ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genpelxp GIF version

Theorem genpelxp 7466
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 3232 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2 nqex 7318 . . . . . 6 Q ∈ V
32elpw2 4141 . . . . 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 3232 . . . . 5 {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
62elpw2 4141 . . . . 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 4641 . . . 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 424 . . 3 ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q)
10 fveq2 5494 . . . . . . . . 9 (𝑤 = 𝐴 → (1st𝑤) = (1st𝐴))
1110eleq2d 2240 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (1st𝑤) ↔ 𝑦 ∈ (1st𝐴)))
12113anbi1d 1311 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
13122rexbidv 2495 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1413rabbidv 2719 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
15 fveq2 5494 . . . . . . . . 9 (𝑤 = 𝐴 → (2nd𝑤) = (2nd𝐴))
1615eleq2d 2240 . . . . . . . 8 (𝑤 = 𝐴 → (𝑦 ∈ (2nd𝑤) ↔ 𝑦 ∈ (2nd𝐴)))
17163anbi1d 1311 . . . . . . 7 (𝑤 = 𝐴 → ((𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
18172rexbidv 2495 . . . . . 6 (𝑤 = 𝐴 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
1918rabbidv 2719 . . . . 5 (𝑤 = 𝐴 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
2014, 19opeq12d 3771 . . . 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 5494 . . . . . . . . 9 (𝑣 = 𝐵 → (1st𝑣) = (1st𝐵))
2221eleq2d 2240 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (1st𝑣) ↔ 𝑧 ∈ (1st𝐵)))
23223anbi2d 1312 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
24232rexbidv 2495 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
2524rabbidv 2719 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
26 fveq2 5494 . . . . . . . . 9 (𝑣 = 𝐵 → (2nd𝑣) = (2nd𝐵))
2726eleq2d 2240 . . . . . . . 8 (𝑣 = 𝐵 → (𝑧 ∈ (2nd𝑣) ↔ 𝑧 ∈ (2nd𝐵)))
28273anbi2d 1312 . . . . . . 7 (𝑣 = 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
29282rexbidv 2495 . . . . . 6 (𝑣 = 𝐵 → (∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
3029rabbidv 2719 . . . . 5 (𝑣 = 𝐵 → {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
3125, 30opeq12d 3771 . . . 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 5985 . . 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 1321 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
3534, 9eqeltrdi 2261 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973   = wceq 1348  wcel 2141  wrex 2449  {crab 2452  wss 3121  𝒫 cpw 3564  cop 3584   × cxp 4607  cfv 5196  (class class class)co 5851  cmpo 5853  1st c1st 6115  2nd c2nd 6116  Qcnq 7235  Pcnp 7246
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-qs 6517  df-ni 7259  df-nqqs 7303
This theorem is referenced by:  addclpr  7492  mulclpr  7527
  Copyright terms: Public domain W3C validator