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Theorem genpv 10107
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 6883 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 rexeq 3320 . . . . 5 (𝑓 = 𝐴 → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
32abbidv 2916 . . . 4 (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
41, 3eqeq12d 2812 . . 3 (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)}))
5 oveq2 6884 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6 rexeq 3320 . . . . . 6 (𝑔 = 𝐵 → (∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
76rexbidv 3231 . . . . 5 (𝑔 = 𝐵 → (∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
87abbidv 2916 . . . 4 (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
95, 8eqeq12d 2812 . . 3 (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)}))
10 elprnq 10099 . . . . . . . . 9 ((𝑓P𝑦𝑓) → 𝑦Q)
11 elprnq 10099 . . . . . . . . 9 ((𝑔P𝑧𝑔) → 𝑧Q)
12 genp.2 . . . . . . . . . 10 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
13 eleq1 2864 . . . . . . . . . 10 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
1412, 13syl5ibrcom 239 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1510, 11, 14syl2an 590 . . . . . . . 8 (((𝑓P𝑦𝑓) ∧ (𝑔P𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1615an4s 651 . . . . . . 7 (((𝑓P𝑔P) ∧ (𝑦𝑓𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1716rexlimdvva 3217 . . . . . 6 ((𝑓P𝑔P) → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1817abssdv 3870 . . . . 5 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19 nqex 10031 . . . . 5 Q ∈ V
20 ssexg 4997 . . . . 5 (({𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ QQ ∈ V) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
2118, 19, 20sylancl 581 . . . 4 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
22 rexeq 3320 . . . . . 6 (𝑤 = 𝑓 → (∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)))
2322abbidv 2916 . . . . 5 (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
24 rexeq 3320 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2524rexbidv 3231 . . . . . 6 (𝑣 = 𝑔 → (∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2625abbidv 2916 . . . . 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 7027 . . . 4 ((𝑓P𝑔P ∧ {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
2921, 28mpd3an3 1587 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
304, 9, 29vtocl2ga 3460 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
31 eqeq1 2801 . . . . 5 (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32312rexbidv 3236 . . . 4 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧)))
33 oveq1 6883 . . . . . 6 (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
3433eqeq2d 2807 . . . . 5 (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35 oveq2 6884 . . . . . 6 (𝑧 = → (𝑔𝐺𝑧) = (𝑔𝐺))
3635eqeq2d 2807 . . . . 5 (𝑧 = → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺)))
3734, 36cbvrex2v 3361 . . . 4 (∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺))
3832, 37syl6bb 279 . . 3 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
3938cbvabv 2922 . 2 {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}
4030, 39syl6eq 2847 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {cab 2783  wrex 3088  Vcvv 3383  wss 3767  (class class class)co 6876  cmpt2 6878  Qcnq 9960  Pcnp 9967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-inf2 8786
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-ni 9980  df-nq 10020  df-np 10089
This theorem is referenced by:  genpelv  10108  plpv  10118  mpv  10119
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