MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  genpv Structured version   Visualization version   GIF version

Theorem genpv 10942
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 7369 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 rexeq 3313 . . . . 5 (𝑓 = 𝐴 → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
32abbidv 2806 . . . 4 (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
41, 3eqeq12d 2753 . . 3 (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)}))
5 oveq2 7370 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6 rexeq 3313 . . . . . 6 (𝑔 = 𝐵 → (∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
76rexbidv 3176 . . . . 5 (𝑔 = 𝐵 → (∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
87abbidv 2806 . . . 4 (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
95, 8eqeq12d 2753 . . 3 (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)}))
10 elprnq 10934 . . . . . . . . 9 ((𝑓P𝑦𝑓) → 𝑦Q)
11 elprnq 10934 . . . . . . . . 9 ((𝑔P𝑧𝑔) → 𝑧Q)
12 genp.2 . . . . . . . . . 10 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
13 eleq1 2826 . . . . . . . . . 10 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
1412, 13syl5ibrcom 247 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1510, 11, 14syl2an 597 . . . . . . . 8 (((𝑓P𝑦𝑓) ∧ (𝑔P𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1615an4s 659 . . . . . . 7 (((𝑓P𝑔P) ∧ (𝑦𝑓𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1716rexlimdvva 3206 . . . . . 6 ((𝑓P𝑔P) → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1817abssdv 4030 . . . . 5 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19 nqex 10866 . . . . 5 Q ∈ V
20 ssexg 5285 . . . . 5 (({𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ QQ ∈ V) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
2118, 19, 20sylancl 587 . . . 4 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
22 rexeq 3313 . . . . . 6 (𝑤 = 𝑓 → (∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)))
2322abbidv 2806 . . . . 5 (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
24 rexeq 3313 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2524rexbidv 3176 . . . . . 6 (𝑣 = 𝑔 → (∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2625abbidv 2806 . . . . 5 (𝑣 = 𝑔 → {𝑥 ∣ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
27 genp.1 . . . . 5 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
2823, 26, 27ovmpog 7519 . . . 4 ((𝑓P𝑔P ∧ {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
2921, 28mpd3an3 1463 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
304, 9, 29vtocl2ga 3538 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
31 eqeq1 2741 . . . . 5 (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32312rexbidv 3214 . . . 4 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧)))
33 oveq1 7369 . . . . . 6 (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
3433eqeq2d 2748 . . . . 5 (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35 oveq2 7370 . . . . . 6 (𝑧 = → (𝑔𝐺𝑧) = (𝑔𝐺))
3635eqeq2d 2748 . . . . 5 (𝑧 = → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺)))
3734, 36cbvrex2vw 3231 . . . 4 (∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺))
3832, 37bitrdi 287 . . 3 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
3938cbvabv 2810 . 2 {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}
4030, 39eqtrdi 2793 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2714  wrex 3074  Vcvv 3448  wss 3915  (class class class)co 7362  cmpo 7364  Qcnq 10795  Pcnp 10802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-ni 10815  df-nq 10855  df-np 10924
This theorem is referenced by:  genpelv  10943  plpv  10953  mpv  10954
  Copyright terms: Public domain W3C validator