Theorem cnfldfunALT 19759

Description: Alternate proof of cnfldfun 19758 (much shorter proof, using cnfldstr 19748 and structn0fun 15869: in addition, it must be shown that (/) e/ ℂ_fld). (Contributed by AV, 18-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cnfldfunALT	|- Fun ℂfld

Proof of Theorem cnfldfunALT

Step	Hyp	Ref	Expression
1		cnfldstr 19748	. 2 |- ℂfld Struct <. 1 , ; 1 3 >.
2		structn0fun 15869	. . 3 |- (ℂfld Struct <. 1 , ; 1 3 >. -> Fun (ℂfld \ { (/) } ) )
3		fvex 6201	. . . . . . . . . . . . 13 |- ( Base ` ndx ) e. _V
4		cnex 10017	. . . . . . . . . . . . 13 |- CC e. _V
5	3, 4	opnzi 4943	. . . . . . . . . . . 12 |- <. ( Base ` ndx ) , CC >. =/= (/)
6	5	nesymi 2851	. . . . . . . . . . 11 |- -. (/) = <. ( Base ` ndx ) , CC >.
7		fvex 6201	. . . . . . . . . . . . 13 |- ( +g ` ndx ) e. _V
8		addex 11830	. . . . . . . . . . . . 13 |- + e. _V
9	7, 8	opnzi 4943	. . . . . . . . . . . 12 |- <. ( +g ` ndx ) , + >. =/= (/)
10	9	nesymi 2851	. . . . . . . . . . 11 |- -. (/) = <. ( +g ` ndx ) , + >.
11		fvex 6201	. . . . . . . . . . . . 13 |- ( .r ` ndx ) e. _V
12		mulex 11831	. . . . . . . . . . . . 13 |- x. e. _V
13	11, 12	opnzi 4943	. . . . . . . . . . . 12 |- <. ( .r ` ndx ) , x. >. =/= (/)
14	13	nesymi 2851	. . . . . . . . . . 11 |- -. (/) = <. ( .r ` ndx ) , x. >.
15		3ioran 1056	. . . . . . . . . . . 12 |- ( -. ( (/) = <. ( Base ` ndx ) , CC >. \/ (/) = <. ( +g ` ndx ) , + >. \/ (/) = <. ( .r ` ndx ) , x. >. ) <-> ( -. (/) = <. ( Base ` ndx ) , CC >. /\ -. (/) = <. ( +g ` ndx ) , + >. /\ -. (/) = <. ( .r ` ndx ) , x. >. ) )
16		0ex 4790	. . . . . . . . . . . . 13 |- (/) e. _V
17	16	eltp 4230	. . . . . . . . . . . 12 |- ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } <-> ( (/) = <. ( Base ` ndx ) , CC >. \/ (/) = <. ( +g ` ndx ) , + >. \/ (/) = <. ( .r ` ndx ) , x. >. ) )
18	15, 17	xchnxbir 323	. . . . . . . . . . 11 |- ( -. (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } <-> ( -. (/) = <. ( Base ` ndx ) , CC >. /\ -. (/) = <. ( +g ` ndx ) , + >. /\ -. (/) = <. ( .r ` ndx ) , x. >. ) )
19	6, 10, 14, 18	mpbir3an 1244	. . . . . . . . . 10 |- -. (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. }
20		fvex 6201	. . . . . . . . . . . . 13 |- ( *r ` ndx ) e. _V
21		cjf 13844	. . . . . . . . . . . . . 14 |- * : CC --> CC
22		fex 6490	. . . . . . . . . . . . . 14 |- ( ( * : CC --> CC /\ CC e. _V ) -> * e. _V )
23	21, 4, 22	mp2an 708	. . . . . . . . . . . . 13 |- * e. _V
24	20, 23	opnzi 4943	. . . . . . . . . . . 12 |- <. ( *r ` ndx ) , * >. =/= (/)
25	24	necomi 2848	. . . . . . . . . . 11 |- (/) =/= <. ( *r ` ndx ) , * >.
