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Theorem genpelxp 7312
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 3177 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2 nqex 7164 . . . . . 6 Q ∈ V
32elpw2 4077 . . . . 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 3177 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
62elpw2 4077 . . . . 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 4566 . . . 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 5414 . . . . . . . . 9 (𝑤 = 𝐴 → (1st𝑤) = (1st𝐴))
1110eleq2d 2207 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (1st𝑤) ↔ 𝑦 ∈ (1st𝐴)))
12113anbi1d 1294 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
13122rexbidv 2458 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1413rabbidv 2670 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
15 fveq2 5414 . . . . . . . . 9 (𝑤 = 𝐴 → (2nd𝑤) = (2nd𝐴))
1615eleq2d 2207 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (2nd𝑤) ↔ 𝑦 ∈ (2nd𝐴)))
17163anbi1d 1294 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
18172rexbidv 2458 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1918rabbidv 2670 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
2014, 19opeq12d 3708 . . . 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 5414 . . . . . . . . 9 (𝑣 = 𝐵 → (1st𝑣) = (1st𝐵))
2221eleq2d 2207 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (1st𝑣) ↔ 𝑧 ∈ (1st𝐵)))
23223anbi2d 1295 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
24232rexbidv 2458 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
2524rabbidv 2670 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
26 fveq2 5414 . . . . . . . . 9 (𝑣 = 𝐵 → (2nd𝑣) = (2nd𝐵))
2726eleq2d 2207 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (2nd𝑣) ↔ 𝑧 ∈ (2nd𝐵)))
28273anbi2d 1295 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
29282rexbidv 2458 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
3029rabbidv 2670 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
3125, 30opeq12d 3708 . . . 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 5898 . . 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 1304 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
3534, 9syl6eqel 2228 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wcel 1480  wrex 2415  {crab 2418  wss 3066  𝒫 cpw 3505  cop 3525   × cxp 4532  cfv 5118  (class class class)co 5767  cmpo 5769  1st c1st 6029  2nd c2nd 6030  Qcnq 7081  Pcnp 7092
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-qs 6428  df-ni 7105  df-nqqs 7149
This theorem is referenced by:  addclpr  7338  mulclpr  7373
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