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Theorem genpelxp 6607
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 3025 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2 nqex 6459 . . . . . 6 Q ∈ V
32elpw2 3911 . . . . 5 ({𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
41, 3mpbir 134 . . . 4 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
5 ssrab2 3025 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
62elpw2 3911 . . . . 5 ({𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
75, 6mpbir 134 . . . 4 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
8 opelxpi 4376 . . . 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 402 . . 3 ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q)
10 fveq2 5178 . . . . . . . . 9 (𝑤 = 𝐴 → (1st𝑤) = (1st𝐴))
1110eleq2d 2107 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (1st𝑤) ↔ 𝑦 ∈ (1st𝐴)))
12113anbi1d 1211 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
13122rexbidv 2349 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1413rabbidv 2549 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
15 fveq2 5178 . . . . . . . . 9 (𝑤 = 𝐴 → (2nd𝑤) = (2nd𝐴))
1615eleq2d 2107 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (2nd𝑤) ↔ 𝑦 ∈ (2nd𝐴)))
17163anbi1d 1211 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
18172rexbidv 2349 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1918rabbidv 2549 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
2014, 19opeq12d 3557 . . . 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 5178 . . . . . . . . 9 (𝑣 = 𝐵 → (1st𝑣) = (1st𝐵))
2221eleq2d 2107 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (1st𝑣) ↔ 𝑧 ∈ (1st𝐵)))
23223anbi2d 1212 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
24232rexbidv 2349 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
2524rabbidv 2549 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
26 fveq2 5178 . . . . . . . . 9 (𝑣 = 𝐵 → (2nd𝑣) = (2nd𝐵))
2726eleq2d 2107 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (2nd𝑣) ↔ 𝑧 ∈ (2nd𝐵)))
28273anbi2d 1212 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
29282rexbidv 2349 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
3029rabbidv 2549 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
3125, 30opeq12d 3557 . . . 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, 32ovmpt2g 5635 . . 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 1221 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
3534, 9syl6eqel 2128 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885   = wceq 1243  wcel 1393  wrex 2307  {crab 2310  wss 2917  𝒫 cpw 3359  cop 3378   × cxp 4343  cfv 4902  (class class class)co 5512  cmpt2 5514  1st c1st 5765  2nd c2nd 5766  Qcnq 6376  Pcnp 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-qs 6112  df-ni 6400  df-nqqs 6444
This theorem is referenced by:  addclpr  6633  mulclpr  6668
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