Theorem qqhcn 24328

Description: The QQHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)

Hypotheses

Ref	Expression
qqhcn.q	|- Q = (ℂfld ↾s QQ )
qqhcn.j	|- J = ( TopOpen ` Q )
qqhcn.z	|- Z = ( ZMod ` R )
qqhcn.k	|- K = ( TopOpen ` R )

Assertion

Ref	Expression
qqhcn	|- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( J Cn K ) )

Proof of Theorem qqhcn

Dummy variables e d q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3522	. . . . . . . 8 |- (NrmRing i^i DivRing ) C_ DivRing
2	1	sseli 3304	. . . . . . 7 |- ( R e. (NrmRing i^i DivRing ) -> R e. DivRing )
3	2	3ad2ant1 978	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. DivRing )
4		simp3 959	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (chr ` R ) = 0 )
5		eqid 2404	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
6		eqid 2404	. . . . . . 7 |- (/r ` R ) = (/r ` R )
7		eqid 2404	. . . . . . 7 |- ( ZRHom ` R ) = ( ZRHom ` R )
8	5, 6, 7	qqhf 24323	. . . . . 6 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
9	3, 4, 8	syl2anc 643	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
10		simpr 448	. . . . . . 7 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) -> e e. RR+ )
11		qsscn 10541	. . . . . . . . . . . . . 14 |- QQ C_ CC
12		simpr 448	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> q e. QQ )
13	11, 12	sseldi 3306	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> q e. CC )
14		0cn 9040	. . . . . . . . . . . . . . 15 |- 0 e. CC
15		eqid 2404	. . . . . . . . . . . . . . . 16 |- ( abs o. - ) = ( abs o. - )
16	15	cnmetdval 18758	. . . . . . . . . . . . . . 15 |- ( ( 0 e. CC /\ q e. CC ) -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
17	14, 16	mpan 652	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
18		df-neg 9250	. . . . . . . . . . . . . . . 16 |- -u q = ( 0 - q )
19	18	fveq2i 5690	. . . . . . . . . . . . . . 15 |- ( abs ` -u q ) = ( abs ` ( 0 - q ) )
20	19	a1i 11	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` ( 0 - q ) ) )
21		absneg 12037	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` q ) )
22	17, 20, 21	3eqtr2d 2442	. . . . . . . . . . . . 13 |- ( q e. CC -> ( 0 ( abs o. - ) q ) = ( abs ` q ) )
23	13, 22	syl 16	. . . . . . . . . . . 12 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( 0 ( abs o. - ) q ) = ( abs ` q ) )
24		zssq 10537	. . . . . . . . . . . . . . 15 |- ZZ C_ QQ
25		0z 10249	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
26	24, 25	sselii 3305	. . . . . . . . . . . . . 14 |- 0 e. QQ
27	26	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> 0 e. QQ )
28	27, 12	ovresd 6173	. . . . . . . . . . . 12 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) = ( 0 ( abs o. - ) q ) )
29		eqid 2404	. . . . . . . . . . . . . 14 |- ( norm ` R ) = ( norm ` R )
30		qqhcn.z	. . . . . . . . . . . . . 14 |- Z = ( ZMod ` R )
31	29, 30	qqhnm 24327	. . . . . . . . . . . . 13 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ q e. QQ ) -> ( ( norm ` R ) ` ( (QQHom ` R ) ` q ) ) = ( abs ` q ) )
32	31	adantlr 696	. . . . . . . . . . . 12 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( norm ` R ) ` ( (QQHom ` R ) ` q ) ) = ( abs ` q ) )
33	23, 28, 32	3eqtr4d 2446	. . . . . . . . . . 11 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) = ( ( norm ` R ) ` ( (QQHom ` R ) ` q ) ) )
34	9	ad2antrr 707	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
35	34, 27	ffvelrnd 5830	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( (QQHom ` R ) ` 0 ) e. ( Base ` R ) )
36	34, 12	ffvelrnd 5830	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( (QQHom ` R ) ` q ) e. ( Base ` R ) )
37	35, 36	ovresd 6173	. . . . . . . . . . . 12 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) = ( ( (QQHom ` R ) ` 0 ) ( dist ` R ) ( (QQHom ` R ) ` q ) ) )
38		inss1 3521	. . . . . . . . . . . . . . . . 17 |- (NrmRing i^i DivRing ) C_ NrmRing
39	38	sseli 3304	. . . . . . . . . . . . . . . 16 |- ( R e. (NrmRing i^i DivRing ) -> R e. NrmRing )
40	39	3ad2ant1 978	. . . . . . . . . . . . . . 15 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. NrmRing )
41	40	ad2antrr 707	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. NrmRing )
42		nrgngp 18651	. . . . . . . . . . . . . 14 |- ( R e. NrmRing -> R e. NrmGrp )
43	41, 42	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. NrmGrp )
