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Theorem mulclprlem 10042
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 10014 . . . . . 6 ((𝐴P𝑔𝐴) → 𝑔Q)
2 elprnq 10014 . . . . . 6 ((𝐵P𝐵) → Q)
3 recclnq 9989 . . . . . . . . 9 (Q → (*Q) ∈ Q)
43adantl 467 . . . . . . . 8 ((𝑔QQ) → (*Q) ∈ Q)
5 vex 3352 . . . . . . . . 9 𝑥 ∈ V
6 ovex 6822 . . . . . . . . 9 (𝑔 ·Q ) ∈ V
7 ltmnq 9995 . . . . . . . . 9 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8 fvex 6342 . . . . . . . . 9 (*Q) ∈ V
9 mulcomnq 9976 . . . . . . . . 9 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
105, 6, 7, 8, 9caovord2 6992 . . . . . . . 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 9982 . . . . . . . . . 10 ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q ( ·Q (*Q)))
13 recidnq 9988 . . . . . . . . . . 11 (Q → ( ·Q (*Q)) = 1Q)
1413oveq2d 6808 . . . . . . . . . 10 (Q → (𝑔 ·Q ( ·Q (*Q))) = (𝑔 ·Q 1Q))
1512, 14syl5eq 2816 . . . . . . . . 9 (Q → ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q 1Q))
16 mulidnq 9986 . . . . . . . . 9 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1715, 16sylan9eqr 2826 . . . . . . . 8 ((𝑔QQ) → ((𝑔 ·Q ) ·Q (*Q)) = 𝑔)
1817breq2d 4796 . . . . . . 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 575 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
21 prcdnq 10016 . . . . . 6 ((𝐴P𝑔𝐴) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2221adantr 466 . . . . 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 10007 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25 mulclnq 9970 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
2624, 25genpprecl 10024 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑥 ·Q (*Q)) ∈ 𝐴𝐵) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
2726exp4b 417 . . . . . . 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 405 . . . . 5 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3029adantlr 686 . . . 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 466 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
332adantl 467 . . 3 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → Q)
34 mulassnq 9982 . . . . . 6 ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q ((*Q) ·Q ))
35 mulcomnq 9976 . . . . . . . 8 ((*Q) ·Q ) = ( ·Q (*Q))
3635, 13syl5eq 2816 . . . . . . 7 (Q → ((*Q) ·Q ) = 1Q)
3736oveq2d 6808 . . . . . 6 (Q → (𝑥 ·Q ((*Q) ·Q )) = (𝑥 ·Q 1Q))
3834, 37syl5eq 2816 . . . . 5 (Q → ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q 1Q))
39 mulidnq 9986 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
4038, 39sylan9eq 2824 . . . 4 ((Q𝑥Q) → ((𝑥 ·Q (*Q)) ·Q ) = 𝑥)
4140eleq1d 2834 . . 3 ((Q𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4233, 41sylan 561 . 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 382  wcel 2144   class class class wbr 4784  cfv 6031  (class class class)co 6792  Qcnq 9875  1Qc1q 9876   ·Q cmq 9879  *Qcrq 9880   <Q cltq 9881  Pcnp 9882   ·P cmp 9885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717  df-er 7895  df-ni 9895  df-mi 9897  df-lti 9898  df-mpq 9932  df-ltpq 9933  df-enq 9934  df-nq 9935  df-erq 9936  df-mq 9938  df-1nq 9939  df-rq 9940  df-ltnq 9941  df-np 10004  df-mp 10007
This theorem is referenced by:  mulclpr  10043
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