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Theorem addclprlem2 11042
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 11041 . . . . 5 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
21adantlr 713 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
3 addclprlem1 11041 . . . . . 6 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q ( +Q 𝑔) → ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵))
4 addcomnq 10976 . . . . . . 7 (𝑔 +Q ) = ( +Q 𝑔)
54breq2i 5157 . . . . . 6 (𝑥 <Q (𝑔 +Q ) ↔ 𝑥 <Q ( +Q 𝑔))
64fveq2i 6899 . . . . . . . . 9 (*Q‘(𝑔 +Q )) = (*Q‘( +Q 𝑔))
76oveq2i 7430 . . . . . . . 8 (𝑥 ·Q (*Q‘(𝑔 +Q ))) = (𝑥 ·Q (*Q‘( +Q 𝑔)))
87oveq1i 7429 . . . . . . 7 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) = ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q )
98eleq1i 2816 . . . . . 6 (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵 ↔ ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵)
103, 5, 93imtr4g 295 . . . . 5 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
1110adantll 712 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
122, 11jcad 511 . . 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 481 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → ((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)))
14 simpl 481 . . . . 5 ((𝐴P𝑔𝐴) → 𝐴P)
15 simpl 481 . . . . 5 ((𝐵P𝐵) → 𝐵P)
1614, 15anim12i 611 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝐴P𝐵P))
17 df-plp 11008 . . . . 5 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
18 addclnq 10970 . . . . 5 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
1917, 18genpprecl 11026 . . . 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 10986 . . . . 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 10984 . . . . 5 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q (𝑔 +Q )) = (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )))
2422, 23eqtr3i 2755 . . . 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 10978 . . . . . . 7 ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = ((𝑔 +Q ) ·Q (*Q‘(𝑔 +Q )))
26 elprnq 11016 . . . . . . . . 9 ((𝐴P𝑔𝐴) → 𝑔Q)
27 elprnq 11016 . . . . . . . . 9 ((𝐵P𝐵) → Q)
2826, 27anim12i 611 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑔QQ))
29 addclnq 10970 . . . . . . . 8 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
30 recidnq 10990 . . . . . . . 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 2777 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = 1Q)
3332oveq2d 7435 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = (𝑥 ·Q 1Q))
34 mulidnq 10988 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
3533, 34sylan9eq 2785 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = 𝑥)
3624, 35eqtrid 2777 . . 3 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) = 𝑥)
3736eleq1d 2810 . 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 394   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  (class class class)co 7419  Qcnq 10877  1Qc1q 10878   +Q cplq 10880   ·Q cmq 10881  *Qcrq 10882   <Q cltq 10883  Pcnp 10884   +P cpp 10886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-ni 10897  df-pli 10898  df-mi 10899  df-lti 10900  df-plpq 10933  df-mpq 10934  df-ltpq 10935  df-enq 10936  df-nq 10937  df-erq 10938  df-plq 10939  df-mq 10940  df-1nq 10941  df-rq 10942  df-ltnq 10943  df-np 11006  df-plp 11008
This theorem is referenced by:  addclpr  11043
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