Theorem genpv 3896

Description: Value of general operation (addition or multiplication) on positive reals.

Hypothesis

Ref	Expression
genp.1	⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}

Assertion

Ref	Expression
genpv	⊢ ((A ∈ P ∧ B ∈ P) → (AFB) = {f∣∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))})

Distinct variable group(s):   x,y,z,f,g,h,A   x,B,y,z,f,g,h   x,w,v,u,G,y,z,f,g,h   f,F,g

Proof of Theorem genpv

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . 4 ⊢ (f = A → (fFg) = (AFg))
2		rexeq 1325	. . . . 5 ⊢ (f = A → (∃y ∈ f ∃z ∈ g x = (yGz) ↔ ∃y ∈ A ∃z ∈ g x = (yGz)))
3	2	biabdv 1183	. . . 4 ⊢ (f = A → {x∣∃y ∈ f ∃z ∈ g x = (yGz)} = {x∣∃y ∈ A ∃z ∈ g x = (yGz)})
4	1, 3	cleq12d 1115	. . 3 ⊢ (f = A → ((fFg) = {x∣∃y ∈ f ∃z ∈ g x = (yGz)} ↔ (AFg) = {x∣∃y ∈ A ∃z ∈ g x = (yGz)}))
5		opreq2 3007	. . . 4 ⊢ (g = B → (AFg) = (AFB))
6		rexeq 1325	. . . . . 6 ⊢ (g = B → (∃z ∈ g x = (yGz) ↔ ∃z ∈ B x = (yGz)))
7	6	birexdv 1220	. . . . 5 ⊢ (g = B → (∃y ∈ A ∃z ∈ g x = (yGz) ↔ ∃y ∈ A ∃z ∈ B x = (yGz)))
8	7	biabdv 1183	. . . 4 ⊢ (g = B → {x∣∃y ∈ A ∃z ∈ g x = (yGz)} = {x∣∃y ∈ A ∃z ∈ B x = (yGz)})
9	5, 8	cleq12d 1115	. . 3 ⊢ (g = B → ((AFg) = {x∣∃y ∈ A ∃z ∈ g x = (yGz)} ↔ (AFB) = {x∣∃y ∈ A ∃z ∈ B x = (yGz)}))
10		visset 1350	. . . . 5 ⊢ f ∈ V
11		visset 1350	. . . . 5 ⊢ g ∈ V
12	10, 11	oprvalex 3055	. . . 4 ⊢ {x∣∃y ∈ f ∃z ∈ g x = (yGz)} ∈ V
13		rexeq 1325	. . . . 5 ⊢ (w = f → (∃y ∈ w ∃z ∈ v x = (yGz) ↔ ∃y ∈ f ∃z ∈ v x = (yGz)))
14	13	biabdv 1183	. . . 4 ⊢ (w = f → {x∣∃y ∈ w ∃z ∈ v x = (yGz)} = {x∣∃y ∈ f ∃z ∈ v x = (yGz)})
15		rexeq 1325	. . . . . 6 ⊢ (v = g → (∃z ∈ v x = (yGz) ↔ ∃z ∈ g x = (yGz)))
16	15	birexdv 1220	. . . . 5 ⊢ (v = g → (∃y ∈ f ∃z ∈ v x = (yGz) ↔ ∃y ∈ f ∃z ∈ g x = (yGz)))
17	16	biabdv 1183	. . . 4 ⊢ (v = g → {x∣∃y ∈ f ∃z ∈ v x = (yGz)} = {x∣∃y ∈ f ∃z ∈ g x = (yGz)})
18		genp.1	. . . 4 ⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}
19	12, 14, 17, 18	oprabval2 3051	. . 3 ⊢ ((f ∈ P ∧ g ∈ P) → (fFg) = {x∣∃y ∈ f ∃z ∈ g x = (yGz)})
20	4, 9, 19	vtocl2ga 1388	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (AFB) = {x∣∃y ∈ A ∃z ∈ B x = (yGz)})
21		cleq1 1107	. . . . . 6 ⊢ (x = f → (x = (gGh) ↔ f = (gGh)))
22	21	anbi2d 468	. . . . 5 ⊢ (x = f → (((g ∈ A ∧ h ∈ B) ∧ x = (gGh)) ↔ ((g ∈ A ∧ h ∈ B) ∧ f = (gGh))))
23	22	bi2exdv 938	. . . 4 ⊢ (x = f → (∃g∃h((g ∈ A ∧ h ∈ B) ∧ x = (gGh)) ↔ ∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))))
24		r2ex 1241	. . . . 5 ⊢ (∃y ∈ A ∃z ∈ B x = (yGz) ↔ ∃y∃z((y ∈ A ∧ z ∈ B) ∧ x = (yGz)))
25		eleq1 1149	. . . . . . . 8 ⊢ (y = g → (y ∈ A ↔ g ∈ A))
26		eleq1 1149	. . . . . . . 8 ⊢ (z = h → (z ∈ B ↔ h ∈ B))
27	25, 26	bi2anan9 478	. . . . . . 7 ⊢ ((y = g ∧ z = h) → ((y ∈ A ∧ z ∈ B) ↔ (g ∈ A ∧ h ∈ B)))
28		opreq12 3008	. . . . . . . 8 ⊢ ((y = g ∧ z = h) → (yGz) = (gGh))
29	28	cleq2d 1112	. . . . . . 7 ⊢ ((y = g ∧ z = h) → (x = (yGz) ↔ x = (gGh)))
30	27, 29	anbi12d 476	. . . . . 6 ⊢ ((y = g ∧ z = h) → (((y ∈ A ∧ z ∈ B) ∧ x = (yGz)) ↔ ((g ∈ A ∧ h ∈ B) ∧ x = (gGh))))
31	30	cbvex2v 976	. . . . 5 ⊢ (∃y∃z((y ∈ A ∧ z ∈ B) ∧ x = (yGz)) ↔ ∃g∃h((g ∈ A ∧ h ∈ B) ∧ x = (gGh)))
32	24, 31	bitr 151	. . . 4 ⊢ (∃y ∈ A ∃z ∈ B x = (yGz) ↔ ∃g∃h((g ∈ A ∧ h ∈ B) ∧ x = (gGh)))
33	23, 32	syl5bb 410	. . 3 ⊢ (x = f → (∃y ∈ A ∃z ∈ B x = (yGz) ↔ ∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))))
34	33	cbvabv 1424	. 2 ⊢ {x∣∃y ∈ A ∃z ∈ B x = (yGz)} = {f∣∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))}
35	20, 34	syl6eq 1140	1 ⊢ ((A ∈ P ∧ B ∈ P) → (AFB) = {f∣∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))})

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  (class class class)co 3001  {copab2 3002  Pcnp 3779

This theorem is referenced by:  genpelv 3897  genpprecl 3898  genpn0 3900  genpss 3901  genpnnp 3902  genpnmax 3904  plpv 3907  mpv 3908

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004

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