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Theorem addclprlem1 10630
Description: Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
addclprlem1 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))

Proof of Theorem addclprlem1
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elprnq 10605 . . 3 ((𝐴P𝑔𝐴) → 𝑔Q)
2 ltrnq 10593 . . . . 5 (𝑥 <Q (𝑔 +Q ) ↔ (*Q‘(𝑔 +Q )) <Q (*Q𝑥))
3 ltmnq 10586 . . . . . 6 (𝑥Q → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ))) <Q (𝑥 ·Q (*Q𝑥))))
4 ovex 7246 . . . . . . 7 (𝑥 ·Q (*Q‘(𝑔 +Q ))) ∈ V
5 ovex 7246 . . . . . . 7 (𝑥 ·Q (*Q𝑥)) ∈ V
6 ltmnq 10586 . . . . . . 7 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
7 vex 3412 . . . . . . 7 𝑔 ∈ V
8 mulcomnq 10567 . . . . . . 7 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
94, 5, 6, 7, 8caovord2 7420 . . . . . 6 (𝑔Q → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) <Q (𝑥 ·Q (*Q𝑥)) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
103, 9sylan9bbr 514 . . . . 5 ((𝑔Q𝑥Q) → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
112, 10syl5bb 286 . . . 4 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
12 recidnq 10579 . . . . . . 7 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1312oveq1d 7228 . . . . . 6 (𝑥Q → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔))
14 mulcomnq 10567 . . . . . . 7 (1Q ·Q 𝑔) = (𝑔 ·Q 1Q)
15 mulidnq 10577 . . . . . . 7 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1614, 15eqtrid 2789 . . . . . 6 (𝑔Q → (1Q ·Q 𝑔) = 𝑔)
1713, 16sylan9eqr 2800 . . . . 5 ((𝑔Q𝑥Q) → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = 𝑔)
1817breq2d 5065 . . . 4 ((𝑔Q𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
1911, 18bitrd 282 . . 3 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
201, 19sylan 583 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
21 prcdnq 10607 . . 3 ((𝐴P𝑔𝐴) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2221adantr 484 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2320, 22sylbid 243 1 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
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 cplq 10469   ·Q cmq 10470  *Qcrq 10471   <Q cltq 10472  Pcnp 10473
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-pr 5322  ax-un 7523
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
This theorem is referenced by:  addclprlem2  10631
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