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Theorem mulclprlem 9694
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 9666 . . . . . 6 ((𝐴P𝑔𝐴) → 𝑔Q)
2 elprnq 9666 . . . . . 6 ((𝐵P𝐵) → Q)
3 recclnq 9641 . . . . . . . . 9 (Q → (*Q) ∈ Q)
43adantl 480 . . . . . . . 8 ((𝑔QQ) → (*Q) ∈ Q)
5 vex 3172 . . . . . . . . 9 𝑥 ∈ V
6 ovex 6552 . . . . . . . . 9 (𝑔 ·Q ) ∈ V
7 ltmnq 9647 . . . . . . . . 9 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8 fvex 6095 . . . . . . . . 9 (*Q) ∈ V
9 mulcomnq 9628 . . . . . . . . 9 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
105, 6, 7, 8, 9caovord2 6718 . . . . . . . 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 9634 . . . . . . . . . 10 ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q ( ·Q (*Q)))
13 recidnq 9640 . . . . . . . . . . 11 (Q → ( ·Q (*Q)) = 1Q)
1413oveq2d 6540 . . . . . . . . . 10 (Q → (𝑔 ·Q ( ·Q (*Q))) = (𝑔 ·Q 1Q))
1512, 14syl5eq 2652 . . . . . . . . 9 (Q → ((𝑔 ·Q ) ·Q (*Q)) = (𝑔 ·Q 1Q))
16 mulidnq 9638 . . . . . . . . 9 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1715, 16sylan9eqr 2662 . . . . . . . 8 ((𝑔QQ) → ((𝑔 ·Q ) ·Q (*Q)) = 𝑔)
1817breq2d 4586 . . . . . . 7 ((𝑔QQ) → ((𝑥 ·Q (*Q)) <Q ((𝑔 ·Q ) ·Q (*Q)) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
1911, 18bitrd 266 . . . . . 6 ((𝑔QQ) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
201, 2, 19syl2an 492 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) ↔ (𝑥 ·Q (*Q)) <Q 𝑔))
21 prcdnq 9668 . . . . . 6 ((𝐴P𝑔𝐴) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2221adantr 479 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) <Q 𝑔 → (𝑥 ·Q (*Q)) ∈ 𝐴))
2320, 22sylbid 228 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) → (𝑥 ·Q (*Q)) ∈ 𝐴))
24 df-mp 9659 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25 mulclnq 9622 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
2624, 25genpprecl 9676 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑥 ·Q (*Q)) ∈ 𝐴𝐵) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
2726exp4b 629 . . . . . . 7 (𝐴P → (𝐵P → ((𝑥 ·Q (*Q)) ∈ 𝐴 → (𝐵 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))))
2827com34 88 . . . . . 6 (𝐴P → (𝐵P → (𝐵 → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))))
2928imp32 447 . . . . 5 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3029adantlr 746 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑥 ·Q (*Q)) ∈ 𝐴 → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3123, 30syld 45 . . 3 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 <Q (𝑔 ·Q ) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
3231adantr 479 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → ((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵)))
332adantl 480 . . 3 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → Q)
34 mulassnq 9634 . . . . . 6 ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q ((*Q) ·Q ))
35 mulcomnq 9628 . . . . . . . 8 ((*Q) ·Q ) = ( ·Q (*Q))
3635, 13syl5eq 2652 . . . . . . 7 (Q → ((*Q) ·Q ) = 1Q)
3736oveq2d 6540 . . . . . 6 (Q → (𝑥 ·Q ((*Q) ·Q )) = (𝑥 ·Q 1Q))
3834, 37syl5eq 2652 . . . . 5 (Q → ((𝑥 ·Q (*Q)) ·Q ) = (𝑥 ·Q 1Q))
39 mulidnq 9638 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
4038, 39sylan9eq 2660 . . . 4 ((Q𝑥Q) → ((𝑥 ·Q (*Q)) ·Q ) = 𝑥)
4140eleq1d 2668 . . 3 ((Q𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4233, 41sylan 486 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q)) ·Q ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
4332, 42sylibd 227 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976   class class class wbr 4574  cfv 5787  (class class class)co 6524  Qcnq 9527  1Qc1q 9528   ·Q cmq 9531  *Qcrq 9532   <Q cltq 9533  Pcnp 9534   ·P cmp 9537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-omul 7426  df-er 7603  df-ni 9547  df-mi 9549  df-lti 9550  df-mpq 9584  df-ltpq 9585  df-enq 9586  df-nq 9587  df-erq 9588  df-mq 9590  df-1nq 9591  df-rq 9592  df-ltnq 9593  df-np 9656  df-mp 9659
This theorem is referenced by:  mulclpr  9695
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