Theorem cnfldfun 19758

Description: The field of complex numbers is a function. (Contributed by AV, 14-Nov-2021.)

Assertion

Ref	Expression
cnfldfun	|- Fun ℂfld

Proof of Theorem cnfldfun

Step	Hyp	Ref	Expression
1		basendxnplusgndx 15989	. . . . . . 7 |- ( Base ` ndx ) =/= ( +g ` ndx )
2		basendxnmulrndx 15999	. . . . . . 7 |- ( Base ` ndx ) =/= ( .r ` ndx )
3		plusgndxnmulrndx 15998	. . . . . . 7 |- ( +g ` ndx ) =/= ( .r ` ndx )
4		fvex 6201	. . . . . . . 8 |- ( Base ` ndx ) e. _V
5		fvex 6201	. . . . . . . 8 |- ( +g ` ndx ) e. _V
6		fvex 6201	. . . . . . . 8 |- ( .r ` ndx ) e. _V
7		cnex 10017	. . . . . . . 8 |- CC e. _V
8		addex 11830	. . . . . . . 8 |- + e. _V
9		mulex 11831	. . . . . . . 8 |- x. e. _V
10	4, 5, 6, 7, 8, 9	funtp 5945	. . . . . . 7 |- ( ( ( Base ` ndx ) =/= ( +g ` ndx ) /\ ( Base ` ndx ) =/= ( .r ` ndx ) /\ ( +g ` ndx ) =/= ( .r ` ndx ) ) -> Fun { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } )
11	1, 2, 3, 10	mp3an 1424	. . . . . 6 |- Fun { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. }
12		fvex 6201	. . . . . . 7 |- ( *r ` ndx ) e. _V
13		cjf 13844	. . . . . . . 8 |- * : CC --> CC
14		fex 6490	. . . . . . . 8 |- ( ( * : CC --> CC /\ CC e. _V ) -> * e. _V )
15	13, 7, 14	mp2an 708	. . . . . . 7 |- * e. _V
16	12, 15	funsn 5939	. . . . . 6 |- Fun { <. ( *r ` ndx ) , * >. }
17	11, 16	pm3.2i 471	. . . . 5 |- ( Fun { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } /\ Fun { <. ( *r ` ndx ) , * >. } )
18	7, 8, 9	dmtpop 5611	. . . . . . 7 |- dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } = { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) }
19	15	dmsnop 5609	. . . . . . 7 |- dom { <. ( *r ` ndx ) , * >. } = { ( *r ` ndx ) }
20	18, 19	ineq12i 3812	. . . . . 6 |- ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. ( *r ` ndx ) , * >. } ) = ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { ( *r ` ndx ) } )
21		basendx 15923	. . . . . . . 8 |- ( Base ` ndx ) = 1
22		1re 10039	. . . . . . . . . 10 |- 1 e. RR
23		1lt4 11199	. . . . . . . . . 10 |- 1 < 4
24	22, 23	ltneii 10150	. . . . . . . . 9 |- 1 =/= 4
25		starvndx 16004	. . . . . . . . 9 |- ( *r ` ndx ) = 4
26	24, 25	neeqtrri 2867	. . . . . . . 8 |- 1 =/= ( *r ` ndx )
27	21, 26	eqnetri 2864	. . . . . . 7 |- ( Base ` ndx ) =/= ( *r ` ndx )
28		plusgndx 15976	. . . . . . . 8 |- ( +g ` ndx ) = 2
29		2re 11090	. . . . . . . . . 10 |- 2 e. RR
30		2lt4 11198	. . . . . . . . . 10 |- 2 < 4
31	29, 30	ltneii 10150	. . . . . . . . 9 |- 2 =/= 4
32	31, 25	neeqtrri 2867	. . . . . . . 8 |- 2 =/= ( *r ` ndx )
33	28, 32	eqnetri 2864	. . . . . . 7 |- ( +g ` ndx ) =/= ( *r ` ndx )
34		mulrndx 15996	. . . . . . . 8 |- ( .r ` ndx ) = 3
35		3re 11094	. . . . . . . . . 10 |- 3 e. RR
36		3lt4 11197	. . . . . . . . . 10 |- 3 < 4
37	35, 36	ltneii 10150	. . . . . . . . 9 |- 3 =/= 4
38	37, 25	neeqtrri 2867	. . . . . . . 8 |- 3 =/= ( *r ` ndx )
39	34, 38	eqnetri 2864	. . . . . . 7 |- ( .r ` ndx ) =/= ( *r ` ndx )
40		disjtpsn 4251	. . . . . . 7 |- ( ( ( Base ` ndx ) =/= ( *r ` ndx ) /\ ( +g ` ndx ) =/= ( *r ` ndx ) /\ ( .r ` ndx ) =/= ( *r ` ndx ) ) -> ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { ( *r ` ndx ) } ) = (/) )
