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Theorem genpv 10399
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 7140 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 rexeq 3393 . . . . 5 (𝑓 = 𝐴 → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
32abbidv 2884 . . . 4 (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
41, 3eqeq12d 2836 . . 3 (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)}))
5 oveq2 7141 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6 rexeq 3393 . . . . . 6 (𝑔 = 𝐵 → (∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
76rexbidv 3284 . . . . 5 (𝑔 = 𝐵 → (∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)))
87abbidv 2884 . . . 4 (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
95, 8eqeq12d 2836 . . 3 (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦𝐴𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)}))
10 elprnq 10391 . . . . . . . . 9 ((𝑓P𝑦𝑓) → 𝑦Q)
11 elprnq 10391 . . . . . . . . 9 ((𝑔P𝑧𝑔) → 𝑧Q)
12 genp.2 . . . . . . . . . 10 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
13 eleq1 2898 . . . . . . . . . 10 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
1412, 13syl5ibrcom 249 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1510, 11, 14syl2an 597 . . . . . . . 8 (((𝑓P𝑦𝑓) ∧ (𝑔P𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1615an4s 658 . . . . . . 7 (((𝑓P𝑔P) ∧ (𝑦𝑓𝑧𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1716rexlimdvva 3281 . . . . . 6 ((𝑓P𝑔P) → (∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
1817abssdv 4024 . . . . 5 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19 nqex 10323 . . . . 5 Q ∈ V
20 ssexg 5203 . . . . 5 (({𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ QQ ∈ V) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
2118, 19, 20sylancl 588 . . . 4 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
22 rexeq 3393 . . . . . 6 (𝑤 = 𝑓 → (∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)))
2322abbidv 2884 . . . . 5 (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
24 rexeq 3393 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2524rexbidv 3284 . . . . . 6 (𝑣 = 𝑔 → (∃𝑦𝑓𝑧𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)))
2625abbidv 2884 . . . . 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 7286 . . . 4 ((𝑓P𝑔P ∧ {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
2921, 28mpd3an3 1458 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦𝑓𝑧𝑔 𝑥 = (𝑦𝐺𝑧)})
304, 9, 29vtocl2ga 3554 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)})
31 eqeq1 2824 . . . . 5 (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32312rexbidv 3287 . . . 4 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧)))
33 oveq1 7140 . . . . . 6 (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
3433eqeq2d 2831 . . . . 5 (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35 oveq2 7141 . . . . . 6 (𝑧 = → (𝑔𝐺𝑧) = (𝑔𝐺))
3635eqeq2d 2831 . . . . 5 (𝑧 = → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺)))
3734, 36cbvrex2vw 3441 . . . 4 (∃𝑦𝐴𝑧𝐵 𝑓 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺))
3832, 37syl6bb 289 . . 3 (𝑥 = 𝑓 → (∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
3938cbvabv 2888 . 2 {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}
4030, 39syl6eq 2871 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  {cab 2798  wrex 3126  Vcvv 3473  wss 3913  (class class class)co 7133  cmpo 7135  Qcnq 10252  Pcnp 10259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-ni 10272  df-nq 10312  df-np 10381
This theorem is referenced by:  genpelv  10400  plpv  10410  mpv  10411
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