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Theorem mulclprlem 11057
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 11029 . . . . . 6 ((𝐴P𝑔𝐴) → 𝑔Q)
2 elprnq 11029 . . . . . 6 ((𝐵P𝐵) → Q)
3 recclnq 11004 . . . . . . . . 9 (Q → (*Q) ∈ Q)
43adantl 481 . . . . . . . 8 ((𝑔QQ) → (*Q) ∈ Q)
5 vex 3482 . . . . . . . . 9 𝑥 ∈ V
6 ovex 7464 . . . . . . . . 9 (𝑔 ·Q ) ∈ V
7 ltmnq 11010 . . . . . . . . 9 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8 fvex 6920 . . . . . . . . 9 (*Q) ∈ V
9 mulcomnq 10991 . . . . . . . . 9 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
105, 6, 7, 8, 9caovord2 7645 . . . . . . . 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 10997 . . . . . . . . . 10 ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q ( ·Q (*Q)))
13 recidnq 11003 . . . . . . . . . . 11 (Q → ( ·Q (*Q)) = 1Q)
1413oveq2d 7447 . . . . . . . . . 10 (Q → (𝑔 ·Q ( ·Q (*Q))) = (𝑔 ·Q 1Q))
1512, 14eqtrid 2787 . . . . . . . . 9 (Q → ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q 1Q))
16 mulidnq 11001 . . . . . . . . 9 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1715, 16sylan9eqr 2797 . . . . . . . 8 ((𝑔QQ) → ((𝑔 ·Q ) ·Q (*Q)) = 𝑔)
1817breq2d 5160 . . . . . . 7 ((𝑔QQ) → ((𝑥 ·Q (*Q)) <Q ((𝑔 ·Q ) ·Q (*Q)) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
1911, 18bitrd 279 . . . . . 6 ((𝑔QQ) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
201, 2, 19syl2an 596 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
21 prcdnq 11031 . . . . . 6 ((𝐴P𝑔𝐴) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2221adantr 480 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2320, 22sylbid 240 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) → (𝑥 ·Q (*Q)) ∈ 𝐴))
24 df-mp 11022 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25 mulclnq 10985 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
2624, 25genpprecl 11039 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑥 ·Q (*Q)) ∈ 𝐴𝐵) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
2726exp4b 430 . . . . . . 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 418 . . . . 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 480 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
332adantl 481 . . 3 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → Q)
34 mulassnq 10997 . . . . . 6 ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q ((*Q) ·Q ))
35 mulcomnq 10991 . . . . . . . 8 ((*Q) ·Q ) = ( ·Q (*Q))
3635, 13eqtrid 2787 . . . . . . 7 (Q → ((*Q) ·Q ) = 1Q)
3736oveq2d 7447 . . . . . 6 (Q → (𝑥 ·Q ((*Q) ·Q )) = (𝑥 ·Q 1Q))
3834, 37eqtrid 2787 . . . . 5 (Q → ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q 1Q))
39 mulidnq 11001 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
4038, 39sylan9eq 2795 . . . 4 ((Q𝑥Q) → ((𝑥 ·Q (*Q)) ·Q ) = 𝑥)
4140eleq1d 2824 . . 3 ((Q𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4233, 41sylan 580 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4332, 42sylibd 239 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Qcnq 10890  1Qc1q 10891   ·Q cmq 10894  *Qcrq 10895   <Q cltq 10896  Pcnp 10897   ·P cmp 10900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ni 10910  df-mi 10912  df-lti 10913  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-mp 11022
This theorem is referenced by:  mulclpr  11058
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