41	27, 33, 39, 40	mp3an 1424	. . . . . 6 |- ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { ( *r ` ndx ) } ) = (/)
42	20, 41	eqtri 2644	. . . . 5 |- ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. ( *r ` ndx ) , * >. } ) = (/)
43		funun 5932	. . . . 5 |- ( ( ( Fun { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } /\ Fun { <. ( *r ` ndx ) , * >. } ) /\ ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. ( *r ` ndx ) , * >. } ) = (/) ) -> Fun ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) )
44	17, 42, 43	mp2an 708	. . . 4 |- Fun ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } )
45		tsetndx 16040	. . . . . . . 8 |- (TopSet ` ndx ) = 9
46		9re 11107	. . . . . . . . . 10 |- 9 e. RR
47		9lt10 11673	. . . . . . . . . 10 |- 9 < ; 1 0
48	46, 47	ltneii 10150	. . . . . . . . 9 |- 9 =/= ; 1 0
49		plendx 16047	. . . . . . . . 9 |- ( le ` ndx ) = ; 1 0
50	48, 49	neeqtrri 2867	. . . . . . . 8 |- 9 =/= ( le ` ndx )
51	45, 50	eqnetri 2864	. . . . . . 7 |- (TopSet ` ndx ) =/= ( le ` ndx )
52		1nn 11031	. . . . . . . . . . 11 |- 1 e. NN
53		2nn0 11309	. . . . . . . . . . 11 |- 2 e. NN0
54		9nn0 11316	. . . . . . . . . . 11 |- 9 e. NN0
55	46	leidi 10562	. . . . . . . . . . 11 |- 9 <_ 9
56	52, 53, 54, 55	decltdi 11547	. . . . . . . . . 10 |- 9 < ; 1 2
57	46, 56	ltneii 10150	. . . . . . . . 9 |- 9 =/= ; 1 2
58		dsndx 16062	. . . . . . . . 9 |- ( dist ` ndx ) = ; 1 2
59	57, 58	neeqtrri 2867	. . . . . . . 8 |- 9 =/= ( dist ` ndx )
60	45, 59	eqnetri 2864	. . . . . . 7 |- (TopSet ` ndx ) =/= ( dist ` ndx )
61		10re 11517	. . . . . . . . . 10 |- ; 1 0 e. RR
62		1nn0 11308	. . . . . . . . . . 11 |- 1 e. NN0
63		0nn0 11307	. . . . . . . . . . 11 |- 0 e. NN0
64		2nn 11185	. . . . . . . . . . 11 |- 2 e. NN
65		2pos 11112	. . . . . . . . . . 11 |- 0 < 2
66	62, 63, 64, 65	declt 11530	. . . . . . . . . 10 |- ; 1 0 < ; 1 2
67	61, 66	ltneii 10150	. . . . . . . . 9 |- ; 1 0 =/= ; 1 2
68	67, 58	neeqtrri 2867	. . . . . . . 8 |- ; 1 0 =/= ( dist ` ndx )
69	49, 68	eqnetri 2864	. . . . . . 7 |- ( le ` ndx ) =/= ( dist ` ndx )
70		fvex 6201	. . . . . . . 8 |- (TopSet ` ndx ) e. _V
71		fvex 6201	. . . . . . . 8 |- ( le ` ndx ) e. _V
72		fvex 6201	. . . . . . . 8 |- ( dist ` ndx ) e. _V
73		fvex 6201	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) e. _V
74		letsr 17227	. . . . . . . . 9 |- <_ e. TosetRel
75	74	elexi 3213	. . . . . . . 8 |- <_ e. _V
76		absf 14077	. . . . . . . . . 10 |- abs : CC --> RR
77		fex 6490	. . . . . . . . . 10 |- ( ( abs : CC --> RR /\ CC e. _V ) -> abs e. _V )
78	76, 7, 77	mp2an 708	. . . . . . . . 9 |- abs e. _V
79		subf 10283	. . . . . . . . . 10 |- - : ( CC X. CC ) --> CC
80	7, 7	xpex 6962	. . . . . . . . . 10 |- ( CC X. CC ) e. _V
81		fex 6490	. . . . . . . . . 10 |- ( ( - : ( CC X. CC ) --> CC /\ ( CC X. CC ) e. _V ) -> - e. _V )
82	79, 80, 81	mp2an 708	. . . . . . . . 9 |- - e. _V
83	78, 82	coex 7118	. . . . . . . 8 |- ( abs o. - ) e. _V
