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Theorem genpmu 6674
 Description: The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpmu ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑞,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑞   𝑥,𝐺,𝑦,𝑧,𝑤,𝑣,𝑞   𝐹,𝑞
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpmu
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6631 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 6634 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑓Q 𝑓 ∈ (2nd𝐴))
3 rexex 2385 . . . 4 (∃𝑓Q 𝑓 ∈ (2nd𝐴) → ∃𝑓 𝑓 ∈ (2nd𝐴))
41, 2, 33syl 17 . . 3 (𝐴P → ∃𝑓 𝑓 ∈ (2nd𝐴))
54adantr 265 . 2 ((𝐴P𝐵P) → ∃𝑓 𝑓 ∈ (2nd𝐴))
6 prop 6631 . . . . 5 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
7 prmu 6634 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑔Q 𝑔 ∈ (2nd𝐵))
8 rexex 2385 . . . . 5 (∃𝑔Q 𝑔 ∈ (2nd𝐵) → ∃𝑔 𝑔 ∈ (2nd𝐵))
96, 7, 83syl 17 . . . 4 (𝐵P → ∃𝑔 𝑔 ∈ (2nd𝐵))
109ad2antlr 466 . . 3 (((𝐴P𝐵P) ∧ 𝑓 ∈ (2nd𝐴)) → ∃𝑔 𝑔 ∈ (2nd𝐵))
11 genpelvl.1 . . . . . . 7 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
12 genpelvl.2 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
1311, 12genppreclu 6671 . . . . . 6 ((𝐴P𝐵P) → ((𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵)) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
1413imp 119 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵)))
15 elprnqu 6638 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
161, 15sylan 271 . . . . . . . . 9 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
17 elprnqu 6638 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (2nd𝐵)) → 𝑔Q)
186, 17sylan 271 . . . . . . . . 9 ((𝐵P𝑔 ∈ (2nd𝐵)) → 𝑔Q)
1916, 18anim12i 325 . . . . . . . 8 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ (𝐵P𝑔 ∈ (2nd𝐵))) → (𝑓Q𝑔Q))
2019an4s 530 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓Q𝑔Q))
2112caovcl 5683 . . . . . . 7 ((𝑓Q𝑔Q) → (𝑓𝐺𝑔) ∈ Q)
2220, 21syl 14 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓𝐺𝑔) ∈ Q)
23 simpr 107 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
2423eleq1d 2122 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
2522, 24rspcedv 2677 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → ((𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵)) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
2614, 25mpd 13 . . . 4 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
2726anassrs 386 . . 3 ((((𝐴P𝐵P) ∧ 𝑓 ∈ (2nd𝐴)) ∧ 𝑔 ∈ (2nd𝐵)) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
2810, 27exlimddv 1794 . 2 (((𝐴P𝐵P) ∧ 𝑓 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
295, 28exlimddv 1794 1 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∧ w3a 896   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ∃wrex 2324  {crab 2327  ⟨cop 3406  ‘cfv 4930  (class class class)co 5540   ↦ cmpt2 5542  1st c1st 5793  2nd c2nd 5794  Qcnq 6436  Pcnp 6447 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-qs 6143  df-ni 6460  df-nqqs 6504  df-inp 6622 This theorem is referenced by:  addclpr  6693  mulclpr  6728
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