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Theorem addclprlem1 11056
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 11031 . . 3 ((𝐴P𝑔𝐴) → 𝑔Q)
2 ltrnq 11019 . . . . 5 (𝑥 <Q (𝑔 +Q ) ↔ (*Q‘(𝑔 +Q )) <Q (*Q𝑥))
3 ltmnq 11012 . . . . . 6 (𝑥Q → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ))) <Q (𝑥 ·Q (*Q𝑥))))
4 ovex 7464 . . . . . . 7 (𝑥 ·Q (*Q‘(𝑔 +Q ))) ∈ V
5 ovex 7464 . . . . . . 7 (𝑥 ·Q (*Q𝑥)) ∈ V
6 ltmnq 11012 . . . . . . 7 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
7 vex 3484 . . . . . . 7 𝑔 ∈ V
8 mulcomnq 10993 . . . . . . 7 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
94, 5, 6, 7, 8caovord2 7645 . . . . . 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 510 . . . . 5 ((𝑔Q𝑥Q) → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
112, 10bitrid 283 . . . 4 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
12 recidnq 11005 . . . . . . 7 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1312oveq1d 7446 . . . . . 6 (𝑥Q → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔))
14 mulcomnq 10993 . . . . . . 7 (1Q ·Q 𝑔) = (𝑔 ·Q 1Q)
15 mulidnq 11003 . . . . . . 7 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1614, 15eqtrid 2789 . . . . . 6 (𝑔Q → (1Q ·Q 𝑔) = 𝑔)
1713, 16sylan9eqr 2799 . . . . 5 ((𝑔Q𝑥Q) → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = 𝑔)
1817breq2d 5155 . . . 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 279 . . 3 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
201, 19sylan 580 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
21 prcdnq 11033 . . 3 ((𝐴P𝑔𝐴) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2221adantr 480 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
2320, 22sylbid 240 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 206  wa 395  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Qcnq 10892  1Qc1q 10893   +Q cplq 10895   ·Q cmq 10896  *Qcrq 10897   <Q cltq 10898  Pcnp 10899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-mi 10914  df-lti 10915  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-mq 10955  df-1nq 10956  df-rq 10957  df-ltnq 10958  df-np 11021
This theorem is referenced by:  addclprlem2  11057
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