84	70, 71, 72, 73, 75, 83	funtp 5945	. . . . . . 7 |- ( ( (TopSet ` ndx ) =/= ( le ` ndx ) /\ (TopSet ` ndx ) =/= ( dist ` ndx ) /\ ( le ` ndx ) =/= ( dist ` ndx ) ) -> Fun { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } )
85	51, 60, 69, 84	mp3an 1424	. . . . . 6 |- Fun { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. }
86		fvex 6201	. . . . . . 7 |- ( UnifSet ` ndx ) e. _V
87		fvex 6201	. . . . . . 7 |- (metUnif ` ( abs o. - ) ) e. _V
88	86, 87	funsn 5939	. . . . . 6 |- Fun { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. }
89	85, 88	pm3.2i 471	. . . . 5 |- ( Fun { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } /\ Fun { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } )
90	73, 75, 83	dmtpop 5611	. . . . . . 7 |- dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } = { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) }
91	87	dmsnop 5609	. . . . . . 7 |- dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } = { ( UnifSet ` ndx ) }
92	90, 91	ineq12i 3812	. . . . . 6 |- ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = ( { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } i^i { ( UnifSet ` ndx ) } )
93		3nn0 11310	. . . . . . . . . . 11 |- 3 e. NN0
94	52, 93, 54, 55	decltdi 11547	. . . . . . . . . 10 |- 9 < ; 1 3
95	46, 94	ltneii 10150	. . . . . . . . 9 |- 9 =/= ; 1 3
96		unifndx 16064	. . . . . . . . 9 |- ( UnifSet ` ndx ) = ; 1 3
97	95, 96	neeqtrri 2867	. . . . . . . 8 |- 9 =/= ( UnifSet ` ndx )
98	45, 97	eqnetri 2864	. . . . . . 7 |- (TopSet ` ndx ) =/= ( UnifSet ` ndx )
99		3nn 11186	. . . . . . . . . . 11 |- 3 e. NN
100		3pos 11114	. . . . . . . . . . 11 |- 0 < 3
101	62, 63, 99, 100	declt 11530	. . . . . . . . . 10 |- ; 1 0 < ; 1 3
102	61, 101	ltneii 10150	. . . . . . . . 9 |- ; 1 0 =/= ; 1 3
103	102, 96	neeqtrri 2867	. . . . . . . 8 |- ; 1 0 =/= ( UnifSet ` ndx )
104	49, 103	eqnetri 2864	. . . . . . 7 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
105	62, 53	deccl 11512	. . . . . . . . . . 11 |- ; 1 2 e. NN0
106	105	nn0rei 11303	. . . . . . . . . 10 |- ; 1 2 e. RR
107		2lt3 11195	. . . . . . . . . . 11 |- 2 < 3
108	62, 53, 99, 107	declt 11530	. . . . . . . . . 10 |- ; 1 2 < ; 1 3
109	106, 108	ltneii 10150	. . . . . . . . 9 |- ; 1 2 =/= ; 1 3
110	109, 96	neeqtrri 2867	. . . . . . . 8 |- ; 1 2 =/= ( UnifSet ` ndx )
111	58, 110	eqnetri 2864	. . . . . . 7 |- ( dist ` ndx ) =/= ( UnifSet ` ndx )
112		disjtpsn 4251	. . . . . . 7 |- ( ( (TopSet ` ndx ) =/= ( UnifSet ` ndx ) /\ ( le ` ndx ) =/= ( UnifSet ` ndx ) /\ ( dist ` ndx ) =/= ( UnifSet ` ndx ) ) -> ( { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } i^i { ( UnifSet ` ndx ) } ) = (/) )
113	98, 104, 111, 112	mp3an 1424	. . . . . 6 |- ( { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } i^i { ( UnifSet ` ndx ) } ) = (/)
114	92, 113	eqtri 2644	. . . . 5 |- ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/)
115		funun 5932	. . . . 5 |- ( ( ( Fun { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } /\ Fun { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) /\ ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/) ) -> Fun ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
