Theorem cnfldfunALT 21441

Description: The field of complex numbers is a function. Alternate proof of cnfldfun 21440 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 21427. (Revised by GG, 31-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cnfldfunALT	⊢ Fun ℂfld

Proof of Theorem cnfldfunALT

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

Step	Hyp	Ref	Expression
1		basendxnplusgndx 17318	. . . . . . 7 ⊢ (Base‘ndx) ≠ (+g‘ndx)
2		basendxnmulrndx 17327	. . . . . . 7 ⊢ (Base‘ndx) ≠ (.r‘ndx)
3		plusgndxnmulrndx 17328	. . . . . . 7 ⊢ (+g‘ndx) ≠ (.r‘ndx)
4		fvex 6882	. . . . . . . 8 ⊢ (Base‘ndx) ∈ V
5		fvex 6882	. . . . . . . 8 ⊢ (+g‘ndx) ∈ V
6		fvex 6882	. . . . . . . 8 ⊢ (.r‘ndx) ∈ V
7		cnex 11156	. . . . . . . 8 ⊢ ℂ ∈ V
8		mpoaddex 12991	. . . . . . . 8 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) ∈ V
9		mpomulex 12993	. . . . . . . 8 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ V
10	4, 5, 6, 7, 8, 9	funtp 6580	. . . . . . 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 1484	. . . . . 6 ⊢ Fun {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩}
12		fvex 6882	. . . . . . 7 ⊢ (*𝑟‘ndx) ∈ V
13		cjf 15133	. . . . . . . 8 ⊢ ∗:ℂ⟶ℂ
14		fex 7212	. . . . . . . 8 ⊢ ((∗:ℂ⟶ℂ ∧ ℂ ∈ V) → ∗ ∈ V)
15	13, 7, 14	mp2an 702	. . . . . . 7 ⊢ ∗ ∈ V
16	12, 15	funsn 6576	. . . . . 6 ⊢ Fun {⟨(*𝑟‘ndx), ∗⟩}
17	11, 16	pm3.2i 474	. . . . 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 6207	. . . . . . 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 6205	. . . . . . 7 ⊢ dom {⟨(*𝑟‘ndx), ∗⟩} = {(*𝑟‘ndx)}
20	18, 19	ineq12i 4172	. . . . . 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 17335	. . . . . . . 8 ⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
22	21	necomi 3013	. . . . . . 7 ⊢ (Base‘ndx) ≠ (*𝑟‘ndx)
23		starvndxnplusgndx 17336	. . . . . . . 8 ⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
24	23	necomi 3013	. . . . . . 7 ⊢ (+g‘ndx) ≠ (*𝑟‘ndx)
25		starvndxnmulrndx 17337	. . . . . . . 8 ⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
26	25	necomi 3013	. . . . . . 7 ⊢ (.r‘ndx) ≠ (*𝑟‘ndx)
27		disjtpsn 4676	. . . . . . 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 1484	. . . . . 6 ⊢ ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(*𝑟‘ndx)}) = ∅
29	20, 28	eqtri 2787	. . . . 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 6569	. . . . 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 702	. . . 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 17406	. . . . . . . 8 ⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
33	32	simpri 489	. . . . . . 7 ⊢ (TopSet‘ndx) ≠ (le‘ndx)
34		dsndxntsetndx 17424	. . . . . . . 8 ⊢ (dist‘ndx) ≠ (TopSet‘ndx)
35	34	necomi 3013	. . . . . . 7 ⊢ (TopSet‘ndx) ≠ (dist‘ndx)
36		slotsdifdsndx 17425	. . . . . . . 8 ⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
37	36	simpri 489	. . . . . . 7 ⊢ (le‘ndx) ≠ (dist‘ndx)
38		fvex 6882	. . . . . . . 8 ⊢ (TopSet‘ndx) ∈ V
39		fvex 6882	. . . . . . . 8 ⊢ (le‘ndx) ∈ V
40		fvex 6882	. . . . . . . 8 ⊢ (dist‘ndx) ∈ V
41		fvex 6882	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) ∈ V
42		letsr 18627	. . . . . . . . 9 ⊢ ≤ ∈ TosetRel
43	42	elexi 3478	. . . . . . . 8 ⊢ ≤ ∈ V
44		absf 15367	. . . . . . . . . 10 ⊢ abs:ℂ⟶ℝ
45		fex 7212	. . . . . . . . . 10 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
