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Theorem genpelxp 7539
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 3255 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2 nqex 7391 . . . . . 6 Q ∈ V
32elpw2 4175 . . . . 5 ({𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
41, 3mpbir 146 . . . 4 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
5 ssrab2 3255 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
62elpw2 4175 . . . . 5 ({𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
75, 6mpbir 146 . . . 4 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
8 opelxpi 4676 . . . 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 426 . . 3 ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q)
10 fveq2 5534 . . . . . . . . 9 (𝑤 = 𝐴 → (1st𝑤) = (1st𝐴))
1110eleq2d 2259 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (1st𝑤) ↔ 𝑦 ∈ (1st𝐴)))
12113anbi1d 1327 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
13122rexbidv 2515 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1413rabbidv 2741 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
15 fveq2 5534 . . . . . . . . 9 (𝑤 = 𝐴 → (2nd𝑤) = (2nd𝐴))
1615eleq2d 2259 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (2nd𝑤) ↔ 𝑦 ∈ (2nd𝐴)))
17163anbi1d 1327 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
18172rexbidv 2515 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1918rabbidv 2741 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
2014, 19opeq12d 3801 . . . 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 5534 . . . . . . . . 9 (𝑣 = 𝐵 → (1st𝑣) = (1st𝐵))
2221eleq2d 2259 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (1st𝑣) ↔ 𝑧 ∈ (1st𝐵)))
23223anbi2d 1328 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
24232rexbidv 2515 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
2524rabbidv 2741 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
26 fveq2 5534 . . . . . . . . 9 (𝑣 = 𝐵 → (2nd𝑣) = (2nd𝐵))
2726eleq2d 2259 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (2nd𝑣) ↔ 𝑧 ∈ (2nd𝐵)))
28273anbi2d 1328 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
29282rexbidv 2515 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
3029rabbidv 2741 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
3125, 30opeq12d 3801 . . . 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 6030 . . 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 1337 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
3534, 9eqeltrdi 2280 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wrex 2469  {crab 2472  wss 3144  𝒫 cpw 3590  cop 3610   × cxp 4642  cfv 5235  (class class class)co 5895  cmpo 5897  1st c1st 6162  2nd c2nd 6163  Qcnq 7308  Pcnp 7319
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-qs 6564  df-ni 7332  df-nqqs 7376
This theorem is referenced by:  addclpr  7565  mulclpr  7600
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