Theorem gg-cnfldfunALT 35485

Description: The field of complex numbers is a function. Alternate proof of cnfldfun 21157 not requiring that the index set of the components is ordered, but using quadratically many inequalities for the indices. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) Revise df-cnfld 21146. (Revised by GG, 31-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
gg-cnfldfunALT	⊢ Fun ℂfld

Proof of Theorem gg-cnfldfunALT

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

Step	Hyp	Ref	Expression
1		basendxnplusgndx 17232	. . . . . . 7 ⊢ (Base‘ndx) ≠ (+g‘ndx)
2		basendxnmulrndx 17245	. . . . . . 7 ⊢ (Base‘ndx) ≠ (.r‘ndx)
3		plusgndxnmulrndx 17247	. . . . . . 7 ⊢ (+g‘ndx) ≠ (.r‘ndx)
4		fvex 6904	. . . . . . . 8 ⊢ (Base‘ndx) ∈ V
5		fvex 6904	. . . . . . . 8 ⊢ (+g‘ndx) ∈ V
6		fvex 6904	. . . . . . . 8 ⊢ (.r‘ndx) ∈ V
7		cnex 11195	. . . . . . . 8 ⊢ ℂ ∈ V
8		mpoaddex 35473	. . . . . . . 8 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) ∈ V
9		mpomulex 35464	. . . . . . . 8 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ V
10	4, 5, 6, 7, 8, 9	funtp 6605	. . . . . . 7 ⊢ (((Base‘ndx) ≠ (+g‘ndx) ∧ (Base‘ndx) ≠ (.r‘ndx) ∧ (+g‘ndx) ≠ (.r‘ndx)) → Fun {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩})
11	1, 2, 3, 10	mp3an 1460	. . . . . 6 ⊢ Fun {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩}
12		fvex 6904	. . . . . . 7 ⊢ (*𝑟‘ndx) ∈ V
13		cjf 15056	. . . . . . . 8 ⊢ ∗:ℂ⟶ℂ
14		fex 7230	. . . . . . . 8 ⊢ ((∗:ℂ⟶ℂ ∧ ℂ ∈ V) → ∗ ∈ V)
15	13, 7, 14	mp2an 689	. . . . . . 7 ⊢ ∗ ∈ V
16	12, 15	funsn 6601	. . . . . 6 ⊢ Fun {⟨(*𝑟‘ndx), ∗⟩}
17	11, 16	pm3.2i 470	. . . . 5 ⊢ (Fun {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ Fun {⟨(*𝑟‘ndx), ∗⟩})
18	7, 8, 9	dmtpop 6217	. . . . . . 7 ⊢ dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} = {(Base‘ndx), (+g‘ndx), (.r‘ndx)}
19	15	dmsnop 6215	. . . . . . 7 ⊢ dom {⟨(*𝑟‘ndx), ∗⟩} = {(*𝑟‘ndx)}
20	18, 19	ineq12i 4210	. . . . . 6 ⊢ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(*𝑟‘ndx), ∗⟩}) = ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(*𝑟‘ndx)})
21		starvndxnbasendx 17254	. . . . . . . 8 ⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
22	21	necomi 2994	. . . . . . 7 ⊢ (Base‘ndx) ≠ (*𝑟‘ndx)
23		starvndxnplusgndx 17255	. . . . . . . 8 ⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
24	23	necomi 2994	. . . . . . 7 ⊢ (+g‘ndx) ≠ (*𝑟‘ndx)
25		starvndxnmulrndx 17256	. . . . . . . 8 ⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
26	25	necomi 2994	. . . . . . 7 ⊢ (.r‘ndx) ≠ (*𝑟‘ndx)
27		disjtpsn 4719	. . . . . . 7 ⊢ (((Base‘ndx) ≠ (*𝑟‘ndx) ∧ (+g‘ndx) ≠ (*𝑟‘ndx) ∧ (.r‘ndx) ≠ (*𝑟‘ndx)) → ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(*𝑟‘ndx)}) = ∅)
28	22, 24, 26, 27	mp3an 1460	. . . . . 6 ⊢ ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(*𝑟‘ndx)}) = ∅
29	20, 28	eqtri 2759	. . . . 5 ⊢ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(*𝑟‘ndx), ∗⟩}) = ∅
30		funun 6594	. . . . 5 ⊢ (((Fun {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∧ Fun {⟨(*𝑟‘ndx), ∗⟩}) ∧ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(*𝑟‘ndx), ∗⟩}) = ∅) → Fun ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}))
31	17, 29, 30	mp2an 689	. . . 4 ⊢ Fun ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩})
32		slotsdifplendx 17325	. . . . . . . 8 ⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
33	32	simpri 485	. . . . . . 7 ⊢ (TopSet‘ndx) ≠ (le‘ndx)
34		dsndxntsetndx 17343	. . . . . . . 8 ⊢ (dist‘ndx) ≠ (TopSet‘ndx)
35	34	necomi 2994	. . . . . . 7 ⊢ (TopSet‘ndx) ≠ (dist‘ndx)
