Theorem cnfldfun 21351

Description: The field of complex numbers is a function. The proof is much shorter than the proof of cnfldfunALT 21352 by using cnfldstr 21339 and structn0fun 17146: in addition, it must be shown that ∅ ∉ ℂ_fld. (Contributed by AV, 18-Nov-2021.) Revise df-cnfld 21338. (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 21339	. 2 ⊢ ℂfld Struct ⟨1, ;13⟩
2		structn0fun 17146	. . 3 ⊢ (ℂfld Struct ⟨1, ;13⟩ → Fun (ℂfld ∖ {∅}))
3		fvex 6904	. . . . . . . . . . . . 13 ⊢ (Base‘ndx) ∈ V
4		cnex 11228	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
5	3, 4	opnzi 5471	. . . . . . . . . . . 12 ⊢ ⟨(Base‘ndx), ℂ⟩ ≠ ∅
6	5	nesymi 2988	. . . . . . . . . . 11 ⊢ ¬ ∅ = ⟨(Base‘ndx), ℂ⟩
7		fvex 6904	. . . . . . . . . . . . 13 ⊢ (+g‘ndx) ∈ V
8		mpoaddex 13016	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) ∈ V
9	7, 8	opnzi 5471	. . . . . . . . . . . 12 ⊢ ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ ≠ ∅
10	9	nesymi 2988	. . . . . . . . . . 11 ⊢ ¬ ∅ = ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩
11		fvex 6904	. . . . . . . . . . . . 13 ⊢ (.r‘ndx) ∈ V
12		mpomulex 13018	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ V
13	11, 12	opnzi 5471	. . . . . . . . . . . 12 ⊢ ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩ ≠ ∅
14	13	nesymi 2988	. . . . . . . . . . 11 ⊢ ¬ ∅ = ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩
15		3ioran 1103	. . . . . . . . . . . 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 5303	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
17	16	eltp 4688	. . . . . . . . . . . 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 332	. . . . . . . . . . 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 1338	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩}
20		fvex 6904	. . . . . . . . . . . . 13 ⊢ (*𝑟‘ndx) ∈ V
21		cjf 15102	. . . . . . . . . . . . . 14 ⊢ ∗:ℂ⟶ℂ
22		fex 7233	. . . . . . . . . . . . . 14 ⊢ ((∗:ℂ⟶ℂ ∧ ℂ ∈ V) → ∗ ∈ V)
23	21, 4, 22	mp2an 690	. . . . . . . . . . . . 13 ⊢ ∗ ∈ V
24	20, 23	opnzi 5471	. . . . . . . . . . . 12 ⊢ ⟨(*𝑟‘ndx), ∗⟩ ≠ ∅
25	24	necomi 2985	. . . . . . . . . . 11 ⊢ ∅ ≠ ⟨(*𝑟‘ndx), ∗⟩
26		nelsn 4664	. . . . . . . . . . 11 ⊢ (∅ ≠ ⟨(*𝑟‘ndx), ∗⟩ → ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩})
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩}
28	19, 27	pm3.2i 469	. . . . . . . . 9 ⊢ (¬ ∅ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ ¬ ∅ ∈ {⟨(*𝑟‘ndx), ∗⟩})
29		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (TopSet‘ndx) ∈ V
30		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (MetOpen‘(abs ∘ − )) ∈ V
31	29, 30	opnzi 5471	. . . . . . . . . . . . 13 ⊢ ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩ ≠ ∅
32	31	nesymi 2988	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩
33		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (le‘ndx) ∈ V
34		letsr 18611	. . . . . . . . . . . . . . 15 ⊢ ≤ ∈ TosetRel
35	34	elexi 3485	. . . . . . . . . . . . . 14 ⊢ ≤ ∈ V
36	33, 35	opnzi 5471	. . . . . . . . . . . . 13 ⊢ ⟨(le‘ndx), ≤ ⟩ ≠ ∅
37	36	nesymi 2988	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨(le‘ndx), ≤ ⟩
38		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (dist‘ndx) ∈ V
39		absf 15335	. . . . . . . . . . . . . . . 16 ⊢ abs:ℂ⟶ℝ
40		fex 7233	. . . . . . . . . . . . . . . 16 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
41	39, 4, 40	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ abs ∈ V
42		subf 11501	. . . . . . . . . . . . . . . 16 ⊢ − :(ℂ × ℂ)⟶ℂ
43	4, 4	xpex 7751	. . . . . . . . . . . . . . . 16 ⊢ (ℂ × ℂ) ∈ V
44		fex 7233	. . . . . . . . . . . . . . . 16 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V) → − ∈ V)
45	42, 43, 44	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ − ∈ V
46	41, 45	coex 7933	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) ∈ V
47	38, 46	opnzi 5471	. . . . . . . . . . . . 13 ⊢ ⟨(dist‘ndx), (abs ∘ − )⟩ ≠ ∅
48	47	nesymi 2988	. . . . . . . . . . . 12 ⊢ ¬ ∅ = ⟨(dist‘ndx), (abs ∘ − )⟩
49	32, 37, 48	3pm3.2ni 1484	. . . . . . . . . . 11 ⊢ ¬ (∅ = ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩ ∨ ∅ = ⟨(le‘ndx), ≤ ⟩ ∨ ∅ = ⟨(dist‘ndx), (abs ∘ − )⟩)
50	16	eltp 4688	. . . . . . . . . . 11 ⊢ (∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ↔ (∅ = ⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩ ∨ ∅ = ⟨(le‘ndx), ≤ ⟩ ∨ ∅ = ⟨(dist‘ndx), (abs ∘ − )⟩))
51	49, 50	mtbir 322	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}
52		fvex 6904	. . . . . . . . . . . . 13 ⊢ (UnifSet‘ndx) ∈ V
53		fvex 6904	. . . . . . . . . . . . 13 ⊢ (metUnif‘(abs ∘ − )) ∈ V
54	52, 53	opnzi 5471	. . . . . . . . . . . 12 ⊢ ⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩ ≠ ∅
55	54	necomi 2985	. . . . . . . . . . 11 ⊢ ∅ ≠ ⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩
56		nelsn 4664	. . . . . . . . . . 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 469	. . . . . . . . 9 ⊢ (¬ ∅ ∈ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ ¬ ∅ ∈ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
59	28, 58	pm3.2i 469	. . . . . . . 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 981	. . . . . . . . 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 981	. . . . . . . . . 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 981	. . . . . . . . . 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 626	. . . . . . . . 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 274	. . . . . . . 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 230	. . . . . . 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 21338	. . . . . . . . 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 2818	. . . . . . . 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 4146	. . . . . . . 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 4146	. . . . . . . . 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 4146	. . . . . . . . 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 912	. . . . . . . 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 296	. . . . . . 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 322	. . . . . 6 ⊢ ¬ ∅ ∈ ℂfld
74		disjsn 4711	. . . . . 6 ⊢ ((ℂfld ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ℂfld)
75	73, 74	mpbir 230	. . . . 5 ⊢ (ℂfld ∩ {∅}) = ∅
76		disjdif2 4475	. . . . 5 ⊢ ((ℂfld ∩ {∅}) = ∅ → (ℂfld ∖ {∅}) = ℂfld)
77	75, 76	ax-mp 5	. . . 4 ⊢ (ℂfld ∖ {∅}) = ℂfld
78	77	funeqi 6570	. . 3 ⊢ (Fun (ℂfld ∖ {∅}) ↔ Fun ℂfld)
79	2, 78	sylib 217	. 2 ⊢ (ℂfld Struct ⟨1, ;13⟩ → Fun ℂfld)
80	1, 79	ax-mp 5	1 ⊢ Fun ℂfld

