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Theorem genpmu 7349
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 7306 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 7309 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑓Q 𝑓 ∈ (2nd𝐴))
3 rexex 2482 . . . 4 (∃𝑓Q 𝑓 ∈ (2nd𝐴) → ∃𝑓 𝑓 ∈ (2nd𝐴))
41, 2, 33syl 17 . . 3 (𝐴P → ∃𝑓 𝑓 ∈ (2nd𝐴))
54adantr 274 . 2 ((𝐴P𝐵P) → ∃𝑓 𝑓 ∈ (2nd𝐴))
6 prop 7306 . . . . 5 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
7 prmu 7309 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑔Q 𝑔 ∈ (2nd𝐵))
8 rexex 2482 . . . . 5 (∃𝑔Q 𝑔 ∈ (2nd𝐵) → ∃𝑔 𝑔 ∈ (2nd𝐵))
96, 7, 83syl 17 . . . 4 (𝐵P → ∃𝑔 𝑔 ∈ (2nd𝐵))
109ad2antlr 481 . . 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 7346 . . . . . 6 ((𝐴P𝐵P) → ((𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵)) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
1413imp 123 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵)))
15 elprnqu 7313 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
161, 15sylan 281 . . . . . . . . 9 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
17 elprnqu 7313 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (2nd𝐵)) → 𝑔Q)
186, 17sylan 281 . . . . . . . . 9 ((𝐵P𝑔 ∈ (2nd𝐵)) → 𝑔Q)
1916, 18anim12i 336 . . . . . . . 8 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ (𝐵P𝑔 ∈ (2nd𝐵))) → (𝑓Q𝑔Q))
2019an4s 578 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓Q𝑔Q))
2112caovcl 5932 . . . . . . 7 ((𝑓Q𝑔Q) → (𝑓𝐺𝑔) ∈ Q)
2220, 21syl 14 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓𝐺𝑔) ∈ Q)
23 simpr 109 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
2423eleq1d 2209 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
2522, 24rspcedv 2796 . . . . 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 398 . . 3 ((((𝐴P𝐵P) ∧ 𝑓 ∈ (2nd𝐴)) ∧ 𝑔 ∈ (2nd𝐵)) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
2810, 27exlimddv 1871 . 2 (((𝐴P𝐵P) ∧ 𝑓 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
295, 28exlimddv 1871 1 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1332  wex 1469  wcel 1481  wrex 2418  {crab 2421  cop 3534  cfv 5130  (class class class)co 5781  cmpo 5783  1st c1st 6043  2nd c2nd 6044  Qcnq 7111  Pcnp 7122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-qs 6442  df-ni 7135  df-nqqs 7179  df-inp 7297
This theorem is referenced by:  addclpr  7368  mulclpr  7403
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