36		slotsdifdsndx 17344	. . . . . . . 8 ⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
37	36	simpri 485	. . . . . . 7 ⊢ (le‘ndx) ≠ (dist‘ndx)
38		fvex 6904	. . . . . . . 8 ⊢ (TopSet‘ndx) ∈ V
39		fvex 6904	. . . . . . . 8 ⊢ (le‘ndx) ∈ V
40		fvex 6904	. . . . . . . 8 ⊢ (dist‘ndx) ∈ V
41		fvex 6904	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) ∈ V
42		letsr 18551	. . . . . . . . 9 ⊢ ≤ ∈ TosetRel
43	42	elexi 3493	. . . . . . . 8 ⊢ ≤ ∈ V
44		absf 15289	. . . . . . . . . 10 ⊢ abs:ℂ⟶ℝ
45		fex 7230	. . . . . . . . . 10 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
46	44, 7, 45	mp2an 689	. . . . . . . . 9 ⊢ abs ∈ V
47		subf 11467	. . . . . . . . . 10 ⊢ − :(ℂ × ℂ)⟶ℂ
48	7, 7	xpex 7744	. . . . . . . . . 10 ⊢ (ℂ × ℂ) ∈ V
49		fex 7230	. . . . . . . . . 10 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V) → − ∈ V)
50	47, 48, 49	mp2an 689	. . . . . . . . 9 ⊢ − ∈ V
51	46, 50	coex 7925	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ V
52	38, 39, 40, 41, 43, 51	funtp 6605	. . . . . . 7 ⊢ (((TopSet‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx)) → Fun {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩})
53	33, 35, 37, 52	mp3an 1460	. . . . . 6 ⊢ Fun {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}
54		fvex 6904	. . . . . . 7 ⊢ (UnifSet‘ndx) ∈ V
55		fvex 6904	. . . . . . 7 ⊢ (metUnif‘(abs ∘ − )) ∈ V
56	54, 55	funsn 6601	. . . . . 6 ⊢ Fun {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}
57	53, 56	pm3.2i 470	. . . . 5 ⊢ (Fun {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ Fun {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
58	41, 43, 51	dmtpop 6217	. . . . . . 7 ⊢ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} = {(TopSet‘ndx), (le‘ndx), (dist‘ndx)}
59	55	dmsnop 6215	. . . . . . 7 ⊢ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} = {(UnifSet‘ndx)}
60	58, 59	ineq12i 4210	. . . . . 6 ⊢ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ({(TopSet‘ndx), (le‘ndx), (dist‘ndx)} ∩ {(UnifSet‘ndx)})
61		slotsdifunifndx 17351	. . . . . . . 8 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
62		unifndxntsetndx 17350	. . . . . . . . . . . 12 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
63	62	necomi 2994	. . . . . . . . . . 11 ⊢ (TopSet‘ndx) ≠ (UnifSet‘ndx)
64	63	a1i 11	. . . . . . . . . 10 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) → (TopSet‘ndx) ≠ (UnifSet‘ndx))
65	64	anim1i 614	. . . . . . . . 9 ⊢ ((((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))) → ((TopSet‘ndx) ≠ (UnifSet‘ndx) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))))
66		3anass 1094	. . . . . . . . 9 ⊢ (((TopSet‘ndx) ≠ (UnifSet‘ndx) ∧ (le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)) ↔ ((TopSet‘ndx) ≠ (UnifSet‘ndx) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))))
67	65, 66	sylibr 233	. . . . . . . 8 ⊢ ((((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))) → ((TopSet‘ndx) ≠ (UnifSet‘ndx) ∧ (le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
68	61, 67	ax-mp 5	. . . . . . 7 ⊢ ((TopSet‘ndx) ≠ (UnifSet‘ndx) ∧ (le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))
69		disjtpsn 4719	. . . . . . 7 ⊢ (((TopSet‘ndx) ≠ (UnifSet‘ndx) ∧ (le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)) → ({(TopSet‘ndx), (le‘ndx), (dist‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅)
70	68, 69	ax-mp 5	. . . . . 6 ⊢ ({(TopSet‘ndx), (le‘ndx), (dist‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅
71	60, 70	eqtri 2759	. . . . 5 ⊢ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅
