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Theorem genpelvl 6667
Description: Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-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
genpelvl ((𝐴P𝐵P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑔,,𝑤,𝑣   𝑔,𝐹   𝐶,𝑔,
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,)

Proof of Theorem genpelvl
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . 7 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2 genpelvl.2 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genipv 6664 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩)
43fveq2d 5209 . . . . 5 ((𝐴P𝐵P) → (1st ‘(𝐴𝐹𝐵)) = (1st ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩))
5 nqex 6518 . . . . . . 7 Q ∈ V
65rabex 3928 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)} ∈ V
75rabex 3928 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ∈ V
86, 7op1st 5800 . . . . 5 (1st ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩) = {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}
94, 8syl6eq 2104 . . . 4 ((𝐴P𝐵P) → (1st ‘(𝐴𝐹𝐵)) = {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)})
109eleq2d 2123 . . 3 ((𝐴P𝐵P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}))
11 elrabi 2717 . . 3 (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)} → 𝐶Q)
1210, 11syl6bi 156 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) → 𝐶Q))
13 prop 6630 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
14 elprnql 6636 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑔 ∈ (1st𝐴)) → 𝑔Q)
1513, 14sylan 271 . . . . . 6 ((𝐴P𝑔 ∈ (1st𝐴)) → 𝑔Q)
16 prop 6630 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
17 elprnql 6636 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∈ (1st𝐵)) → Q)
1816, 17sylan 271 . . . . . 6 ((𝐵P ∈ (1st𝐵)) → Q)
192caovcl 5682 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
2015, 18, 19syl2an 277 . . . . 5 (((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) → (𝑔𝐺) ∈ Q)
2120an4s 530 . . . 4 (((𝐴P𝐵P) ∧ (𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵))) → (𝑔𝐺) ∈ Q)
22 eleq1 2116 . . . 4 (𝐶 = (𝑔𝐺) → (𝐶Q ↔ (𝑔𝐺) ∈ Q))
2321, 22syl5ibrcom 150 . . 3 (((𝐴P𝐵P) ∧ (𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵))) → (𝐶 = (𝑔𝐺) → 𝐶Q))
2423rexlimdvva 2457 . 2 ((𝐴P𝐵P) → (∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺) → 𝐶Q))
25 eqeq1 2062 . . . . . 6 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
26252rexbidv 2366 . . . . 5 (𝑓 = 𝐶 → (∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺)))
2726elrab3 2721 . . . 4 (𝐶Q → (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)} ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺)))
2810, 27sylan9bb 443 . . 3 (((𝐴P𝐵P) ∧ 𝐶Q) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺)))
2928ex 112 . 2 ((𝐴P𝐵P) → (𝐶Q → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺))))
3012, 24, 29pm5.21ndd 631 1 ((𝐴P𝐵P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wrex 2324  {crab 2327  cop 3405  cfv 4929  (class class class)co 5539  cmpt2 5541  1st c1st 5792  2nd c2nd 5793  Qcnq 6435  Pcnp 6446
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 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
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 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-qs 6142  df-ni 6459  df-nqqs 6503  df-inp 6621
This theorem is referenced by:  genpprecll  6669  genpcdl  6674  genprndl  6676  genpdisj  6678  genpassl  6679  addnqprlemrl  6712  mulnqprlemrl  6728  distrlem1prl  6737  distrlem5prl  6741  1idprl  6745  ltexprlemfl  6764  recexprlem1ssl  6788  recexprlemss1l  6790  cauappcvgprlemladdfl  6810
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