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Theorem genpelxp 7287
Description: Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
Hypothesis
Ref Expression
genpelvl.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
Assertion
Ref Expression
genpelxp ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑤,𝑣
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpelxp
StepHypRef Expression
1 ssrab2 3152 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2 nqex 7139 . . . . . 6 Q ∈ V
32elpw2 4052 . . . . 5 ({𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
41, 3mpbir 145 . . . 4 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
5 ssrab2 3152 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
62elpw2 4052 . . . . 5 ({𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
75, 6mpbir 145 . . . 4 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
8 opelxpi 4541 . . . 4 (({𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ∧ {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q) → ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q))
94, 7, 8mp2an 422 . . 3 ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q)
10 fveq2 5389 . . . . . . . . 9 (𝑤 = 𝐴 → (1st𝑤) = (1st𝐴))
1110eleq2d 2187 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (1st𝑤) ↔ 𝑦 ∈ (1st𝐴)))
12113anbi1d 1279 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
13122rexbidv 2437 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1413rabbidv 2649 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
15 fveq2 5389 . . . . . . . . 9 (𝑤 = 𝐴 → (2nd𝑤) = (2nd𝐴))
1615eleq2d 2187 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (2nd𝑤) ↔ 𝑦 ∈ (2nd𝐴)))
17163anbi1d 1279 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
18172rexbidv 2437 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1918rabbidv 2649 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
2014, 19opeq12d 3683 . . . 4 (𝑤 = 𝐴 → ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
21 fveq2 5389 . . . . . . . . 9 (𝑣 = 𝐵 → (1st𝑣) = (1st𝐵))
2221eleq2d 2187 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (1st𝑣) ↔ 𝑧 ∈ (1st𝐵)))
23223anbi2d 1280 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
24232rexbidv 2437 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
2524rabbidv 2649 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
26 fveq2 5389 . . . . . . . . 9 (𝑣 = 𝐵 → (2nd𝑣) = (2nd𝐵))
2726eleq2d 2187 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (2nd𝑣) ↔ 𝑧 ∈ (2nd𝐵)))
28273anbi2d 1280 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
29282rexbidv 2437 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
3029rabbidv 2649 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
3125, 30opeq12d 3683 . . . 4 (𝑣 = 𝐵 → ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
32 genpelvl.1 . . . 4 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
3320, 31, 32ovmpog 5873 . . 3 ((𝐴P𝐵P ∧ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q)) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
349, 33mp3an3 1289 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
3534, 9syl6eqel 2208 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wcel 1465  wrex 2394  {crab 2397  wss 3041  𝒫 cpw 3480  cop 3500   × cxp 4507  cfv 5093  (class class class)co 5742  cmpo 5744  1st c1st 6004  2nd c2nd 6005  Qcnq 7056  Pcnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-qs 6403  df-ni 7080  df-nqqs 7124
This theorem is referenced by:  addclpr  7313  mulclpr  7348
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