72		funun 6594	. . . . 5 ⊢ (((Fun {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ Fun {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ∧ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅) → Fun ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
73	57, 71, 72	mp2an 689	. . . 4 ⊢ Fun ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
74	31, 73	pm3.2i 470	. . 3 ⊢ (Fun ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∧ Fun ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
75		dmun 5910	. . . . 5 ⊢ dom ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) = (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ dom {⟨(*𝑟‘ndx), ∗⟩})
76		dmun 5910	. . . . 5 ⊢ dom ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
77	75, 76	ineq12i 4210	. . . 4 ⊢ (dom ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∩ dom ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ((dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ dom {⟨(*𝑟‘ndx), ∗⟩}) ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
78	18, 58	ineq12i 4210	. . . . . . . . 9 ⊢ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)})
79		tsetndxnbasendx 17306	. . . . . . . . . . . 12 ⊢ (TopSet‘ndx) ≠ (Base‘ndx)
80	79	necomi 2994	. . . . . . . . . . 11 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
81		tsetndxnplusgndx 17307	. . . . . . . . . . . 12 ⊢ (TopSet‘ndx) ≠ (+g‘ndx)
82	81	necomi 2994	. . . . . . . . . . 11 ⊢ (+g‘ndx) ≠ (TopSet‘ndx)
83		tsetndxnmulrndx 17308	. . . . . . . . . . . 12 ⊢ (TopSet‘ndx) ≠ (.r‘ndx)
84	83	necomi 2994	. . . . . . . . . . 11 ⊢ (.r‘ndx) ≠ (TopSet‘ndx)
85	80, 82, 84	3pm3.2i 1338	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ∧ (+g‘ndx) ≠ (TopSet‘ndx) ∧ (.r‘ndx) ≠ (TopSet‘ndx))
86		plendxnbasendx 17320	. . . . . . . . . . . 12 ⊢ (le‘ndx) ≠ (Base‘ndx)
87	86	necomi 2994	. . . . . . . . . . 11 ⊢ (Base‘ndx) ≠ (le‘ndx)
88		plendxnplusgndx 17321	. . . . . . . . . . . 12 ⊢ (le‘ndx) ≠ (+g‘ndx)
89	88	necomi 2994	. . . . . . . . . . 11 ⊢ (+g‘ndx) ≠ (le‘ndx)
90		plendxnmulrndx 17322	. . . . . . . . . . . 12 ⊢ (le‘ndx) ≠ (.r‘ndx)
91	90	necomi 2994	. . . . . . . . . . 11 ⊢ (.r‘ndx) ≠ (le‘ndx)
92	87, 89, 91	3pm3.2i 1338	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (le‘ndx) ∧ (+g‘ndx) ≠ (le‘ndx) ∧ (.r‘ndx) ≠ (le‘ndx))
93		dsndxnbasendx 17339	. . . . . . . . . . . 12 ⊢ (dist‘ndx) ≠ (Base‘ndx)
94	93	necomi 2994	. . . . . . . . . . 11 ⊢ (Base‘ndx) ≠ (dist‘ndx)
95		dsndxnplusgndx 17340	. . . . . . . . . . . 12 ⊢ (dist‘ndx) ≠ (+g‘ndx)
96	95	necomi 2994	. . . . . . . . . . 11 ⊢ (+g‘ndx) ≠ (dist‘ndx)
97		dsndxnmulrndx 17341	. . . . . . . . . . . 12 ⊢ (dist‘ndx) ≠ (.r‘ndx)
98	97	necomi 2994	. . . . . . . . . . 11 ⊢ (.r‘ndx) ≠ (dist‘ndx)
99	94, 96, 98	3pm3.2i 1338	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (dist‘ndx) ∧ (+g‘ndx) ≠ (dist‘ndx) ∧ (.r‘ndx) ≠ (dist‘ndx))
100		disjtp2 4720	. . . . . . . . . 10 ⊢ ((((Base‘ndx) ≠ (TopSet‘ndx) ∧ (+g‘ndx) ≠ (TopSet‘ndx) ∧ (.r‘ndx) ≠ (TopSet‘ndx)) ∧ ((Base‘ndx) ≠ (le‘ndx) ∧ (+g‘ndx) ≠ (le‘ndx) ∧ (.r‘ndx) ≠ (le‘ndx)) ∧ ((Base‘ndx) ≠ (dist‘ndx) ∧ (+g‘ndx) ≠ (dist‘ndx) ∧ (.r‘ndx) ≠ (dist‘ndx))) → ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)}) = ∅)
101	85, 92, 99, 100	mp3an 1460	. . . . . . . . 9 ⊢ ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)}) = ∅
102	78, 101	eqtri 2759	. . . . . . . 8 ⊢ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅
103	18, 59	ineq12i 4210	. . . . . . . . 9 ⊢ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(UnifSet‘ndx)})
104		unifndxnbasendx 17349	. . . . . . . . . . . . . . 15 ⊢ (UnifSet‘ndx) ≠ (Base‘ndx)