Syntax hints:  ¬ wn 3   ∧ wa 394   ∨ wo 845   ∨ w3o 1083   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  Vcvv 3463   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946  ∅c0 4323  {csn 4624  {ctp 4628  ⟨cop 4630   class class class wbr 5144   × cxp 5671   ∘ ccom 5677  Fun wfun 6538  ⟶wf 6540  ‘cfv 6544  (class class class)co 7414   ∈ cmpo 7416  ℂcc 11145  ℝcr 11146  1c1 11148   + caddc 11150   · cmul 11152   ≤ cle 11288   − cmin 11483  3c3 12312  ;cdc 12721  ∗ccj 15094  abscabs 15232   Struct cstr 17141  ndxcnx 17188  Basecbs 17206  +_gcplusg 17259  ._rcmulr 17260  *_𝑟cstv 17261  TopSetcts 17265  lecple 17266  distcds 17268  UnifSetcunif 17269   TosetRel ctsr 18583  MetOpencmopn 21327  metUnifcmetu 21328  ℂ_fldccnfld 21337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224  ax-pre-sup 11225

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9476  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-div 11911  df-nn 12257  df-2 12319  df-3 12320  df-4 12321  df-5 12322  df-6 12323  df-7 12324  df-8 12325  df-9 12326  df-n0 12517  df-z 12603  df-dec 12722  df-uz 12867  df-rp 13021  df-fz 13531  df-seq 14014  df-exp 14074  df-cj 15097  df-re 15098  df-im 15099  df-sqrt 15233  df-abs 15234  df-struct 17142  df-slot 17177  df-ndx 17189  df-base 17207  df-plusg 17272  df-mulr 17273  df-starv 17274  df-tset 17278  df-ple 17279  df-ds 17281  df-unif 17282  df-ps 18584  df-tsr 18585  df-cnfld 21338

This theorem is referenced by: (None)