116	89, 114, 115	mp2an 708	. . . 4 |- Fun ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } )
117	44, 116	pm3.2i 471	. . 3 |- ( Fun ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) /\ Fun ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
118		dmun 5331	. . . . 5 |- dom ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) = ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. dom { <. ( *r ` ndx ) , * >. } )
119		dmun 5331	. . . . 5 |- dom ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } )
120	118, 119	ineq12i 3812	. . . 4 |- ( dom ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) i^i dom ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = ( ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. dom { <. ( *r ` ndx ) , * >. } ) i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
121	18, 90	ineq12i 3812	. . . . . . . . 9 |- ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } )
122		1lt9 11229	. . . . . . . . . . . . . 14 |- 1 < 9
123	22, 122	ltneii 10150	. . . . . . . . . . . . 13 |- 1 =/= 9
124	123, 45	neeqtrri 2867	. . . . . . . . . . . 12 |- 1 =/= (TopSet ` ndx )
125	21, 124	eqnetri 2864	. . . . . . . . . . 11 |- ( Base ` ndx ) =/= (TopSet ` ndx )
126		2lt9 11228	. . . . . . . . . . . . . 14 |- 2 < 9
127	29, 126	ltneii 10150	. . . . . . . . . . . . 13 |- 2 =/= 9
128	127, 45	neeqtrri 2867	. . . . . . . . . . . 12 |- 2 =/= (TopSet ` ndx )
129	28, 128	eqnetri 2864	. . . . . . . . . . 11 |- ( +g ` ndx ) =/= (TopSet ` ndx )
130		3lt9 11227	. . . . . . . . . . . . . 14 |- 3 < 9
131	35, 130	ltneii 10150	. . . . . . . . . . . . 13 |- 3 =/= 9
132	131, 45	neeqtrri 2867	. . . . . . . . . . . 12 |- 3 =/= (TopSet ` ndx )
133	34, 132	eqnetri 2864	. . . . . . . . . . 11 |- ( .r ` ndx ) =/= (TopSet ` ndx )
134	125, 129, 133	3pm3.2i 1239	. . . . . . . . . 10 |- ( ( Base ` ndx ) =/= (TopSet ` ndx ) /\ ( +g ` ndx ) =/= (TopSet ` ndx ) /\ ( .r ` ndx ) =/= (TopSet ` ndx ) )
135		1lt10 11681	. . . . . . . . . . . . . 14 |- 1 < ; 1 0
136	22, 135	ltneii 10150	. . . . . . . . . . . . 13 |- 1 =/= ; 1 0
137	136, 49	neeqtrri 2867	. . . . . . . . . . . 12 |- 1 =/= ( le ` ndx )
138	21, 137	eqnetri 2864	. . . . . . . . . . 11 |- ( Base ` ndx ) =/= ( le ` ndx )
139		2lt10 11680	. . . . . . . . . . . . . 14 |- 2 < ; 1 0
140	29, 139	ltneii 10150	. . . . . . . . . . . . 13 |- 2 =/= ; 1 0
141	140, 49	neeqtrri 2867	. . . . . . . . . . . 12 |- 2 =/= ( le ` ndx )
142	28, 141	eqnetri 2864	. . . . . . . . . . 11 |- ( +g ` ndx ) =/= ( le ` ndx )
143		3lt10 11679	. . . . . . . . . . . . . 14 |- 3 < ; 1 0
144	35, 143	ltneii 10150	. . . . . . . . . . . . 13 |- 3 =/= ; 1 0
145	144, 49	neeqtrri 2867	. . . . . . . . . . . 12 |- 3 =/= ( le ` ndx )
146	34, 145	eqnetri 2864	. . . . . . . . . . 11 |- ( .r ` ndx ) =/= ( le ` ndx )
147	138, 142, 146	3pm3.2i 1239	. . . . . . . . . 10 |- ( ( Base ` ndx ) =/= ( le ` ndx ) /\ ( +g ` ndx ) =/= ( le ` ndx ) /\ ( .r ` ndx ) =/= ( le ` ndx ) )
148	22, 46, 122	ltleii 10160	. . . . . . . . . . . . . . 15 |- 1 <_ 9