26		nelsn 4212	. . . . . . . . . . 11 |- ( (/) =/= <. ( *r ` ndx ) , * >. -> -. (/) e. { <. ( *r ` ndx ) , * >. } )
27	25, 26	ax-mp 5	. . . . . . . . . 10 |- -. (/) e. { <. ( *r ` ndx ) , * >. }
28	19, 27	pm3.2i 471	. . . . . . . . 9 |- ( -. (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } /\ -. (/) e. { <. ( *r ` ndx ) , * >. } )
29		fvex 6201	. . . . . . . . . . . . . 14 |- (TopSet ` ndx ) e. _V
30		fvex 6201	. . . . . . . . . . . . . 14 |- ( MetOpen ` ( abs o. - ) ) e. _V
31	29, 30	opnzi 4943	. . . . . . . . . . . . 13 |- <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. =/= (/)
32	31	nesymi 2851	. . . . . . . . . . . 12 |- -. (/) = <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >.
33		fvex 6201	. . . . . . . . . . . . . 14 |- ( le ` ndx ) e. _V
34		letsr 17227	. . . . . . . . . . . . . . 15 |- <_ e. TosetRel
35	34	elexi 3213	. . . . . . . . . . . . . 14 |- <_ e. _V
36	33, 35	opnzi 4943	. . . . . . . . . . . . 13 |- <. ( le ` ndx ) , <_ >. =/= (/)
37	36	nesymi 2851	. . . . . . . . . . . 12 |- -. (/) = <. ( le ` ndx ) , <_ >.
38		fvex 6201	. . . . . . . . . . . . . 14 |- ( dist ` ndx ) e. _V
39		absf 14077	. . . . . . . . . . . . . . . 16 |- abs : CC --> RR
40		fex 6490	. . . . . . . . . . . . . . . 16 |- ( ( abs : CC --> RR /\ CC e. _V ) -> abs e. _V )
41	39, 4, 40	mp2an 708	. . . . . . . . . . . . . . 15 |- abs e. _V
42		subf 10283	. . . . . . . . . . . . . . . 16 |- - : ( CC X. CC ) --> CC
43	4, 4	xpex 6962	. . . . . . . . . . . . . . . 16 |- ( CC X. CC ) e. _V
44		fex 6490	. . . . . . . . . . . . . . . 16 |- ( ( - : ( CC X. CC ) --> CC /\ ( CC X. CC ) e. _V ) -> - e. _V )
45	42, 43, 44	mp2an 708	. . . . . . . . . . . . . . 15 |- - e. _V
46	41, 45	coex 7118	. . . . . . . . . . . . . 14 |- ( abs o. - ) e. _V
47	38, 46	opnzi 4943	. . . . . . . . . . . . 13 |- <. ( dist ` ndx ) , ( abs o. - ) >. =/= (/)
48	47	nesymi 2851	. . . . . . . . . . . 12 |- -. (/) = <. ( dist ` ndx ) , ( abs o. - ) >.
49		3ioran 1056	. . . . . . . . . . . 12 |- ( -. ( (/) = <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. \/ (/) = <. ( le ` ndx ) , <_ >. \/ (/) = <. ( dist ` ndx ) , ( abs o. - ) >. ) <-> ( -. (/) = <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. /\ -. (/) = <. ( le ` ndx ) , <_ >. /\ -. (/) = <. ( dist ` ndx ) , ( abs o. - ) >. ) )
50	32, 37, 48, 49	mpbir3an 1244	. . . . . . . . . . 11 |- -. ( (/) = <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. \/ (/) = <. ( le ` ndx ) , <_ >. \/ (/) = <. ( dist ` ndx ) , ( abs o. - ) >. )
51	16	eltp 4230	. . . . . . . . . . 11 |- ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } <-> ( (/) = <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. \/ (/) = <. ( le ` ndx ) , <_ >. \/ (/) = <. ( dist ` ndx ) , ( abs o. - ) >. ) )
52	50, 51	mtbir 313	. . . . . . . . . 10 |- -. (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. }
53		fvex 6201	. . . . . . . . . . . . 13 |- ( UnifSet ` ndx ) e. _V
54		fvex 6201	. . . . . . . . . . . . 13 |- (metUnif ` ( abs o. - ) ) e. _V
55	53, 54	opnzi 4943	. . . . . . . . . . . 12 |- <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. =/= (/)
56	55	necomi 2848	. . . . . . . . . . 11 |- (/) =/= <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >.
