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Theorem mulclprlem 9801
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 9773 . . . . . 6 ((𝐴P𝑔𝐴) → 𝑔Q)
2 elprnq 9773 . . . . . 6 ((𝐵P𝐵) → Q)
3 recclnq 9748 . . . . . . . . 9 (Q → (*Q) ∈ Q)
43adantl 482 . . . . . . . 8 ((𝑔QQ) → (*Q) ∈ Q)
5 vex 3193 . . . . . . . . 9 𝑥 ∈ V
6 ovex 6643 . . . . . . . . 9 (𝑔 ·Q ) ∈ V
7 ltmnq 9754 . . . . . . . . 9 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8 fvex 6168 . . . . . . . . 9 (*Q) ∈ V
9 mulcomnq 9735 . . . . . . . . 9 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
105, 6, 7, 8, 9caovord2 6811 . . . . . . . 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 9741 . . . . . . . . . 10 ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q ( ·Q (*Q)))
13 recidnq 9747 . . . . . . . . . . 11 (Q → ( ·Q (*Q)) = 1Q)
1413oveq2d 6631 . . . . . . . . . 10 (Q → (𝑔 ·Q ( ·Q (*Q))) = (𝑔 ·Q 1Q))
1512, 14syl5eq 2667 . . . . . . . . 9 (Q → ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q 1Q))
16 mulidnq 9745 . . . . . . . . 9 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1715, 16sylan9eqr 2677 . . . . . . . 8 ((𝑔QQ) → ((𝑔 ·Q ) ·Q (*Q)) = 𝑔)
1817breq2d 4635 . . . . . . 7 ((𝑔QQ) → ((𝑥 ·Q (*Q)) <Q ((𝑔 ·Q ) ·Q (*Q)) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
1911, 18bitrd 268 . . . . . 6 ((𝑔QQ) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
201, 2, 19syl2an 494 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
21 prcdnq 9775 . . . . . 6 ((𝐴P𝑔𝐴) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2221adantr 481 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2320, 22sylbid 230 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) → (𝑥 ·Q (*Q)) ∈ 𝐴))
24 df-mp 9766 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25 mulclnq 9729 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
2624, 25genpprecl 9783 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑥 ·Q (*Q)) ∈ 𝐴𝐵) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
2726exp4b 631 . . . . . . 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 449 . . . . 5 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3029adantlr 750 . . . 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 481 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
332adantl 482 . . 3 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → Q)
34 mulassnq 9741 . . . . . 6 ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q ((*Q) ·Q ))
35 mulcomnq 9735 . . . . . . . 8 ((*Q) ·Q ) = ( ·Q (*Q))
3635, 13syl5eq 2667 . . . . . . 7 (Q → ((*Q) ·Q ) = 1Q)
3736oveq2d 6631 . . . . . 6 (Q → (𝑥 ·Q ((*Q) ·Q )) = (𝑥 ·Q 1Q))
3834, 37syl5eq 2667 . . . . 5 (Q → ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q 1Q))
39 mulidnq 9745 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
4038, 39sylan9eq 2675 . . . 4 ((Q𝑥Q) → ((𝑥 ·Q (*Q)) ·Q ) = 𝑥)
4140eleq1d 2683 . . 3 ((Q𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4233, 41sylan 488 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4332, 42sylibd 229 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987   class class class wbr 4623  cfv 5857  (class class class)co 6615  Qcnq 9634  1Qc1q 9635   ·Q cmq 9638  *Qcrq 9639   <Q cltq 9640  Pcnp 9641   ·P cmp 9644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ni 9654  df-mi 9656  df-lti 9657  df-mpq 9691  df-ltpq 9692  df-enq 9693  df-nq 9694  df-erq 9695  df-mq 9697  df-1nq 9698  df-rq 9699  df-ltnq 9700  df-np 9763  df-mp 9766
This theorem is referenced by:  mulclpr  9802
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