N (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q )
6 simpll 469 . . . . . 6 (((A Q B Q) x N) → A Q)
7 1pi 6175 . . . . . . . . . . 11 1𝑜 N
8 opelxpi 4303 . . . . . . . . . . 11 ((x N 1𝑜 N) → ⟨x, 1𝑜 (N × N))
97, 8mpan2 403 . . . . . . . . . 10 (x N → ⟨x, 1𝑜 (N × N))
10 enqex 6219 . . . . . . . . . . 11 ~Q V
1110ecelqsi 6071 . . . . . . . . . 10 (⟨x, 1𝑜 (N × N) → [⟨x, 1𝑜⟩] ~Q ((N × N) / ~Q ))
129, 11syl 14 . . . . . . . . 9 (x N → [⟨x, 1𝑜⟩] ~Q ((N × N) / ~Q ))
13 df-nqqs 6207 . . . . . . . . 9 Q = ((N × N) / ~Q )
1412, 13syl6eleqr 2113 . . . . . . . 8 (x N → [⟨x, 1𝑜⟩] ~Q Q)
1514adantl 262 . . . . . . 7 (((A Q B Q) x N) → [⟨x, 1𝑜⟩] ~Q Q)
16 simplr 470 . . . . . . 7 (((A Q B Q) x N) → B Q)
17 mulclnq 6235 . . . . . . 7 (([⟨x, 1𝑜⟩] ~Q Q B Q) → ([⟨x, 1𝑜⟩] ~Q ·Q B) Q)
1815, 16, 17syl2anc 393 . . . . . 6 (((A Q B Q) x N) → ([⟨x, 1𝑜⟩] ~Q ·Q B) Q)
1916, 1syl 14 . . . . . 6 (((A Q B Q) x N) → (*QB) Q)
20 ltmnqg 6260 . . . . . 6 ((A Q ([⟨x, 1𝑜⟩] ~Q ·Q B) Q (*QB) Q) → (A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B) ↔ ((*QB) ·Q A) <Q ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B))))
216, 18, 19, 20syl3anc 1121 . . . . 5 (((A Q B Q) x N) → (A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B) ↔ ((*QB) ·Q A) <Q ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B))))
22 mulcomnqg 6242 . . . . . . 7 (((*QB) Q A Q) → ((*QB) ·Q A) = (A ·Q (*QB)))
2319, 6, 22syl2anc 393 . . . . . 6 (((A Q B Q) x N) → ((*QB) ·Q A) = (A ·Q (*QB)))
24 mulcomnqg 6242 . . . . . . . 8 (((*QB) Q ([⟨x, 1𝑜⟩] ~Q ·Q B) Q) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)))
2519, 18, 24syl2anc 393 . . . . . . 7 (((A Q B Q) x N) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)))
26 mulassnqg 6243 . . . . . . . . 9 (([⟨x, 1𝑜⟩] ~Q Q B Q (*QB) Q) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))))
2715, 16, 19, 26syl3anc 1121 . . . . . . . 8 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))))
28 recidnq 6252 . . . . . . . . . 10 (B Q → (B ·Q (*QB)) = 1Q)
2928oveq2d 5452 . . . . . . . . 9 (B Q → ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))) = ([⟨x, 1𝑜⟩] ~Q ·Q 1Q))
3016, 29syl 14 . . . . . . . 8 (((A Q B Q) x N) → ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))) = ([⟨x, 1𝑜⟩] ~Q ·Q 1Q))
31 mulidnq 6248 . . . . . . . . 9 ([⟨x, 1𝑜⟩] ~Q Q → ([⟨x, 1𝑜⟩] ~Q ·Q 1Q) = [⟨x, 1𝑜⟩] ~Q )
3215, 31syl 14 . . . . . . . 8 (((A Q B Q) x N) → ([⟨x, 1𝑜⟩] ~Q ·Q 1Q) = [⟨x, 1𝑜⟩] ~Q )
3327, 30, 323eqtrd 2058 . . . . . . 7 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = [⟨x, 1𝑜⟩] ~Q )
3425, 33eqtrd 2054 . . . . . 6 (((A Q B Q) x N) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = [⟨x, 1𝑜⟩] ~Q )
3523, 34breq12d 3751 . . . . 5 (((A Q B Q) x N) → (((*QB) ·Q A) <Q ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) ↔ (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q ))
3621, 35bitrd 177 . . . 4 (((A Q B Q) x N) → (A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B) ↔ (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q ))
3736biimprd 147 . . 3 (((A Q B Q) x N) → ((A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~QA <Q ([⟨x, 1𝑜⟩] ~Q ·Q B)))
3837reximdva 2399 . 2 ((A Q B Q) → (x N (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Qx N A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B)))
395, 38mpd 13 1 ((A Q B Q) → x N A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  ⟨cop 3353   class class class wbr 3738   × cxp 4270  ‘cfv 4829  (class class class)co 5436  1𝑜c1o 5909  [cec 6015   / cqs 6016  Ncnpi 6130   ~Q ceq 6137  Qcnq 6138  1Qc1q 6139   ·Q cmq 6141  *Qcrq 6142
 Copyright terms: Public domain W3C validator