MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulclprlem Structured version   Visualization version   GIF version

Theorem mulclprlem 10633
Description: Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 14-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
mulclprlem ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
Distinct variable groups:   𝑥,𝑔,   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑔,)   𝐵(𝑔,)

Proof of Theorem mulclprlem
Dummy variables 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elprnq 10605 . . . . . 6 ((𝐴P𝑔𝐴) → 𝑔Q)
2 elprnq 10605 . . . . . 6 ((𝐵P𝐵) → Q)
3 recclnq 10580 . . . . . . . . 9 (Q → (*Q) ∈ Q)
43adantl 485 . . . . . . . 8 ((𝑔QQ) → (*Q) ∈ Q)
5 vex 3412 . . . . . . . . 9 𝑥 ∈ V
6 ovex 7246 . . . . . . . . 9 (𝑔 ·Q ) ∈ V
7 ltmnq 10586 . . . . . . . . 9 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8 fvex 6730 . . . . . . . . 9 (*Q) ∈ V
9 mulcomnq 10567 . . . . . . . . 9 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
105, 6, 7, 8, 9caovord2 7420 . . . . . . . 8 ((*Q) ∈ Q → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q ((𝑔 ·Q ) ·Q (*Q))))
114, 10syl 17 . . . . . . 7 ((𝑔QQ) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q ((𝑔 ·Q ) ·Q (*Q))))
12 mulassnq 10573 . . . . . . . . . 10 ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q ( ·Q (*Q)))
13 recidnq 10579 . . . . . . . . . . 11 (Q → ( ·Q (*Q)) = 1Q)
1413oveq2d 7229 . . . . . . . . . 10 (Q → (𝑔 ·Q ( ·Q (*Q))) = (𝑔 ·Q 1Q))
1512, 14eqtrid 2789 . . . . . . . . 9 (Q → ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q 1Q))
16 mulidnq 10577 . . . . . . . . 9 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1715, 16sylan9eqr 2800 . . . . . . . 8 ((𝑔QQ) → ((𝑔 ·Q ) ·Q (*Q)) = 𝑔)
1817breq2d 5065 . . . . . . 7 ((𝑔QQ) → ((𝑥 ·Q (*Q)) <Q ((𝑔 ·Q ) ·Q (*Q)) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
1911, 18bitrd 282 . . . . . 6 ((𝑔QQ) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
201, 2, 19syl2an 599 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
21 prcdnq 10607 . . . . . 6 ((𝐴P𝑔𝐴) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2221adantr 484 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2320, 22sylbid 243 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) → (𝑥 ·Q (*Q)) ∈ 𝐴))
24 df-mp 10598 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25 mulclnq 10561 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
2624, 25genpprecl 10615 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑥 ·Q (*Q)) ∈ 𝐴𝐵) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
2726exp4b 434 . . . . . . 7 (𝐴P → (𝐵P → ((𝑥 ·Q (*Q)) ∈ 𝐴 → (𝐵 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))))
2827com34 91 . . . . . 6 (𝐴P → (𝐵P → (𝐵 → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))))
2928imp32 422 . . . . 5 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3029adantlr 715 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3123, 30syld 47 . . 3 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3231adantr 484 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
332adantl 485 . . 3 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → Q)
34 mulassnq 10573 . . . . . 6 ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q ((*Q) ·Q ))
35 mulcomnq 10567 . . . . . . . 8 ((*Q) ·Q ) = ( ·Q (*Q))
3635, 13eqtrid 2789 . . . . . . 7 (Q → ((*Q) ·Q ) = 1Q)
3736oveq2d 7229 . . . . . 6 (Q → (𝑥 ·Q ((*Q) ·Q )) = (𝑥 ·Q 1Q))
3834, 37eqtrid 2789 . . . . 5 (Q → ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q 1Q))
39 mulidnq 10577 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
4038, 39sylan9eq 2798 . . . 4 ((Q𝑥Q) → ((𝑥 ·Q (*Q)) ·Q ) = 𝑥)
4140eleq1d 2822 . . 3 ((Q𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4233, 41sylan 583 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4332, 42sylibd 242 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2110   class class class wbr 5053  cfv 6380  (class class class)co 7213  Qcnq 10466  1Qc1q 10467   ·Q cmq 10470  *Qcrq 10471   <Q cltq 10472  Pcnp 10473   ·P cmp 10476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-ni 10486  df-mi 10488  df-lti 10489  df-mpq 10523  df-ltpq 10524  df-enq 10525  df-nq 10526  df-erq 10527  df-mq 10529  df-1nq 10530  df-rq 10531  df-ltnq 10532  df-np 10595  df-mp 10598
This theorem is referenced by:  mulclpr  10634
  Copyright terms: Public domain W3C validator