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Theorem addclprlem2 11055
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 11054 . . . . 5 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
21adantlr 715 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
3 addclprlem1 11054 . . . . . 6 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q ( +Q 𝑔) → ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵))
4 addcomnq 10989 . . . . . . 7 (𝑔 +Q ) = ( +Q 𝑔)
54breq2i 5156 . . . . . 6 (𝑥 <Q (𝑔 +Q ) ↔ 𝑥 <Q ( +Q 𝑔))
64fveq2i 6910 . . . . . . . . 9 (*Q‘(𝑔 +Q )) = (*Q‘( +Q 𝑔))
76oveq2i 7442 . . . . . . . 8 (𝑥 ·Q (*Q‘(𝑔 +Q ))) = (𝑥 ·Q (*Q‘( +Q 𝑔)))
87oveq1i 7441 . . . . . . 7 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) = ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q )
98eleq1i 2830 . . . . . 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 714 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
122, 11jcad 512 . . 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 482 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → ((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)))
14 simpl 482 . . . . 5 ((𝐴P𝑔𝐴) → 𝐴P)
15 simpl 482 . . . . 5 ((𝐵P𝐵) → 𝐵P)
1614, 15anim12i 613 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝐴P𝐵P))
17 df-plp 11021 . . . . 5 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
18 addclnq 10983 . . . . 5 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
1917, 18genpprecl 11039 . . . 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 10999 . . . . 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 10997 . . . . 5 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q (𝑔 +Q )) = (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )))
2422, 23eqtr3i 2765 . . . 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 10991 . . . . . . 7 ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = ((𝑔 +Q ) ·Q (*Q‘(𝑔 +Q )))
26 elprnq 11029 . . . . . . . . 9 ((𝐴P𝑔𝐴) → 𝑔Q)
27 elprnq 11029 . . . . . . . . 9 ((𝐵P𝐵) → Q)
2826, 27anim12i 613 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑔QQ))
29 addclnq 10983 . . . . . . . 8 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
30 recidnq 11003 . . . . . . . 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 2787 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = 1Q)
3332oveq2d 7447 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = (𝑥 ·Q 1Q))
34 mulidnq 11001 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
3533, 34sylan9eq 2795 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = 𝑥)
3624, 35eqtrid 2787 . . 3 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) = 𝑥)
3736eleq1d 2824 . 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 239 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (𝐴 +P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Qcnq 10890  1Qc1q 10891   +Q cplq 10893   ·Q cmq 10894  *Qcrq 10895   <Q cltq 10896  Pcnp 10897   +P cpp 10899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-plp 11021
This theorem is referenced by:  addclpr  11056
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