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Theorem addclprlem1 10173
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 10148 . . 3 ((𝐴P𝑔𝐴) → 𝑔Q)
2 ltrnq 10136 . . . . 5 (𝑥 <Q (𝑔 +Q ) ↔ (*Q‘(𝑔 +Q )) <Q (*Q𝑥))
3 ltmnq 10129 . . . . . 6 (𝑥Q → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ))) <Q (𝑥 ·Q (*Q𝑥))))
4 ovex 6954 . . . . . . 7 (𝑥 ·Q (*Q‘(𝑔 +Q ))) ∈ V
5 ovex 6954 . . . . . . 7 (𝑥 ·Q (*Q𝑥)) ∈ V
6 ltmnq 10129 . . . . . . 7 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
7 vex 3400 . . . . . . 7 𝑔 ∈ V
8 mulcomnq 10110 . . . . . . 7 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
94, 5, 6, 7, 8caovord2 7123 . . . . . 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 506 . . . . 5 ((𝑔Q𝑥Q) → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
112, 10syl5bb 275 . . . 4 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
12 recidnq 10122 . . . . . . 7 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1312oveq1d 6937 . . . . . 6 (𝑥Q → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔))
14 mulcomnq 10110 . . . . . . 7 (1Q ·Q 𝑔) = (𝑔 ·Q 1Q)
15 mulidnq 10120 . . . . . . 7 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1614, 15syl5eq 2825 . . . . . 6 (𝑔Q → (1Q ·Q 𝑔) = 𝑔)
1713, 16sylan9eqr 2835 . . . . 5 ((𝑔Q𝑥Q) → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = 𝑔)
1817breq2d 4898 . . . 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 271 . . 3 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
201, 19sylan 575 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
21 prcdnq 10150 . . 3 ((𝐴P𝑔𝐴) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2221adantr 474 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2320, 22sylbid 232 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 198  wa 386  wcel 2106   class class class wbr 4886  cfv 6135  (class class class)co 6922  Qcnq 10009  1Qc1q 10010   +Q cplq 10012   ·Q cmq 10013  *Qcrq 10014   <Q cltq 10015  Pcnp 10016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-omul 7848  df-er 8026  df-ni 10029  df-mi 10031  df-lti 10032  df-mpq 10066  df-ltpq 10067  df-enq 10068  df-nq 10069  df-erq 10070  df-mq 10072  df-1nq 10073  df-rq 10074  df-ltnq 10075  df-np 10138
This theorem is referenced by:  addclprlem2  10174
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