Theorem cnfldfun 21515

Description: The field of complex numbers is a function. The proof is much shorter than the proof of cnfldfunALT 21516 by using cnfldstr 21503 and structn0fun 17210: in addition, it must be shown that ∅ ∉ ℂ_fld. (Contributed by AV, 18-Nov-2021.) Revise df-cnfld 21502. (Revised by GG, 31-Mar-2025.)

Assertion

Ref	Expression
cnfldfun	⊢ Fun ℂfld

Proof of Theorem cnfldfun

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnfldstr 21503	. 2 ⊢ ℂfld Struct ⟨1, ;13⟩
2		structn0fun 17210	. . 3 ⊢ (ℂfld Struct ⟨1, ;13⟩ → Fun (ℂfld ∖ {∅}))
3		fvex 6894	. . . . . . . . . . . . 13 ⊢ (Base‘ndx) ∈ V
4		cnex 11180	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
5	3, 4	opnzi 5456	. . . . . . . . . . . 12 ⊢ ⟨(Base‘ndx), ℂ⟩ ≠ ∅
6	5	nesymi 3013	. . . . . . . . . . 11 ⊢ ¬ ∅ = ⟨(Base‘ndx), ℂ⟩
7		fvex 6894	. . . . . . . . . . . . 13 ⊢ (+g‘ndx) ∈ V
8		mpoaddex 13011	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) ∈ V
9	7, 8	opnzi 5456	. . . . . . . . . . . 12 ⊢ ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ ≠ ∅
10	9	nesymi 3013	. . . . . . . . . . 11 ⊢ ¬ ∅ = ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩
11		fvex 6894	. . . . . . . . . . . . 13 ⊢ (.r‘ndx) ∈ V
12		mpomulex 13013	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ V
13	11, 12	opnzi 5456	. . . . . . . . . . . 12 ⊢ ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩ ≠ ∅
14	13	nesymi 3013	. . . . . . . . . . 11 ⊢ ¬ ∅ = ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩
15		3ioran 1121	. . . . . . . . . . . 12 ⊢ (¬ (∅ = ⟨(Base‘ndx), ℂ⟩ ∨ ∅ = ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ ∨ ∅ = ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩) ↔ (¬ ∅ = ⟨(Base‘ndx), ℂ⟩ ∧ ¬ ∅ = ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ ∧ ¬ ∅ = ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩))
16		0ex 5269	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
17	16	eltp 4654	. . . . . . . . . . . 12 ⊢ (∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ↔ (∅ = ⟨(Base‘ndx), ℂ⟩ ∨ ∅ = ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ ∨ ∅ = ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩))
18	15, 17	xchnxbir 336	. . . . . . . . . . 11 ⊢ (¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ↔ (¬ ∅ = ⟨(Base‘ndx), ℂ⟩ ∧ ¬ ∅ = ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ ∧ ¬ ∅ = ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩))
19	6, 10, 14, 18	mpbir3an 1358	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩}
20		fvex 6894	. . . . . . . . . . . . 13 ⊢ (*𝑟‘ndx) ∈ V
21		cjf 15154	. . . . . . . . . . . . . 14 ⊢ ∗:ℂ⟶ℂ
22		fex 7224	. . . . . . . . . . . . . 14 ⊢ ((∗:ℂ⟶ℂ ∧ ℂ ∈ V) → ∗ ∈ V)
23	21, 4, 22	mp2an 704	. . . . . . . . . . . . 13 ⊢ ∗ ∈ V
24	20, 23	opnzi 5456	. . . . . . . . . . . 12 ⊢ ⟨(*𝑟‘ndx), ∗⟩ ≠ ∅
25	24	necomi 3010	. . . . . . . . . . 11 ⊢ ∅ ≠ ⟨(*𝑟‘ndx), ∗⟩
26		nelsn 4631	. . . . . . . . . . 11 ⊢ (∅ ≠ ⟨(*𝑟‘ndx), ∗⟩ → ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩})
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}
28	19, 27	pm3.2i 475	. . . . . . . . 9 ⊢ (¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩})
29		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (TopSet‘ndx) ∈ V
30		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (MetOpen‘(abs ∘ − )) ∈ V
31	29, 30	opnzi 5456	. . . . . . . . . . . . 13 ⊢ ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩ ≠ ∅
32	31	nesymi 3013	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩
33		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (le‘ndx) ∈ V
34		letsr 18648	. . . . . . . . . . . . . . 15 ⊢ ≤ ∈ TosetRel
35	34	elexi 3475	. . . . . . . . . . . . . 14 ⊢ ≤ ∈ V
36	33, 35	opnzi 5456	. . . . . . . . . . . . 13 ⊢ ⟨(le‘ndx), ≤ ⟩ ≠ ∅
37	36	nesymi 3013	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨(le‘ndx), ≤ ⟩
38		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (dist‘ndx) ∈ V
39		absf 15388	. . . . . . . . . . . . . . . 16 ⊢ abs:ℂ⟶ℝ
40		fex 7224	. . . . . . . . . . . . . . . 16 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
41	39, 4, 40	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ abs ∈ V
42		subf 11458	. . . . . . . . . . . . . . . 16 ⊢ − :(ℂ × ℂ)⟶ℂ
43	4, 4	xpex 7751	. . . . . . . . . . . . . . . 16 ⊢ (ℂ × ℂ) ∈ V
44		fex 7224	. . . . . . . . . . . . . . . 16 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V) → − ∈ V)
45	42, 43, 44	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ − ∈ V
46	41, 45	coex 7926	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) ∈ V
47	38, 46	opnzi 5456	. . . . . . . . . . . . 13 ⊢ ⟨(dist‘ndx), (abs ∘ − )⟩ ≠ ∅
48	47	nesymi 3013	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨(dist‘ndx), (abs ∘ − )⟩
49	32, 37, 48	3pm3.2ni 1517	. . . . . . . . . . 11 ⊢ ¬ (∅ = ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩ ∨ ∅ = ⟨(le‘ndx), ≤ ⟩ ∨ ∅ = ⟨(dist‘ndx), (abs ∘ − )⟩)
