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Theorem addnqprl 6627
Description: Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addnqprl ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))

Proof of Theorem addnqprl
Dummy variables 𝑥 𝑦 𝑟 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6573 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 addnqprllem 6625 . . . . . 6 (((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴)))
31, 2sylanl1 382 . . . . 5 (((𝐴P𝐺 ∈ (1st𝐴)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴)))
43adantlr 446 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴)))
5 prop 6573 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 addnqprllem 6625 . . . . . 6 (((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (1st𝐵)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)))
75, 6sylanl1 382 . . . . 5 (((𝐵P𝐻 ∈ (1st𝐵)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)))
87adantll 445 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)))
94, 8jcad 291 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵))))
10 simpl 102 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))))
11 simpl 102 . . . . 5 ((𝐴P𝐺 ∈ (1st𝐴)) → 𝐴P)
12 simpl 102 . . . . 5 ((𝐵P𝐻 ∈ (1st𝐵)) → 𝐵P)
1311, 12anim12i 321 . . . 4 (((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) → (𝐴P𝐵P))
14 df-iplp 6566 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
15 addclnq 6473 . . . . 5 ((𝑟Q𝑠Q) → (𝑟 +Q 𝑠) ∈ Q)
1614, 15genpprecll 6612 . . . 4 ((𝐴P𝐵P) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵))))
1710, 13, 163syl 17 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵))))
189, 17syld 40 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵))))
19 simpr 103 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝑋Q)
20 elprnql 6579 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → 𝐺Q)
211, 20sylan 267 . . . . . . . 8 ((𝐴P𝐺 ∈ (1st𝐴)) → 𝐺Q)
2221ad2antrr 457 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐺Q)
23 elprnql 6579 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (1st𝐵)) → 𝐻Q)
245, 23sylan 267 . . . . . . . 8 ((𝐵P𝐻 ∈ (1st𝐵)) → 𝐻Q)
2524ad2antlr 458 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐻Q)
26 addclnq 6473 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 +Q 𝐻) ∈ Q)
2722, 25, 26syl2anc 391 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 +Q 𝐻) ∈ Q)
28 recclnq 6490 . . . . . 6 ((𝐺 +Q 𝐻) ∈ Q → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
2927, 28syl 14 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
30 mulassnqg 6482 . . . . 5 ((𝑋Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
3119, 29, 27, 30syl3anc 1135 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
32 mulclnq 6474 . . . . . 6 ((𝑋Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
3319, 29, 32syl2anc 391 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
34 distrnqg 6485 . . . . 5 (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q𝐺Q𝐻Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
3533, 22, 25, 34syl3anc 1135 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
36 mulcomnqg 6481 . . . . . . . 8 (((*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
3729, 27, 36syl2anc 391 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
38 recidnq 6491 . . . . . . . 8 ((𝐺 +Q 𝐻) ∈ Q → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
3927, 38syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
4037, 39eqtrd 2072 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = 1Q)
4140oveq2d 5528 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = (𝑋 ·Q 1Q))
42 mulidnq 6487 . . . . . 6 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
4342adantl 262 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
4441, 43eqtrd 2072 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = 𝑋)
4531, 35, 443eqtr3d 2080 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) = 𝑋)
4645eleq1d 2106 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵)) ↔ 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))
4718, 46sylibd 138 1 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  cop 3378   class class class wbr 3764  cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Qcnq 6378  1Qc1q 6379   +Q cplq 6380   ·Q cmq 6381  *Qcrq 6382   <Q cltq 6383  Pcnp 6389   +P cpp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-iplp 6566
This theorem is referenced by:  addlocprlemlt  6629  addclpr  6635
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