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Theorem genpv 9859
Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpv ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑓,𝑔,   𝑓,𝐹,𝑔
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,)

Proof of Theorem genpv
StepHypRef Expression
1 oveq1 6697 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 rexeq 3169 . . . . 5 (𝑓 = 𝐴 → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
32abbidv 2770 . . . 4 (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
41, 3eqeq12d 2666 . . 3 (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)}))
5 oveq2 6698 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6 rexeq 3169 . . . . . 6 (𝑔 = 𝐵 → (∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
76rexbidv 3081 . . . . 5 (𝑔 = 𝐵 → (∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
87abbidv 2770 . . . 4 (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
95, 8eqeq12d 2666 . . 3 (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)}))
10 elprnq 9851 . . . . . . . . 9 ((𝑓P𝑦𝑓) → 𝑦Q)
11 elprnq 9851 . . . . . . . . 9 ((𝑔P𝑧𝑔) → 𝑧Q)
12 genp.2 . . . . . . . . . 10 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
13 eleq1 2718 . . . . . . . . . 10 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
1412, 13syl5ibrcom 237 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1510, 11, 14syl2an 493 . . . . . . . 8 (((𝑓P𝑦𝑓) ∧ (𝑔P𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1615an4s 886 . . . . . . 7 (((𝑓P𝑔P) ∧ (𝑦𝑓𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1716rexlimdvva 3067 . . . . . 6 ((𝑓P𝑔P) → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1817abssdv 3709 . . . . 5 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19 nqex 9783 . . . . 5 Q ∈ V
20 ssexg 4837 . . . . 5 (({𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ QQ ∈ V) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
2118, 19, 20sylancl 695 . . . 4 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
22 rexeq 3169 . . . . . 6 (𝑤 = 𝑓 → (∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)))
2322abbidv 2770 . . . . 5 (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
24 rexeq 3169 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2524rexbidv 3081 . . . . . 6 (𝑣 = 𝑔 → (∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2625abbidv 2770 . . . . 5 (𝑣 = 𝑔 → {𝑥 ∣ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
27 genp.1 . . . . 5 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
2823, 26, 27ovmpt2g 6837 . . . 4 ((𝑓P𝑔P ∧ {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
2921, 28mpd3an3 1465 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
304, 9, 29vtocl2ga 3305 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
31 eqeq1 2655 . . . . 5 (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32312rexbidv 3086 . . . 4 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧)))
33 oveq1 6697 . . . . . 6 (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
3433eqeq2d 2661 . . . . 5 (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35 oveq2 6698 . . . . . 6 (𝑧 = → (𝑔𝐺𝑧) = (𝑔𝐺))
3635eqeq2d 2661 . . . . 5 (𝑧 = → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺)))
3734, 36cbvrex2v 3210 . . . 4 (∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺))
3832, 37syl6bb 276 . . 3 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
3938cbvabv 2776 . 2 {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}
4030, 39syl6eq 2701 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231  wss 3607  (class class class)co 6690  cmpt2 6692  Qcnq 9712  Pcnp 9719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-ni 9732  df-nq 9772  df-np 9841
This theorem is referenced by:  genpelv  9860  plpv  9870  mpv  9871
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