149	52, 53, 62, 148	decltdi 11547	. . . . . . . . . . . . . 14 |- 1 < ; 1 2
150	22, 149	ltneii 10150	. . . . . . . . . . . . 13 |- 1 =/= ; 1 2
151	150, 58	neeqtrri 2867	. . . . . . . . . . . 12 |- 1 =/= ( dist ` ndx )
152	21, 151	eqnetri 2864	. . . . . . . . . . 11 |- ( Base ` ndx ) =/= ( dist ` ndx )
153	29, 46, 126	ltleii 10160	. . . . . . . . . . . . . . 15 |- 2 <_ 9
154	52, 53, 53, 153	decltdi 11547	. . . . . . . . . . . . . 14 |- 2 < ; 1 2
155	29, 154	ltneii 10150	. . . . . . . . . . . . 13 |- 2 =/= ; 1 2
156	155, 58	neeqtrri 2867	. . . . . . . . . . . 12 |- 2 =/= ( dist ` ndx )
157	28, 156	eqnetri 2864	. . . . . . . . . . 11 |- ( +g ` ndx ) =/= ( dist ` ndx )
158	35, 46, 130	ltleii 10160	. . . . . . . . . . . . . . 15 |- 3 <_ 9
159	52, 53, 93, 158	decltdi 11547	. . . . . . . . . . . . . 14 |- 3 < ; 1 2
160	35, 159	ltneii 10150	. . . . . . . . . . . . 13 |- 3 =/= ; 1 2
161	160, 58	neeqtrri 2867	. . . . . . . . . . . 12 |- 3 =/= ( dist ` ndx )
162	34, 161	eqnetri 2864	. . . . . . . . . . 11 |- ( .r ` ndx ) =/= ( dist ` ndx )
163	152, 157, 162	3pm3.2i 1239	. . . . . . . . . 10 |- ( ( Base ` ndx ) =/= ( dist ` ndx ) /\ ( +g ` ndx ) =/= ( dist ` ndx ) /\ ( .r ` ndx ) =/= ( dist ` ndx ) )
164		disjtp2 4252	. . . . . . . . . 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 ) } i^i { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } ) = (/) )
165	134, 147, 163, 164	mp3an 1424	. . . . . . . . 9 |- ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } ) = (/)
166	121, 165	eqtri 2644	. . . . . . . 8 |- ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = (/)
167	18, 91	ineq12i 3812	. . . . . . . . 9 |- ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { ( UnifSet ` ndx ) } )
168	52, 93, 62, 148	decltdi 11547	. . . . . . . . . . . . 13 |- 1 < ; 1 3
169	22, 168	ltneii 10150	. . . . . . . . . . . 12 |- 1 =/= ; 1 3
170	169, 96	neeqtrri 2867	. . . . . . . . . . 11 |- 1 =/= ( UnifSet ` ndx )
171	21, 170	eqnetri 2864	. . . . . . . . . 10 |- ( Base ` ndx ) =/= ( UnifSet ` ndx )
172	52, 93, 53, 153	decltdi 11547	. . . . . . . . . . . . 13 |- 2 < ; 1 3
173	29, 172	ltneii 10150	. . . . . . . . . . . 12 |- 2 =/= ; 1 3
174	173, 96	neeqtrri 2867	. . . . . . . . . . 11 |- 2 =/= ( UnifSet ` ndx )
175	28, 174	eqnetri 2864	. . . . . . . . . 10 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
176	52, 93, 93, 158	decltdi 11547	. . . . . . . . . . . . 13 |- 3 < ; 1 3
177	35, 176	ltneii 10150	. . . . . . . . . . . 12 |- 3 =/= ; 1 3
178	177, 96	neeqtrri 2867	. . . . . . . . . . 11 |- 3 =/= ( UnifSet ` ndx )
179	34, 178	eqnetri 2864	. . . . . . . . . 10 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
180		disjtpsn 4251	. . . . . . . . . 10 |- ( ( ( Base ` ndx ) =/= ( UnifSet ` ndx ) /\ ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) ) -> ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { ( UnifSet ` ndx ) } ) = (/) )
181	171, 175, 179, 180	mp3an 1424	. . . . . . . . 9 |- ( { ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) } i^i { ( UnifSet ` ndx ) } ) = (/)
182	167, 181	eqtri 2644	. . . . . . . 8 |- ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/)