57		nelsn 4212	. . . . . . . . . . 11 |- ( (/) =/= <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. -> -. (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } )
58	56, 57	ax-mp 5	. . . . . . . . . 10 |- -. (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. }
59	52, 58	pm3.2i 471	. . . . . . . . 9 |- ( -. (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } /\ -. (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } )
60	28, 59	pm3.2i 471	. . . . . . . 8 |- ( ( -. (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } /\ -. (/) e. { <. ( *r ` ndx ) , * >. } ) /\ ( -. (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } /\ -. (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
61		ioran 511	. . . . . . . . 9 |- ( -. ( ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) \/ ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) <-> ( -. ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) /\ -. ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
62		ioran 511	. . . . . . . . . 10 |- ( -. ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) <-> ( -. (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } /\ -. (/) e. { <. ( *r ` ndx ) , * >. } ) )
63		ioran 511	. . . . . . . . . 10 |- ( -. ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) <-> ( -. (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } /\ -. (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
64	62, 63	anbi12i 733	. . . . . . . . 9 |- ( ( -. ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) /\ -. ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) <-> ( ( -. (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } /\ -. (/) e. { <. ( *r ` ndx ) , * >. } ) /\ ( -. (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } /\ -. (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
65	61, 64	bitri 264	. . . . . . . 8 |- ( -. ( ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) \/ ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) <-> ( ( -. (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } /\ -. (/) e. { <. ( *r ` ndx ) , * >. } ) /\ ( -. (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } /\ -. (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
66	60, 65	mpbir 221	. . . . . . 7 |- -. ( ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) \/ ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
67		df-cnfld 19747	. . . . . . . . 9 |- 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. - ) ) >. } ) )
68	67	eleq2i 2693	. . . . . . . 8 |- ( (/) e. ℂfld <-> (/) e. ( ( { <. ( 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. - ) ) >. } ) ) )
69		elun 3753	. . . . . . . 8 |- ( (/) e. ( ( { <. ( 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. - ) ) >. } ) ) <-> ( (/) e. ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) \/ (/) e. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
70		elun 3753	. . . . . . . . 9 |- ( (/) e. ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) <-> ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) )
71		elun 3753	. . . . . . . . 9 |- ( (/) e. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) <-> ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
72	70, 71	orbi12i 543	. . . . . . . 8 |- ( ( (/) e. ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) \/ (/) e. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) <-> ( ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) \/ ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
73	68, 69, 72	3bitri 286	. . . . . . 7 |- ( (/) e. ℂfld <-> ( ( (/) e. { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } \/ (/) e. { <. ( *r ` ndx ) , * >. } ) \/ ( (/) e. { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } \/ (/) e. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) ) )
74	66, 73	mtbir 313	. . . . . 6 |- -. (/) e. ℂfld
75		disjsn 4246	. . . . . 6 |- ( (ℂfld i^i { (/) } ) = (/) <-> -. (/) e. ℂfld )
76	74, 75	mpbir 221	. . . . 5 |- (ℂfld i^i { (/) } ) = (/)
77		disjdif2 4047	. . . . 5 |- ( (ℂfld i^i { (/) } ) = (/) -> (ℂfld \ { (/) } ) = ℂfld )
78	76, 77	ax-mp 5	. . . 4 |- (ℂfld \ { (/) } ) = ℂfld
79	78	funeqi 5909	. . 3 |- ( Fun (ℂfld \ { (/) } ) <-> Fun ℂfld )
80	2, 79	sylib 208	. 2 |- (ℂfld Struct <. 1 , ; 1 3 >. -> Fun ℂfld )
81	1, 80	ax-mp 5	1 |- Fun ℂfld

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {ctp 4181   <.cop 4183   class class class wbr 4653   X. cxp 5112   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   3c3 11071  ;cdc 11493   *ccj 13836   abscabs 13974   Struct cstr 15853   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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  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)