46	44, 7, 45	mp2an 702	. . . . . . . . 9 ⊢ abs ∈ V
47		subf 11434	. . . . . . . . . 10 ⊢ − :(ℂ × ℂ)⟶ℂ
48	7, 7	xpex 7738	. . . . . . . . . 10 ⊢ (ℂ × ℂ) ∈ V
49		fex 7212	. . . . . . . . . 10 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V) → − ∈ V)
50	47, 48, 49	mp2an 702	. . . . . . . . 9 ⊢ − ∈ V
51	46, 50	coex 7913	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ V
52	38, 39, 40, 41, 43, 51	funtp 6580	. . . . . . 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 1484	. . . . . 6 ⊢ Fun {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}
54		fvex 6882	. . . . . . 7 ⊢ (UnifSet‘ndx) ∈ V
55		fvex 6882	. . . . . . 7 ⊢ (metUnif‘(abs ∘ − )) ∈ V
56	54, 55	funsn 6576	. . . . . 6 ⊢ Fun {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}
57	53, 56	pm3.2i 474	. . . . 5 ⊢ (Fun {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∧ Fun {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
58	41, 43, 51	dmtpop 6207	. . . . . . 7 ⊢ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} = {(TopSet‘ndx), (le‘ndx), (dist‘ndx)}
59	55	dmsnop 6205	. . . . . . 7 ⊢ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} = {(UnifSet‘ndx)}
60	58, 59	ineq12i 4172	. . . . . 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 17432	. . . . . . . 8 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
62		unifndxntsetndx 17431	. . . . . . . . . . . 12 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
63	62	necomi 3013	. . . . . . . . . . 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 624	. . . . . . . . 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 1107	. . . . . . . . 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 236	. . . . . . . 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 4676	. . . . . . 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 2787	. . . . 5 ⊢ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅
72		funun 6569	. . . . 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 702	. . . 4 ⊢ Fun ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
74	31, 73	pm3.2i 474	. . 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 5888	. . . . 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 5888	. . . . 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 4172	. . . 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 4172	. . . . . . . . 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 17387	. . . . . . . . . . . 12 ⊢ (TopSet‘ndx) ≠ (Base‘ndx)
80	79	necomi 3013	. . . . . . . . . . 11 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
81		tsetndxnplusgndx 17388	. . . . . . . . . . . 12 ⊢ (TopSet‘ndx) ≠ (+g‘ndx)
82	81	necomi 3013	. . . . . . . . . . 11 ⊢ (+g‘ndx) ≠ (TopSet‘ndx)
83		tsetndxnmulrndx 17389	. . . . . . . . . . . 12 ⊢ (TopSet‘ndx) ≠ (.r‘ndx)
84	83	necomi 3013	. . . . . . . . . . 11 ⊢ (.r‘ndx) ≠ (TopSet‘ndx)
85	80, 82, 84	3pm3.2i 1354	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ∧ (+g‘ndx) ≠ (TopSet‘ndx) ∧ (.r‘ndx) ≠ (TopSet‘ndx))
86		plendxnbasendx 17401	. . . . . . . . . . . 12 ⊢ (le‘ndx) ≠ (Base‘ndx)
87	86	necomi 3013	. . . . . . . . . . 11 ⊢ (Base‘ndx) ≠ (le‘ndx)
88		plendxnplusgndx 17402	. . . . . . . . . . . 12 ⊢ (le‘ndx) ≠ (+g‘ndx)
89	88	necomi 3013	. . . . . . . . . . 11 ⊢ (+g‘ndx) ≠ (le‘ndx)
90		plendxnmulrndx 17403	. . . . . . . . . . . 12 ⊢ (le‘ndx) ≠ (.r‘ndx)
91	90	necomi 3013	. . . . . . . . . . 11 ⊢ (.r‘ndx) ≠ (le‘ndx)