183	166, 182	pm3.2i 471	. . . . . . 7 |- ( ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = (/) /\ ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/) )
184		undisj2 4030	. . . . . . 7 |- ( ( ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = (/) /\ ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/) ) <-> ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) )
185	183, 184	mpbi 220	. . . . . 6 |- ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/)
186	19, 90	ineq12i 3812	. . . . . . . . 9 |- ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = ( { ( *r ` ndx ) } i^i { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } )
187		incom 3805	. . . . . . . . . 10 |- ( { ( *r ` ndx ) } i^i { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } ) = ( { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } i^i { ( *r ` ndx ) } )
188		4re 11097	. . . . . . . . . . . . . 14 |- 4 e. RR
189		4lt9 11226	. . . . . . . . . . . . . 14 |- 4 < 9
190	188, 189	gtneii 10149	. . . . . . . . . . . . 13 |- 9 =/= 4
191	190, 25	neeqtrri 2867	. . . . . . . . . . . 12 |- 9 =/= ( *r ` ndx )
192	45, 191	eqnetri 2864	. . . . . . . . . . 11 |- (TopSet ` ndx ) =/= ( *r ` ndx )
193		4lt10 11678	. . . . . . . . . . . . . 14 |- 4 < ; 1 0
194	188, 193	gtneii 10149	. . . . . . . . . . . . 13 |- ; 1 0 =/= 4
195	194, 25	neeqtrri 2867	. . . . . . . . . . . 12 |- ; 1 0 =/= ( *r ` ndx )
196	49, 195	eqnetri 2864	. . . . . . . . . . 11 |- ( le ` ndx ) =/= ( *r ` ndx )
197		4nn0 11311	. . . . . . . . . . . . . . 15 |- 4 e. NN0
198	188, 46, 189	ltleii 10160	. . . . . . . . . . . . . . 15 |- 4 <_ 9
199	52, 53, 197, 198	decltdi 11547	. . . . . . . . . . . . . 14 |- 4 < ; 1 2
200	188, 199	gtneii 10149	. . . . . . . . . . . . 13 |- ; 1 2 =/= 4
201	200, 25	neeqtrri 2867	. . . . . . . . . . . 12 |- ; 1 2 =/= ( *r ` ndx )
202	58, 201	eqnetri 2864	. . . . . . . . . . 11 |- ( dist ` ndx ) =/= ( *r ` ndx )
203		disjtpsn 4251	. . . . . . . . . . 11 |- ( ( (TopSet ` ndx ) =/= ( *r ` ndx ) /\ ( le ` ndx ) =/= ( *r ` ndx ) /\ ( dist ` ndx ) =/= ( *r ` ndx ) ) -> ( { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } i^i { ( *r ` ndx ) } ) = (/) )
204	192, 196, 202, 203	mp3an 1424	. . . . . . . . . 10 |- ( { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } i^i { ( *r ` ndx ) } ) = (/)
205	187, 204	eqtri 2644	. . . . . . . . 9 |- ( { ( *r ` ndx ) } i^i { (TopSet ` ndx ) , ( le ` ndx ) , ( dist ` ndx ) } ) = (/)
206	186, 205	eqtri 2644	. . . . . . . 8 |- ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = (/)
207	19, 91	ineq12i 3812	. . . . . . . . 9 |- ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = ( { ( *r ` ndx ) } i^i { ( UnifSet ` ndx ) } )
208	52, 93, 197, 198	decltdi 11547	. . . . . . . . . . . . 13 |- 4 < ; 1 3
209	188, 208	ltneii 10150	. . . . . . . . . . . 12 |- 4 =/= ; 1 3
210	209, 96	neeqtrri 2867	. . . . . . . . . . 11 |- 4 =/= ( UnifSet ` ndx )
211	25, 210	eqnetri 2864	. . . . . . . . . 10 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
