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Theorem addclprlem1 9526
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 9501 . . 3 ((𝐴P𝑔𝐴) → 𝑔Q)
2 ltrnq 9489 . . . . 5 (𝑥 <Q (𝑔 +Q ) ↔ (*Q‘(𝑔 +Q )) <Q (*Q𝑥))
3 ltmnq 9482 . . . . . 6 (𝑥Q → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ))) <Q (𝑥 ·Q (*Q𝑥))))
4 ovex 6391 . . . . . . 7 (𝑥 ·Q (*Q‘(𝑔 +Q ))) ∈ V
5 ovex 6391 . . . . . . 7 (𝑥 ·Q (*Q𝑥)) ∈ V
6 ltmnq 9482 . . . . . . 7 (𝑤Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
7 vex 3069 . . . . . . 7 𝑔 ∈ V
8 mulcomnq 9463 . . . . . . 7 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
94, 5, 6, 7, 8caovord2 6556 . . . . . 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 724 . . . . 5 ((𝑔Q𝑥Q) → ((*Q‘(𝑔 +Q )) <Q (*Q𝑥) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
112, 10syl5bb 267 . . . 4 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔)))
12 recidnq 9475 . . . . . . 7 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1312oveq1d 6378 . . . . . 6 (𝑥Q → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔))
14 mulcomnq 9463 . . . . . . 7 (1Q ·Q 𝑔) = (𝑔 ·Q 1Q)
15 mulidnq 9473 . . . . . . 7 (𝑔Q → (𝑔 ·Q 1Q) = 𝑔)
1614, 15syl5eq 2551 . . . . . 6 (𝑔Q → (1Q ·Q 𝑔) = 𝑔)
1713, 16sylan9eqr 2561 . . . . 5 ((𝑔Q𝑥Q) → ((𝑥 ·Q (*Q𝑥)) ·Q 𝑔) = 𝑔)
1817breq2d 4446 . . . 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 263 . . 3 ((𝑔Q𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
201, 19sylan 481 . 2 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) <Q 𝑔))
21 prcdnq 9503 . . 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 225 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 191  wa 378  wcel 1937   class class class wbr 4434  cfv 5633  (class class class)co 6363  Qcnq 9362  1Qc1q 9363   +Q cplq 9365   ·Q cmq 9366  *Qcrq 9367   <Q cltq 9368  Pcnp 9369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-1st 6870  df-2nd 6871  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-1o 7259  df-oadd 7263  df-omul 7264  df-er 7440  df-ni 9382  df-mi 9384  df-lti 9385  df-mpq 9419  df-ltpq 9420  df-enq 9421  df-nq 9422  df-erq 9423  df-mq 9425  df-1nq 9426  df-rq 9427  df-ltnq 9428  df-np 9491
This theorem is referenced by:  addclprlem2  9527
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