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Theorem addclprlem2 10761
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
addclprlem2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (𝐴 +P 𝐵)))
Distinct variable groups:   𝑥,𝑔,   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑔,)   𝐵(𝑔,)

Proof of Theorem addclprlem2
Dummy variables 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclprlem1 10760 . . . . 5 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
21adantlr 712 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
3 addclprlem1 10760 . . . . . 6 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q ( +Q 𝑔) → ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵))
4 addcomnq 10695 . . . . . . 7 (𝑔 +Q ) = ( +Q 𝑔)
54breq2i 5082 . . . . . 6 (𝑥 <Q (𝑔 +Q ) ↔ 𝑥 <Q ( +Q 𝑔))
64fveq2i 6770 . . . . . . . . 9 (*Q‘(𝑔 +Q )) = (*Q‘( +Q 𝑔))
76oveq2i 7279 . . . . . . . 8 (𝑥 ·Q (*Q‘(𝑔 +Q ))) = (𝑥 ·Q (*Q‘( +Q 𝑔)))
87oveq1i 7278 . . . . . . 7 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) = ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q )
98eleq1i 2829 . . . . . 6 (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵 ↔ ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵)
103, 5, 93imtr4g 296 . . . . 5 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
1110adantll 711 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
122, 11jcad 513 . . 3 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴 ∧ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵)))
13 simpl 483 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → ((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)))
14 simpl 483 . . . . 5 ((𝐴P𝑔𝐴) → 𝐴P)
15 simpl 483 . . . . 5 ((𝐵P𝐵) → 𝐵P)
1614, 15anim12i 613 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝐴P𝐵P))
17 df-plp 10727 . . . . 5 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
18 addclnq 10689 . . . . 5 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
1917, 18genpprecl 10745 . . . 4 ((𝐴P𝐵P) → ((((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴 ∧ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) ∈ (𝐴 +P 𝐵)))
2013, 16, 193syl 18 . . 3 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → ((((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴 ∧ ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) ∈ (𝐴 +P 𝐵)))
2112, 20syld 47 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) ∈ (𝐴 +P 𝐵)))
22 distrnq 10705 . . . . 5 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q (𝑔 +Q )) = (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ))
23 mulassnq 10703 . . . . 5 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q (𝑔 +Q )) = (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )))
2422, 23eqtr3i 2768 . . . 4 (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) = (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )))
25 mulcomnq 10697 . . . . . . 7 ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = ((𝑔 +Q ) ·Q (*Q‘(𝑔 +Q )))
26 elprnq 10735 . . . . . . . . 9 ((𝐴P𝑔𝐴) → 𝑔Q)
27 elprnq 10735 . . . . . . . . 9 ((𝐵P𝐵) → Q)
2826, 27anim12i 613 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑔QQ))
29 addclnq 10689 . . . . . . . 8 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
30 recidnq 10709 . . . . . . . 8 ((𝑔 +Q ) ∈ Q → ((𝑔 +Q ) ·Q (*Q‘(𝑔 +Q ))) = 1Q)
3128, 29, 303syl 18 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑔 +Q ) ·Q (*Q‘(𝑔 +Q ))) = 1Q)
3225, 31eqtrid 2790 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = 1Q)
3332oveq2d 7284 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = (𝑥 ·Q 1Q))
34 mulidnq 10707 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
3533, 34sylan9eq 2798 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = 𝑥)
3624, 35eqtrid 2790 . . 3 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) = 𝑥)
3736eleq1d 2823 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → ((((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) ∈ (𝐴 +P 𝐵) ↔ 𝑥 ∈ (𝐴 +P 𝐵)))
3821, 37sylibd 238 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (𝐴 +P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106   class class class wbr 5074  cfv 6427  (class class class)co 7268  Qcnq 10596  1Qc1q 10597   +Q cplq 10599   ·Q cmq 10600  *Qcrq 10601   <Q cltq 10602  Pcnp 10603   +P cpp 10605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579  ax-inf2 9387
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7704  df-1st 7821  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-rdg 8229  df-1o 8285  df-oadd 8289  df-omul 8290  df-er 8486  df-ni 10616  df-pli 10617  df-mi 10618  df-lti 10619  df-plpq 10652  df-mpq 10653  df-ltpq 10654  df-enq 10655  df-nq 10656  df-erq 10657  df-plq 10658  df-mq 10659  df-1nq 10660  df-rq 10661  df-ltnq 10662  df-np 10725  df-plp 10727
This theorem is referenced by:  addclpr  10762
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