Theorem erngset 41951

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

Hypotheses

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

Assertion

Ref	Expression
erngset	⊢ ((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 ↦ (𝑠 ∘ 𝑡))⟩})

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

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erngset.d	. . 3 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
2		erngset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3	2	erngfset 41950	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (EDRing‘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) ↦ (𝑠 ∘ 𝑡))⟩}))
4	3	fveq1d 6888	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((EDRing‘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) ↦ (𝑠 ∘ 𝑡))⟩})‘𝑊))
5	1, 4	eqtrid 2807	. 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) ↦ (𝑠 ∘ 𝑡))⟩})‘𝑊))
6		fveq2 6886	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
7	6	opeq2d 4846	. . . . 5 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩)
8		tpeq1 4699	. . . . . 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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(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) ↦ (𝑠 ∘ 𝑡))⟩})
9		erngset.e	. . . . . . . 8 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
10	9	opeq2i 4843	. . . . . . 7 ⊢ ⟨(Base‘ndx), 𝐸⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩
11		tpeq1 4699	. . . . . . 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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(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) ↦ (𝑠 ∘ 𝑡))⟩})
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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(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) ↦ (𝑠 ∘ 𝑡))⟩}
13	8, 12	eqtr4di 2813	. . . . 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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(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) ↦ (𝑠 ∘ 𝑡))⟩})
14	7, 13	syl 18	. . . 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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(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) ↦ (𝑠 ∘ 𝑡))⟩})
15	6, 9	eqtr4di 2813	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
16		fveq2 6886	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
17		erngset.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
19		eqidd 2761	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))
20	18, 19	mpteq12dv 5191	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
21	15, 15, 20	mpoeq123dv 7496	. . . . . 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 4846	. . . . 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 4703	. . . 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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(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) ↦ (𝑠 ∘ 𝑡))⟩})
24		eqidd 2761	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠 ∘ 𝑡) = (𝑠 ∘ 𝑡))
25	15, 15, 24	mpoeq123dv 7496	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠 ∘ 𝑡)) = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡)))
26	25	opeq2d 4846	. . . . 5 ⊢ (𝑤 = 𝑊 → ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠 ∘ 𝑡))⟩ = ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))⟩)
27	26	tpeq3d 4704	. . . 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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))⟩})
28	14, 23, 27	3eqtrd 2799	. . 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) ↦ (𝑠 ∘ 𝑡))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))⟩})
29		eqid 2760	. . 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) ↦ (𝑠 ∘ 𝑡))⟩}) = (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) ↦ (𝑠 ∘ 𝑡))⟩})
30		tpex 7763	. . 3 ⊢ {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))⟩} ∈ V
31	28, 29, 30	fvmpt 6994	. 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) ↦ (𝑠 ∘ 𝑡))⟩})‘𝑊) = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))⟩})
32	5, 31	sylan9eq 2815	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 ↦ (𝑠 ∘ 𝑡))⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {ctp 4588  ⟨cop 4834   ↦ cmpt 5185   ∘ ccom 5654  ‘cfv 6539   ∈ cmpo 7423  ndxcnx 17377  Basecbs 17393  +_gcplusg 17434  ._rcmulr 17435  LHypclh 41135  LTrncltrn 41252  TEndoctendo 41903  EDRingcedring 41904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-oprab 7425  df-mpo 7426  df-edring 41908

This theorem is used by:  erngbase  41952  erngfplus  41953  erngfmul  41956