92	87, 89, 91	3pm3.2i 1354	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (le‘ndx) ∧ (+g‘ndx) ≠ (le‘ndx) ∧ (.r‘ndx) ≠ (le‘ndx))
93		dsndxnbasendx 17420	. . . . . . . . . . . 12 ⊢ (dist‘ndx) ≠ (Base‘ndx)
94	93	necomi 3013	. . . . . . . . . . 11 ⊢ (Base‘ndx) ≠ (dist‘ndx)
95		dsndxnplusgndx 17421	. . . . . . . . . . . 12 ⊢ (dist‘ndx) ≠ (+g‘ndx)
96	95	necomi 3013	. . . . . . . . . . 11 ⊢ (+g‘ndx) ≠ (dist‘ndx)
97		dsndxnmulrndx 17422	. . . . . . . . . . . 12 ⊢ (dist‘ndx) ≠ (.r‘ndx)
98	97	necomi 3013	. . . . . . . . . . 11 ⊢ (.r‘ndx) ≠ (dist‘ndx)
99	94, 96, 98	3pm3.2i 1354	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (dist‘ndx) ∧ (+g‘ndx) ≠ (dist‘ndx) ∧ (.r‘ndx) ≠ (dist‘ndx))
100		disjtp2 4677	. . . . . . . . . 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 1484	. . . . . . . . 9 ⊢ ({(Base‘ndx), (+g‘ndx), (.r‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)}) = ∅
102	78, 101	eqtri 2787	. . . . . . . 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 4172	. . . . . . . . 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 17430	. . . . . . . . . . . . . . 15 ⊢ (UnifSet‘ndx) ≠ (Base‘ndx)
105	104	necomi 3013	. . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . 13 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) → ((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx)))
108		3anass 1107	. . . . . . . . . . . . 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 592	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 4676	. . . . . . . . . 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 2787	. . . . . . . 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 474	. . . . . . 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 4419	. . . . . . 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 232	. . . . . 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 4172	. . . . . . . . 9 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ({(*𝑟‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)})
119		tsetndxnstarvndx 17390	. . . . . . . . . . 11 ⊢ (TopSet‘ndx) ≠ (*𝑟‘ndx)
120		necom 3012	. . . . . . . . . . . . 13 ⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ↔ (le‘ndx) ≠ (*𝑟‘ndx))
121	120	birani 507	. . . . . . . . . . . 12 ⊢ (((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx)) → (le‘ndx) ≠ (*𝑟‘ndx))
122	32, 121	ax-mp 5	. . . . . . . . . . 11 ⊢ (le‘ndx) ≠ (*𝑟‘ndx)
123		necom 3012	. . . . . . . . . . . . 13 ⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) ↔ (dist‘ndx) ≠ (*𝑟‘ndx))
124	123	birani 507	. . . . . . . . . . . 12 ⊢ (((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx)) → (dist‘ndx) ≠ (*𝑟‘ndx))
125	36, 124	ax-mp 5	. . . . . . . . . . 11 ⊢ (dist‘ndx) ≠ (*𝑟‘ndx)
126		disjtpsn 4676	. . . . . . . . . . 11 ⊢ (((TopSet‘ndx) ≠ (*𝑟‘ndx) ∧ (le‘ndx) ≠ (*𝑟‘ndx) ∧ (dist‘ndx) ≠ (*𝑟‘ndx)) → ({(TopSet‘ndx), (le‘ndx), (dist‘ndx)} ∩ {(*𝑟‘ndx)}) = ∅)
127	119, 122, 125, 126	mp3an 1484	. . . . . . . . . 10 ⊢ ({(TopSet‘ndx), (le‘ndx), (dist‘ndx)} ∩ {(*𝑟‘ndx)}) = ∅
128	127	ineqcomi 4165	. . . . . . . . 9 ⊢ ({(*𝑟‘ndx)} ∩ {(TopSet‘ndx), (le‘ndx), (dist‘ndx)}) = ∅
129	118, 128	eqtri 2787	. . . . . . . 8 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅
130	19, 59	ineq12i 4172	. . . . . . . . 9 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ({(*𝑟‘ndx)} ∩ {(UnifSet‘ndx)})
