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Theorem addclprlem1 9783
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 9758 . . 3 ((𝐴P𝑔𝐴) → 𝑔Q)
2 ltrnq 9746 . . . . 5 (𝑥 <Q (𝑔 +Q ) ↔ (*Q‘(𝑔 +Q )) <Q (*Q𝑥))
3 ltmnq 9739 . . . . . 6 (𝑥Q → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ))) <Q (𝑥 ·Q (*Q𝑥))))
4 ovex 6633 . . . . . . 7 (𝑥 ·Q (*Q‘(𝑔 +Q ))) ∈ V
5 ovex 6633 . . . . . . 7 (𝑥 ·Q (*Q𝑥)) ∈ V
6 ltmnq 9739 . . . . . . 7 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
7 vex 3194 . . . . . . 7 𝑔 ∈ V
8 mulcomnq 9720 . . . . . . 7 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
94, 5, 6, 7, 8caovord2 6800 . . . . . 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 736 . . . . 5 ((𝑔Q𝑥Q) → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
112, 10syl5bb 272 . . . 4 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
12 recidnq 9732 . . . . . . 7 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1312oveq1d 6620 . . . . . 6 (𝑥Q → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔))
14 mulcomnq 9720 . . . . . . 7 (1Q ·Q 𝑔) = (𝑔 ·Q 1Q)
15 mulidnq 9730 . . . . . . 7 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1614, 15syl5eq 2672 . . . . . 6 (𝑔Q → (1Q ·Q 𝑔) = 𝑔)
1713, 16sylan9eqr 2682 . . . . 5 ((𝑔Q𝑥Q) → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = 𝑔)
1817breq2d 4630 . . . 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 268 . . 3 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
201, 19sylan 488 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
21 prcdnq 9760 . . 3 ((𝐴P𝑔𝐴) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2221adantr 481 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2320, 22sylbid 230 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 196  wa 384  wcel 1992   class class class wbr 4618  cfv 5850  (class class class)co 6605  Qcnq 9619  1Qc1q 9620   +Q cplq 9622   ·Q cmq 9623  *Qcrq 9624   <Q cltq 9625  Pcnp 9626
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-omul 7511  df-er 7688  df-ni 9639  df-mi 9641  df-lti 9642  df-mpq 9676  df-ltpq 9677  df-enq 9678  df-nq 9679  df-erq 9680  df-mq 9682  df-1nq 9683  df-rq 9684  df-ltnq 9685  df-np 9748
This theorem is referenced by:  addclprlem2  9784
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