Theorem erngset-rN 40803

Description: The division ring on trace-preserving endomorphisms for a fiducial co-atom 𝑊. (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
erngset.h-r	⊢ 𝐻 = (LHyp‘𝐾)
erngset.t-r	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
erngset.e-r	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
erngset.d-r	⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊)

Assertion

Ref	Expression
erngset-rN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))⟩})

Distinct variable groups:   𝑓,𝑠,𝑡,𝐾   𝑓,𝑊,𝑠,𝑡

Allowed substitution hints:   𝐷(𝑡,𝑓,𝑠)   𝑇(𝑡,𝑓,𝑠)   𝐸(𝑡,𝑓,𝑠)   𝐻(𝑡,𝑓,𝑠)   𝑉(𝑡,𝑓,𝑠)

Proof of Theorem erngset-rN

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erngset.d-r	. . 3 ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊)
2		erngset.h-r	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3	2	erngfset-rN 40802	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (EDRingR‘𝐾) = (𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩}))
4	3	fveq1d 6913	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((EDRingR‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})‘𝑊))
5	1, 4	eqtrid 2788	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐷 = ((𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})‘𝑊))
6		fveq2 6911	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
7	6	opeq2d 4886	. . . . 5 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩)
8		tpeq1 4748	. . . . . 6 ⊢ (⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩ → {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})
9		erngset.e-r	. . . . . . . 8 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
10	9	opeq2i 4883	. . . . . . 7 ⊢ ⟨(Base‘ndx), 𝐸⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩
11		tpeq1 4748	. . . . . . 7 ⊢ (⟨(Base‘ndx), 𝐸⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩ → {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})
12	10, 11	ax-mp 5	. . . . . 6 ⊢ {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩}
13	8, 12	eqtr4di 2794	. . . . 5 ⊢ (⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩ → {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})
14	7, 13	syl 17	. . . 4 ⊢ (𝑤 = 𝑊 → {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})
15	6, 9	eqtr4di 2794	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
16		fveq2 6911	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
17		erngset.t-r	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
19		eqidd 2737	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))
20	18, 19	mpteq12dv 5240	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
21	15, 15, 20	mpoeq123dv 7512	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))))
22	21	opeq2d 4886	. . . . 5 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩ = ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩)
23	22	tpeq2d 4752	. . . 4 ⊢ (𝑤 = 𝑊 → {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})
24		eqidd 2737	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑡 ∘ 𝑠) = (𝑡 ∘ 𝑠))
25	15, 15, 24	mpoeq123dv 7512	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠)) = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠)))
26	25	opeq2d 4886	. . . . 5 ⊢ (𝑤 = 𝑊 → ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩ = ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))⟩)
27	26	tpeq3d 4753	. . . 4 ⊢ (𝑤 = 𝑊 → {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))⟩})
28	14, 23, 27	3eqtrd 2780	. . 3 ⊢ (𝑤 = 𝑊 → {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))⟩})
29		eqid 2736	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩}) = (𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})
30		tpex 7769	. . 3 ⊢ {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))⟩} ∈ V
31	28, 29, 30	fvmpt 7020	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩})‘𝑊) = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))⟩})
32	5, 31	sylan9eq 2796	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1538   ∈ wcel 2107  {ctp 4636  ⟨cop 4638   ↦ cmpt 5232   ∘ ccom 5694  ‘cfv 6566   ∈ cmpo 7437  ndxcnx 17233  Basecbs 17251  +_gcplusg 17304  ._rcmulr 17305  LHypclh 39979  LTrncltrn 40096  TEndoctendo 40747  EDRing_Rcedring-rN 40749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5286  ax-sep 5303  ax-nul 5313  ax-pr 5439  ax-un 7758

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3435  df-v 3481  df-sbc 3793  df-csb 3910  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4914  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-res 5702  df-ima 5703  df-iota 6519  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-oprab 7439  df-mpo 7440  df-edring-rN 40751

This theorem is referenced by:  erngbase-rN  40804  erngfplus-rN  40805  erngfmul-rN  40808