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Theorem addclprlem2 9695
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 9694 . . . . 5 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
21adantlr 746 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
3 addclprlem1 9694 . . . . . 6 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q ( +Q 𝑔) → ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵))
4 addcomnq 9629 . . . . . . 7 (𝑔 +Q ) = ( +Q 𝑔)
54breq2i 4585 . . . . . 6 (𝑥 <Q (𝑔 +Q ) ↔ 𝑥 <Q ( +Q 𝑔))
64fveq2i 6091 . . . . . . . . 9 (*Q‘(𝑔 +Q )) = (*Q‘( +Q 𝑔))
76oveq2i 6538 . . . . . . . 8 (𝑥 ·Q (*Q‘(𝑔 +Q ))) = (𝑥 ·Q (*Q‘( +Q 𝑔)))
87oveq1i 6537 . . . . . . 7 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) = ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q )
98eleq1i 2678 . . . . . 6 (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵 ↔ ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵)
103, 5, 93imtr4g 283 . . . . 5 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
1110adantll 745 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
122, 11jcad 553 . . 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 471 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → ((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)))
14 simpl 471 . . . . 5 ((𝐴P𝑔𝐴) → 𝐴P)
15 simpl 471 . . . . 5 ((𝐵P𝐵) → 𝐵P)
1614, 15anim12i 587 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝐴P𝐵P))
17 df-plp 9661 . . . . 5 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
18 addclnq 9623 . . . . 5 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
1917, 18genpprecl 9679 . . . 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 45 . 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 9639 . . . . 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 9637 . . . . 5 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q (𝑔 +Q )) = (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )))
2422, 23eqtr3i 2633 . . . 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 9631 . . . . . . 7 ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = ((𝑔 +Q ) ·Q (*Q‘(𝑔 +Q )))
26 elprnq 9669 . . . . . . . . 9 ((𝐴P𝑔𝐴) → 𝑔Q)
27 elprnq 9669 . . . . . . . . 9 ((𝐵P𝐵) → Q)
2826, 27anim12i 587 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑔QQ))
29 addclnq 9623 . . . . . . . 8 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
30 recidnq 9643 . . . . . . . 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, 31syl5eq 2655 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = 1Q)
3332oveq2d 6543 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = (𝑥 ·Q 1Q))
34 mulidnq 9641 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
3533, 34sylan9eq 2663 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = 𝑥)
3624, 35syl5eq 2655 . . 3 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) = 𝑥)
3736eleq1d 2671 . 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 227 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (𝐴 +P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976   class class class wbr 4577  cfv 5790  (class class class)co 6527  Qcnq 9530  1Qc1q 9531   +Q cplq 9533   ·Q cmq 9534  *Qcrq 9535   <Q cltq 9536  Pcnp 9537   +P cpp 9539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-pli 9551  df-mi 9552  df-lti 9553  df-plpq 9586  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-plq 9592  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-plp 9661
This theorem is referenced by:  addclpr  9696
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