105	104	necomi 2994	. . . . . . . . . . . . . 14 ⊢ (Base‘ndx) ≠ (UnifSet‘ndx)
106	105	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) → (Base‘ndx) ≠ (UnifSet‘ndx))
107		3simpa 1147	. . . . . . . . . . . . 13 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) → ((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx)))
108		3anass 1094	. . . . . . . . . . . . 13 ⊢ (((Base‘ndx) ≠ (UnifSet‘ndx) ∧ (+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx)) ↔ ((Base‘ndx) ≠ (UnifSet‘ndx) ∧ ((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx))))
109	106, 107, 108	sylanbrc 582	. . . . . . . . . . . 12 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) → ((Base‘ndx) ≠ (UnifSet‘ndx) ∧ (+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx)))
110	109	adantr 480	. . . . . . . . . . 11 ⊢ ((((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))) → ((Base‘ndx) ≠ (UnifSet‘ndx) ∧ (+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx)))
111	61, 110	ax-mp 5	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (UnifSet‘ndx) ∧ (+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx))
112		disjtpsn 4719	. . . . . . . . . 10 ⊢ (((Base‘ndx) ≠ (UnifSet‘ndx) ∧ (+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx)) → ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅)
113	111, 112	ax-mp 5	. . . . . . . . 9 ⊢ ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅
114	103, 113	eqtri 2759	. . . . . . . 8 ⊢ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅
115	102, 114	pm3.2i 470	. . . . . . 7 ⊢ ((dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅ ∧ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅)
116		undisj2 4462	. . . . . . 7 ⊢ (((dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅ ∧ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅) ↔ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅)
117	115, 116	mpbi 229	. . . . . 6 ⊢ (dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅
118	19, 58	ineq12i 4210	. . . . . . . . 9 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ({(*𝑟‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)})
119		tsetndxnstarvndx 17309	. . . . . . . . . . 11 ⊢ (TopSet‘ndx) ≠ (*𝑟‘ndx)
120		necom 2993	. . . . . . . . . . . . . 14 ⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ↔ (le‘ndx) ≠ (*𝑟‘ndx))
121	120	biimpi 215	. . . . . . . . . . . . 13 ⊢ ((*𝑟‘ndx) ≠ (le‘ndx) → (le‘ndx) ≠ (*𝑟‘ndx))
122	121	adantr 480	. . . . . . . . . . . 12 ⊢ (((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx)) → (le‘ndx) ≠ (*𝑟‘ndx))
123	32, 122	ax-mp 5	. . . . . . . . . . 11 ⊢ (le‘ndx) ≠ (*𝑟‘ndx)
124		necom 2993	. . . . . . . . . . . . . 14 ⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) ↔ (dist‘ndx) ≠ (*𝑟‘ndx))
125	124	biimpi 215	. . . . . . . . . . . . 13 ⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) → (dist‘ndx) ≠ (*𝑟‘ndx))
126	125	adantr 480	. . . . . . . . . . . 12 ⊢ (((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx)) → (dist‘ndx) ≠ (*𝑟‘ndx))
127	36, 126	ax-mp 5	. . . . . . . . . . 11 ⊢ (dist‘ndx) ≠ (*𝑟‘ndx)
128		disjtpsn 4719	. . . . . . . . . . 11 ⊢ (((TopSet‘ndx) ≠ (*𝑟‘ndx) ∧ (le‘ndx) ≠ (*𝑟‘ndx) ∧ (dist‘ndx) ≠ (*𝑟‘ndx)) → ({(TopSet‘ndx), (le‘ndx), (dist‘ndx)} ∩ {(*𝑟‘ndx)}) = ∅)
129	119, 123, 127, 128	mp3an 1460	. . . . . . . . . 10 ⊢ ({(TopSet‘ndx), (le‘ndx), (dist‘ndx)} ∩ {(*𝑟‘ndx)}) = ∅
130	129	ineqcomi 4203	. . . . . . . . 9 ⊢ ({(*𝑟‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)}) = ∅
131	118, 130	eqtri 2759	. . . . . . . 8 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅
