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Theorem genpv 10033
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 6821 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 rexeq 3278 . . . . 5 (𝑓 = 𝐴 → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
32abbidv 2879 . . . 4 (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
41, 3eqeq12d 2775 . . 3 (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)}))
5 oveq2 6822 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6 rexeq 3278 . . . . . 6 (𝑔 = 𝐵 → (∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
76rexbidv 3190 . . . . 5 (𝑔 = 𝐵 → (∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
87abbidv 2879 . . . 4 (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
95, 8eqeq12d 2775 . . 3 (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)}))
10 elprnq 10025 . . . . . . . . 9 ((𝑓P𝑦𝑓) → 𝑦Q)
11 elprnq 10025 . . . . . . . . 9 ((𝑔P𝑧𝑔) → 𝑧Q)
12 genp.2 . . . . . . . . . 10 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
13 eleq1 2827 . . . . . . . . . 10 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
1412, 13syl5ibrcom 237 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1510, 11, 14syl2an 495 . . . . . . . 8 (((𝑓P𝑦𝑓) ∧ (𝑔P𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1615an4s 904 . . . . . . 7 (((𝑓P𝑔P) ∧ (𝑦𝑓𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1716rexlimdvva 3176 . . . . . 6 ((𝑓P𝑔P) → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1817abssdv 3817 . . . . 5 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19 nqex 9957 . . . . 5 Q ∈ V
20 ssexg 4956 . . . . 5 (({𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ QQ ∈ V) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
2118, 19, 20sylancl 697 . . . 4 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
22 rexeq 3278 . . . . . 6 (𝑤 = 𝑓 → (∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)))
2322abbidv 2879 . . . . 5 (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
24 rexeq 3278 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2524rexbidv 3190 . . . . . 6 (𝑣 = 𝑔 → (∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2625abbidv 2879 . . . . 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 6961 . . . 4 ((𝑓P𝑔P ∧ {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
2921, 28mpd3an3 1574 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
304, 9, 29vtocl2ga 3414 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
31 eqeq1 2764 . . . . 5 (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32312rexbidv 3195 . . . 4 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧)))
33 oveq1 6821 . . . . . 6 (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
3433eqeq2d 2770 . . . . 5 (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35 oveq2 6822 . . . . . 6 (𝑧 = → (𝑔𝐺𝑧) = (𝑔𝐺))
3635eqeq2d 2770 . . . . 5 (𝑧 = → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺)))
3734, 36cbvrex2v 3319 . . . 4 (∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺))
3832, 37syl6bb 276 . . 3 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
3938cbvabv 2885 . 2 {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}
4030, 39syl6eq 2810 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  wss 3715  (class class class)co 6814  cmpt2 6816  Qcnq 9886  Pcnp 9893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-ni 9906  df-nq 9946  df-np 10015
This theorem is referenced by:  genpelv  10034  plpv  10044  mpv  10045
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