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Theorem genpmu 7514
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 7471 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 7474 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑓Q 𝑓 ∈ (2nd𝐴))
3 rexex 2523 . . . 4 (∃𝑓Q 𝑓 ∈ (2nd𝐴) → ∃𝑓 𝑓 ∈ (2nd𝐴))
41, 2, 33syl 17 . . 3 (𝐴P → ∃𝑓 𝑓 ∈ (2nd𝐴))
54adantr 276 . 2 ((𝐴P𝐵P) → ∃𝑓 𝑓 ∈ (2nd𝐴))
6 prop 7471 . . . . 5 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
7 prmu 7474 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑔Q 𝑔 ∈ (2nd𝐵))
8 rexex 2523 . . . . 5 (∃𝑔Q 𝑔 ∈ (2nd𝐵) → ∃𝑔 𝑔 ∈ (2nd𝐵))
96, 7, 83syl 17 . . . 4 (𝐵P → ∃𝑔 𝑔 ∈ (2nd𝐵))
109ad2antlr 489 . . 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 7511 . . . . . 6 ((𝐴P𝐵P) → ((𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵)) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
1413imp 124 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵)))
15 elprnqu 7478 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
161, 15sylan 283 . . . . . . . . 9 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
17 elprnqu 7478 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (2nd𝐵)) → 𝑔Q)
186, 17sylan 283 . . . . . . . . 9 ((𝐵P𝑔 ∈ (2nd𝐵)) → 𝑔Q)
1916, 18anim12i 338 . . . . . . . 8 (((𝐴P𝑓 ∈ (2nd𝐴)) ∧ (𝐵P𝑔 ∈ (2nd𝐵))) → (𝑓Q𝑔Q))
2019an4s 588 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓Q𝑔Q))
2112caovcl 6026 . . . . . . 7 ((𝑓Q𝑔Q) → (𝑓𝐺𝑔) ∈ Q)
2220, 21syl 14 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) → (𝑓𝐺𝑔) ∈ Q)
23 simpr 110 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
2423eleq1d 2246 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑔 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
2522, 24rspcedv 2845 . . . . 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 400 . . 3 ((((𝐴P𝐵P) ∧ 𝑓 ∈ (2nd𝐴)) ∧ 𝑔 ∈ (2nd𝐵)) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
2810, 27exlimddv 1898 . 2 (((𝐴P𝐵P) ∧ 𝑓 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
295, 28exlimddv 1898 1 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wex 1492  wcel 2148  wrex 2456  {crab 2459  cop 3595  cfv 5215  (class class class)co 5872  cmpo 5874  1st c1st 6136  2nd c2nd 6137  Qcnq 7276  Pcnp 7287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-qs 6538  df-ni 7300  df-nqqs 7344  df-inp 7462
This theorem is referenced by:  addclpr  7533  mulclpr  7568
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