212		disjsn2 4247	. . . . . . . . . 10 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) -> ( { ( *r ` ndx ) } i^i { ( UnifSet ` ndx ) } ) = (/) )
213	211, 212	ax-mp 5	. . . . . . . . 9 |- ( { ( *r ` ndx ) } i^i { ( UnifSet ` ndx ) } ) = (/)
214	207, 213	eqtri 2644	. . . . . . . 8 |- ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/)
215	206, 214	pm3.2i 471	. . . . . . 7 |- ( ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = (/) /\ ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/) )
216		undisj2 4030	. . . . . . 7 |- ( ( ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } ) = (/) /\ ( dom { <. ( *r ` ndx ) , * >. } i^i dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) = (/) ) <-> ( dom { <. ( *r ` ndx ) , * >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) )
217	215, 216	mpbi 220	. . . . . 6 |- ( dom { <. ( *r ` ndx ) , * >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/)
218	185, 217	pm3.2i 471	. . . . 5 |- ( ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) /\ ( dom { <. ( *r ` ndx ) , * >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) )
219		undisj1 4029	. . . . 5 |- ( ( ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) /\ ( dom { <. ( *r ` ndx ) , * >. } i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) ) <-> ( ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. dom { <. ( *r ` ndx ) , * >. } ) i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) )
220	218, 219	mpbi 220	. . . 4 |- ( ( dom { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. dom { <. ( *r ` ndx ) , * >. } ) i^i ( dom { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. dom { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/)
221	120, 220	eqtri 2644	. . 3 |- ( dom ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) i^i dom ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/)
222		funun 5932	. . 3 |- ( ( ( Fun ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) /\ Fun ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) /\ ( dom ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) i^i dom ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) = (/) ) -> Fun ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
223	117, 221, 222	mp2an 708	. 2 |- Fun ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
224		df-cnfld 19747	. . 3 |- ℂfld = ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
225	224	funeqi 5909	. 2 |- ( Fun ℂfld <-> Fun ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
226	223, 225	mpbir 221	1 |- Fun ℂfld

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {ctp 4181   <.cop 4183   X. cxp 5112   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   2c2 11070   3c3 11071   4c4 11072   9c9 11077  ;cdc 11493   *ccj 13836   abscabs 13974   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   *rcstv 15943  TopSetcts 15947   lecple 15948   distcds 15950   UnifSetcunif 15951   TosetRel ctsr 17199   MetOpencmopn 19736  metUnifcmetu 19737  ℂ_fldccnfld 19746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-ps 17200  df-tsr 17201  df-cnfld 19747

This theorem is referenced by: (None)