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Theorem genpelvu 7285
Description: Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-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
genpelvu ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)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 genpelvu
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 7281 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩)
43fveq2d 5391 . . . . 5 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩))
5 nqex 7135 . . . . . . 7 Q ∈ V
65rabex 4040 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)} ∈ V
75rabex 4040 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ∈ V
86, 7op2nd 6011 . . . . 5 (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}
94, 8syl6eq 2164 . . . 4 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)})
109eleq2d 2185 . . 3 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}))
11 elrabi 2808 . . 3 (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} → 𝐶Q)
1210, 11syl6bi 162 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝐶Q))
13 prop 7247 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
14 elprnqu 7254 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
1513, 14sylan 279 . . . . . 6 ((𝐴P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
16 prop 7247 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
17 elprnqu 7254 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∈ (2nd𝐵)) → Q)
1816, 17sylan 279 . . . . . 6 ((𝐵P ∈ (2nd𝐵)) → Q)
192caovcl 5891 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
2015, 18, 19syl2an 285 . . . . 5 (((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
2120an4s 560 . . . 4 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
22 eleq1 2178 . . . 4 (𝐶 = (𝑔𝐺) → (𝐶Q ↔ (𝑔𝐺) ∈ Q))
2321, 22syl5ibrcom 156 . . 3 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝐶 = (𝑔𝐺) → 𝐶Q))
2423rexlimdvva 2532 . 2 ((𝐴P𝐵P) → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺) → 𝐶Q))
25 eqeq1 2122 . . . . . 6 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
26252rexbidv 2435 . . . . 5 (𝑓 = 𝐶 → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2726elrab3 2812 . . . 4 (𝐶Q → (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2810, 27sylan9bb 455 . . 3 (((𝐴P𝐵P) ∧ 𝐶Q) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2928ex 114 . 2 ((𝐴P𝐵P) → (𝐶Q → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺))))
3012, 24, 29pm5.21ndd 677 1 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945   = wceq 1314  wcel 1463  wrex 2392  {crab 2395  cop 3498  cfv 5091  (class class class)co 5740  cmpo 5742  1st c1st 6002  2nd c2nd 6003  Qcnq 7052  Pcnp 7063
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-qs 6401  df-ni 7076  df-nqqs 7120  df-inp 7238
This theorem is referenced by:  genppreclu  7287  genpcuu  7292  genprndu  7294  genpdisj  7295  genpassu  7297  addnqprlemru  7330  mulnqprlemru  7346  distrlem1pru  7355  distrlem5pru  7359  1idpru  7363  ltexprlemfu  7383  recexprlem1ssu  7406  recexprlemss1u  7408  cauappcvgprlemladdfu  7426  caucvgprlemladdfu  7449
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