Theorem erngset-rN 41437

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 41436	. . . 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 6871	. . 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 2811	. 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 6869	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
7	6	opeq2d 4840	. . . . 5 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩)
8		tpeq1 4703	. . . . . 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 4837	. . . . . . 7 ⊢ ⟨(Base‘ndx), 𝐸⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩
11		tpeq1 4703	. . . . . . 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 2817	. . . . 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 2817	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
16		fveq2 6869	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
17		erngset.t-r	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2817	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
19		eqidd 2765	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))
20	18, 19	mpteq12dv 5189	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
21	15, 15, 20	mpoeq123dv 7473	. . . . . 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 4840	. . . . 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 4707	. . . 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 2765	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑡 ∘ 𝑠) = (𝑡 ∘ 𝑠))
25	15, 15, 24	mpoeq123dv 7473	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠)) = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠)))
26	25	opeq2d 4840	. . . . 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 4708	. . . 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 2803	. . 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 2764	. . 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 7731	. . 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 6977	. 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 2819	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 399   = wceq 1562   ∈ wcel 2144  {ctp 4588  ⟨cop 4590   ↦ cmpt 5183   ∘ ccom 5653  ‘cfv 6523   ∈ cmpo 7400  ndxcnx 17231  Basecbs 17247  +_gcplusg 17288  ._rcmulr 17289  LHypclh 40613  LTrncltrn 40730  TEndoctendo 41381  EDRing_Rcedring-rN 41383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-oprab 7402  df-mpo 7403  df-edring-rN 41385

This theorem is referenced by:  erngbase-rN  41438  erngfplus-rN  41439  erngfmul-rN  41442