50	16	eltp 4654	. . . . . . . . . . 11 ⊢ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ↔ (∅ = ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩ ∨ ∅ = ⟨(le‘ndx), ≤ ⟩ ∨ ∅ = ⟨(dist‘ndx), (abs ∘ − )⟩))
51	49, 50	mtbir 326	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}
52		fvex 6894	. . . . . . . . . . . . 13 ⊢ (UnifSet‘ndx) ∈ V
53		fvex 6894	. . . . . . . . . . . . 13 ⊢ (metUnif‘(abs ∘ − )) ∈ V
54	52, 53	opnzi 5456	. . . . . . . . . . . 12 ⊢ ⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩ ≠ ∅
55	54	necomi 3010	. . . . . . . . . . 11 ⊢ ∅ ≠ ⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩
56		nelsn 4631	. . . . . . . . . . 11 ⊢ (∅ ≠ ⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩ → ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
57	55, 56	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}
58	51, 57	pm3.2i 475	. . . . . . . . 9 ⊢ (¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
59	28, 58	pm3.2i 475	. . . . . . . 8 ⊢ ((¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∧ (¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
60		ioran 999	. . . . . . . . 9 ⊢ (¬ ((∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∨ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ↔ (¬ (∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∧ ¬ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
61		ioran 999	. . . . . . . . . 10 ⊢ (¬ (∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ↔ (¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}))
62		ioran 999	. . . . . . . . . 10 ⊢ (¬ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ↔ (¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
63	61, 62	anbi12i 639	. . . . . . . . 9 ⊢ ((¬ (∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∧ ¬ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ↔ ((¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∧ (¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
64	60, 63	bitri 278	. . . . . . . 8 ⊢ (¬ ((∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∨ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ↔ ((¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∧ (¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
65	59, 64	mpbir 234	. . . . . . 7 ⊢ ¬ ((∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∨ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
66		df-cnfld 21502	. . . . . . . . 9 ⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
67	66	eleq2i 2853	. . . . . . . 8 ⊢ (∅ ∈ ℂfld ↔ ∅ ∈ (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
68		elun 4106	. . . . . . . 8 ⊢ (∅ ∈ (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ↔ (∅ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ∅ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
69		elun 4106	. . . . . . . . 9 ⊢ (∅ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ↔ (∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}))
70		elun 4106	. . . . . . . . 9 ⊢ (∅ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ↔ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
71	69, 70	orbi12i 927	. . . . . . . 8 ⊢ ((∅ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ∅ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ↔ ((∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∨ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
72	67, 68, 71	3bitri 300	. . . . . . 7 ⊢ (∅ ∈ ℂfld ↔ ((∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}) ∨ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∨ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
73	65, 72	mtbir 326	. . . . . 6 ⊢ ¬ ∅ ∈ ℂfld
74		disjsn 4676	. . . . . 6 ⊢ ((ℂfld ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ℂfld)
75	73, 74	mpbir 234	. . . . 5 ⊢ (ℂfld ∩ {∅}) = ∅
76		disjdif2 4440	. . . . 5 ⊢ ((ℂfld ∩ {∅}) = ∅ → (ℂfld ∖ {∅}) = ℂfld)
77	75, 76	ax-mp 5	. . . 4 ⊢ (ℂfld ∖ {∅}) = ℂfld
78	77	funeqi 6557	. . 3 ⊢ (Fun (ℂfld ∖ {∅}) ↔ Fun ℂfld)
79	2, 78	sylib 221	. 2 ⊢ (ℂfld Struct ⟨1, ;13⟩ → Fun ℂfld)
80	1, 79	ax-mp 5	1 ⊢ Fun ℂfld

Syntax hints:  ¬ wn 3   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903  ∅c0 4285  {csn 4588  {ctp 4592  ⟨cop 4594   class class class wbr 5108   × cxp 5659   ∘ ccom 5665  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  ℂcc 11097  ℝcr 11098  1c1 11100   + caddc 11102   · cmul 11104   ≤ cle 11243   − cmin 11440  3c3 12295  ;cdc 12710  ∗ccj 15146  abscabs 15284   Struct cstr 17205  ndxcnx 17252  Basecbs 17268  +_gcplusg 17309  ._rcmulr 17310  *_𝑟cstv 17311  TopSetcts 17315  lecple 17316  distcds 17318  UnifSetcunif 17319   TosetRel ctsr 18620  MetOpencmopn 21491  metUnifcmetu 21492  ℂ_fldccnfld 21501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-rp 13016  df-fz 13535  df-seq 14037  df-exp 14097  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-struct 17206  df-slot 17241  df-ndx 17253  df-base 17269  df-plusg 17322  df-mulr 17323  df-starv 17324  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-ps 18621  df-tsr 18622  df-cnfld 21502

This theorem is referenced by: (None)