132	19, 59	ineq12i 4210	. . . . . . . . 9 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ({(*𝑟‘ndx)} ∩ {(UnifSet‘ndx)})
133		simpl3 1192	. . . . . . . . . . 11 ⊢ ((((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))) → (*𝑟‘ndx) ≠ (UnifSet‘ndx))
134	61, 133	ax-mp 5	. . . . . . . . . 10 ⊢ (*𝑟‘ndx) ≠ (UnifSet‘ndx)
135		disjsn2 4716	. . . . . . . . . 10 ⊢ ((*𝑟‘ndx) ≠ (UnifSet‘ndx) → ({(*𝑟‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅)
136	134, 135	ax-mp 5	. . . . . . . . 9 ⊢ ({(*𝑟‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅
137	132, 136	eqtri 2759	. . . . . . . 8 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅
138	131, 137	pm3.2i 470	. . . . . . 7 ⊢ ((dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅ ∧ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅)
139		undisj2 4462	. . . . . . 7 ⊢ (((dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅ ∧ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅) ↔ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅)
140	138, 139	mpbi 229	. . . . . 6 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅
141	117, 140	pm3.2i 470	. . . . 5 ⊢ ((dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅ ∧ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅)
142		undisj1 4461	. . . . 5 ⊢ (((dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅ ∧ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅) ↔ ((dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ dom {⟨(*𝑟‘ndx), ∗⟩}) ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅)
143	141, 142	mpbi 229	. . . 4 ⊢ ((dom {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ dom {⟨(*𝑟‘ndx), ∗⟩}) ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅
144	77, 143	eqtri 2759	. . 3 ⊢ (dom ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∩ dom ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅
145		funun 6594	. . 3 ⊢ (((Fun ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∧ Fun ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ∧ (dom ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∩ dom ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅) → Fun (({⟨(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 ∘ − ))⟩})))
146	74, 144, 145	mp2an 689	. 2 ⊢ Fun (({⟨(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 ∘ − ))⟩}))
147		gg-dfcnfld 35474	. . 3 ⊢ 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 ∘ − ))⟩}))
148	147	funeqi 6569	. 2 ⊢ (Fun ℂfld ↔ Fun (({⟨(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 ∘ − ))⟩})))
149	146, 148	mpbir 230	1 ⊢ Fun ℂfld

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  Vcvv 3473   ∪ cun 3946   ∩ cin 3947  ∅c0 4322  {csn 4628  {ctp 4632  ⟨cop 4634   × cxp 5674  dom cdm 5676   ∘ ccom 5680  Fun wfun 6537  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ∈ cmpo 7414  ℂcc 11112  ℝcr 11113   + caddc 11117   · cmul 11119   ≤ cle 11254   − cmin 11449  ∗ccj 15048  abscabs 15186  ndxcnx 17131  Basecbs 17149  +_gcplusg 17202  ._rcmulr 17203  *_𝑟cstv 17204  TopSetcts 17208  lecple 17209  distcds 17211  UnifSetcunif 17212   TosetRel ctsr 18523  MetOpencmopn 21135  metUnifcmetu 21136  ℂ_fldccnfld 21145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192  ax-addf 11193  ax-mulf 11194

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-rp 12980  df-seq 13972  df-exp 14033  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-mulr 17216  df-starv 17217  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-ps 18524  df-tsr 18525  df-cnfld 21146

This theorem is referenced by: (None)