Theorem genpprecl 3898

Description: Pre-closure law for general operation 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
genpprecl	⊢ ((A ∈ P ∧ B ∈ P) → ((C ∈ A ∧ D ∈ B) → (CGD) ∈ (AFB)))

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

Proof of Theorem genpprecl

Step	Hyp	Ref	Expression
1		cleqid 1102	. 2 ⊢ (CGD) = (CGD)
2		genp.1	. . . . . 6 ⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}
3	2	genpv 3896	. . . . 5 ⊢ ((A ∈ P ∧ B ∈ P) → (AFB) = {f∣∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))})
4	3	eleq2d 1156	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → ((CGD) ∈ (AFB) ↔ (CGD) ∈ {f∣∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))}))
5		oprex 3018	. . . . 5 ⊢ (CGD) ∈ V
6		cleq1 1107	. . . . . . 7 ⊢ (f = (CGD) → (f = (gGh) ↔ (CGD) = (gGh)))
7	6	anbi2d 468	. . . . . 6 ⊢ (f = (CGD) → (((g ∈ A ∧ h ∈ B) ∧ f = (gGh)) ↔ ((g ∈ A ∧ h ∈ B) ∧ (CGD) = (gGh))))
8	7	bi2exdv 938	. . . . 5 ⊢ (f = (CGD) → (∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh)) ↔ ∃g∃h((g ∈ A ∧ h ∈ B) ∧ (CGD) = (gGh))))
9	5, 8	elab 1415	. . . 4 ⊢ ((CGD) ∈ {f∣∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))} ↔ ∃g∃h((g ∈ A ∧ h ∈ B) ∧ (CGD) = (gGh)))
10	4, 9	syl6bb 414	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → ((CGD) ∈ (AFB) ↔ ∃g∃h((g ∈ A ∧ h ∈ B) ∧ (CGD) = (gGh))))
11		eleq1 1149	. . . . . . 7 ⊢ (g = C → (g ∈ A ↔ C ∈ A))
12		eleq1 1149	. . . . . . 7 ⊢ (h = D → (h ∈ B ↔ D ∈ B))
13	11, 12	bi2anan9 478	. . . . . 6 ⊢ ((g = C ∧ h = D) → ((g ∈ A ∧ h ∈ B) ↔ (C ∈ A ∧ D ∈ B)))
14		opreq12 3008	. . . . . . 7 ⊢ ((g = C ∧ h = D) → (gGh) = (CGD))
15	14	cleq2d 1112	. . . . . 6 ⊢ ((g = C ∧ h = D) → ((CGD) = (gGh) ↔ (CGD) = (CGD)))
16	13, 15	anbi12d 476	. . . . 5 ⊢ ((g = C ∧ h = D) → (((g ∈ A ∧ h ∈ B) ∧ (CGD) = (gGh)) ↔ ((C ∈ A ∧ D ∈ B) ∧ (CGD) = (CGD))))
17	16	cla4e2gv 1398	. . . 4 ⊢ ((C ∈ A ∧ D ∈ B) → (((C ∈ A ∧ D ∈ B) ∧ (CGD) = (CGD)) → ∃g∃h((g ∈ A ∧ h ∈ B) ∧ (CGD) = (gGh))))
18	17	anabsi5 377	. . 3 ⊢ (((C ∈ A ∧ D ∈ B) ∧ (CGD) = (CGD)) → ∃g∃h((g ∈ A ∧ h ∈ B) ∧ (CGD) = (gGh)))
19	10, 18	syl5bir 184	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (((C ∈ A ∧ D ∈ B) ∧ (CGD) = (CGD)) → (CGD) ∈ (AFB)))
20	1, 19	mpan2i 522	1 ⊢ ((A ∈ P ∧ B ∈ P) → ((C ∈ A ∧ D ∈ B) → (CGD) ∈ (AFB)))

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

This theorem is referenced by:  genpnmax 3904  addclprlem2 3913  mulclprlem 3915  distrlem1pr 3921  distrlem2pr 3922  ltaddpr 3934  ltexprlem7 3942

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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