131		simpl3 1208	. . . . . . . . . . 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))
132	61, 131	ax-mp 5	. . . . . . . . . 10 ⊢ (*𝑟‘ndx) ≠ (UnifSet‘ndx)
133		disjsn2 4673	. . . . . . . . . 10 ⊢ ((*𝑟‘ndx) ≠ (UnifSet‘ndx) → ({(*𝑟‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅)
134	132, 133	ax-mp 5	. . . . . . . . 9 ⊢ ({(*𝑟‘ndx)} ∩ {(UnifSet‘ndx)}) = ∅
135	130, 134	eqtri 2787	. . . . . . . 8 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅
136	129, 135	pm3.2i 474	. . . . . . 7 ⊢ ((dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}) = ∅ ∧ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) = ∅)
137		undisj2 4419	. . . . . . 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 ∘ − ))⟩})) = ∅)
138	136, 137	mpbi 232	. . . . . 6 ⊢ (dom {⟨(*𝑟‘ndx), ∗⟩} ∩ (dom {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ dom {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) = ∅
139	117, 138	pm3.2i 474	. . . . 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 ∘ − ))⟩})) = ∅)
140		undisj1 4418	. . . . 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 ∘ − ))⟩})) = ∅)
141	139, 140	mpbi 232	. . . 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 ∘ − ))⟩})) = ∅
142	77, 141	eqtri 2787	. . 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 ∘ − ))⟩})) = ∅
143		funun 6569	. . 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 ∘ − ))⟩})))
144	74, 142, 143	mp2an 702	. 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 ∘ − ))⟩}))
145		df-cnfld 21427	. . 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 ∘ − ))⟩}))
146	145	funeqi 6544	. 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 ∘ − ))⟩})))
147	144, 146	mpbir 233	1 ⊢ Fun ℂfld

Syntax hints:   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  Vcvv 3456   ∪ cun 3904   ∩ cin 3905  ∅c0 4287  {csn 4584  {ctp 4588  ⟨cop 4590   × cxp 5647  dom cdm 5649   ∘ ccom 5653  Fun wfun 6517  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  ℂcc 11073  ℝcr 11074   + caddc 11078   · cmul 11080   ≤ cle 11219   − cmin 11416  ∗ccj 15125  abscabs 15263  ndxcnx 17231  Basecbs 17247  +_gcplusg 17288  ._rcmulr 17289  *_𝑟cstv 17290  TopSetcts 17294  lecple 17295  distcds 17297  UnifSetcunif 17298   TosetRel ctsr 18599  MetOpencmopn 21416  metUnifcmetu 21417  ℂ_fldccnfld 21426

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-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  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-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  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-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  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-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  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-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  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-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-sup 9390  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-rp 12996  df-seq 14017  df-exp 14077  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-slot 17220  df-ndx 17232  df-base 17248  df-plusg 17301  df-mulr 17302  df-starv 17303  df-tset 17307  df-ple 17308  df-ds 17310  df-unif 17311  df-ps 18600  df-tsr 18601  df-cnfld 21427

This theorem is referenced by: (None)