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Theorem addclprlem2 10155
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 10154 . . . . 5 (((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
21adantlr 708 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
3 addclprlem1 10154 . . . . . 6 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q ( +Q 𝑔) → ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵))
4 addcomnq 10089 . . . . . . 7 (𝑔 +Q ) = ( +Q 𝑔)
54breq2i 4882 . . . . . 6 (𝑥 <Q (𝑔 +Q ) ↔ 𝑥 <Q ( +Q 𝑔))
64fveq2i 6437 . . . . . . . . 9 (*Q‘(𝑔 +Q )) = (*Q‘( +Q 𝑔))
76oveq2i 6917 . . . . . . . 8 (𝑥 ·Q (*Q‘(𝑔 +Q ))) = (𝑥 ·Q (*Q‘( +Q 𝑔)))
87oveq1i 6916 . . . . . . 7 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) = ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q )
98eleq1i 2898 . . . . . 6 (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵 ↔ ((𝑥 ·Q (*Q‘( +Q 𝑔))) ·Q ) ∈ 𝐵)
103, 5, 93imtr4g 288 . . . . 5 (((𝐵P𝐵) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
1110adantll 707 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q ) ∈ 𝐵))
122, 11jcad 510 . . 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 476 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → ((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)))
14 simpl 476 . . . . 5 ((𝐴P𝑔𝐴) → 𝐴P)
15 simpl 476 . . . . 5 ((𝐵P𝐵) → 𝐵P)
1614, 15anim12i 608 . . . 4 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝐴P𝐵P))
17 df-plp 10121 . . . . 5 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
18 addclnq 10083 . . . . 5 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
1917, 18genpprecl 10139 . . . 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 10099 . . . . 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 10097 . . . . 5 ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q (𝑔 +Q )) = (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )))
2422, 23eqtr3i 2852 . . . 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 10091 . . . . . . 7 ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = ((𝑔 +Q ) ·Q (*Q‘(𝑔 +Q )))
26 elprnq 10129 . . . . . . . . 9 ((𝐴P𝑔𝐴) → 𝑔Q)
27 elprnq 10129 . . . . . . . . 9 ((𝐵P𝐵) → Q)
2826, 27anim12i 608 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑔QQ))
29 addclnq 10083 . . . . . . . 8 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
30 recidnq 10103 . . . . . . . 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 2874 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q )) = 1Q)
3332oveq2d 6922 . . . . 5 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = (𝑥 ·Q 1Q))
34 mulidnq 10101 . . . . 5 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
3533, 34sylan9eq 2882 . . . 4 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 ·Q ((*Q‘(𝑔 +Q )) ·Q (𝑔 +Q ))) = 𝑥)
3624, 35syl5eq 2874 . . 3 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q )) = 𝑥)
3736eleq1d 2892 . 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 231 1 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (𝐴 +P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166   class class class wbr 4874  cfv 6124  (class class class)co 6906  Qcnq 9990  1Qc1q 9991   +Q cplq 9993   ·Q cmq 9994  *Qcrq 9995   <Q cltq 9996  Pcnp 9997   +P cpp 9999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-inf2 8816
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-omul 7832  df-er 8010  df-ni 10010  df-pli 10011  df-mi 10012  df-lti 10013  df-plpq 10046  df-mpq 10047  df-ltpq 10048  df-enq 10049  df-nq 10050  df-erq 10051  df-plq 10052  df-mq 10053  df-1nq 10054  df-rq 10055  df-ltnq 10056  df-np 10119  df-plp 10121
This theorem is referenced by:  addclpr  10156
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