44		eqid 2404	. . . . . . . . . . . . . 14 |- ( -g ` R ) = ( -g ` R )
45		eqid 2404	. . . . . . . . . . . . . 14 |- ( dist ` R ) = ( dist ` R )
46	29, 5, 44, 45	ngpdsr 18604	. . . . . . . . . . . . 13 |- ( ( R e. NrmGrp /\ ( (QQHom ` R ) ` 0 ) e. ( Base ` R ) /\ ( (QQHom ` R ) ` q ) e. ( Base ` R ) ) -> ( ( (QQHom ` R ) ` 0 ) ( dist ` R ) ( (QQHom ` R ) ` q ) ) = ( ( norm ` R ) ` ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` 0 ) ) ) )
47	43, 35, 36, 46	syl3anc 1184	. . . . . . . . . . . 12 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` 0 ) ( dist ` R ) ( (QQHom ` R ) ` q ) ) = ( ( norm ` R ) ` ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` 0 ) ) ) )
48	3	ad2antrr 707	. . . . . . . . . . . . . . . 16 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. DivRing )
49	4	ad2antrr 707	. . . . . . . . . . . . . . . 16 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> (chr ` R ) = 0 )
50	5, 6, 7	qqh0 24321	. . . . . . . . . . . . . . . 16 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 0 ) = ( 0g ` R ) )
51	48, 49, 50	syl2anc 643	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( (QQHom ` R ) ` 0 ) = ( 0g ` R ) )
52	51	oveq2d 6056	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` 0 ) ) = ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( 0g ` R ) ) )
53		ngpgrp 18599	. . . . . . . . . . . . . . . 16 |- ( R e. NrmGrp -> R e. Grp )
54	43, 53	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. Grp )
55		eqid 2404	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) = ( 0g ` R )
56	5, 55, 44	grpsubid1 14829	. . . . . . . . . . . . . . 15 |- ( ( R e. Grp /\ ( (QQHom ` R ) ` q ) e. ( Base ` R ) ) -> ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( 0g ` R ) ) = ( (QQHom ` R ) ` q ) )
57	54, 36, 56	syl2anc 643	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( 0g ` R ) ) = ( (QQHom ` R ) ` q ) )
58	52, 57	eqtrd 2436	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` 0 ) ) = ( (QQHom ` R ) ` q ) )
59	58	fveq2d 5691	. . . . . . . . . . . 12 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( norm ` R ) ` ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` 0 ) ) ) = ( ( norm ` R ) ` ( (QQHom ` R ) ` q ) ) )
60	37, 47, 59	3eqtrd 2440	. . . . . . . . . . 11 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) = ( ( norm ` R ) ` ( (QQHom ` R ) ` q ) ) )
61	33, 60	eqtr4d 2439	. . . . . . . . . 10 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) = ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) )
62	61	breq1d 4182	. . . . . . . . 9 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e <-> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) )
63	62	biimpd 199	. . . . . . . 8 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) )
64	63	ralrimiva 2749	. . . . . . 7 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) -> A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) )
65		breq2 4176	. . . . . . . . . 10 |- ( d = e -> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d <-> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e ) )
66	65	imbi1d 309	. . . . . . . . 9 |- ( d = e -> ( ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) <-> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) ) )
67	66	ralbidv 2686	. . . . . . . 8 |- ( d = e -> ( A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) <-> A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) ) )
68	67	rspcev 3012	. . . . . . 7 |- ( ( e e. RR+ /\ A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) ) -> E. d e. RR+ A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) )
69	10, 64, 68	syl2anc 643	. . . . . 6 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) -> E. d e. RR+ A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) )
70	69	ralrimiva 2749	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> A. e e. RR+ E. d e. RR+ A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) )
71		qqhcn.q	. . . . . . . 8 |- Q = (ℂfld ↾s QQ )
72		cnfldxms 18764	. . . . . . . . 9 |- ℂfld e. * MetSp
73		qex 10542	. . . . . . . . 9 |- QQ e. _V
74		ressxms 18508	. . . . . . . . 9 |- ( (ℂfld e. * MetSp /\ QQ e. _V ) -> (ℂfld ↾s QQ ) e. * MetSp )
75	72, 73, 74	mp2an 654	. . . . . . . 8 |- (ℂfld ↾s QQ ) e. * MetSp
76	71, 75	eqeltri 2474	. . . . . . 7 |- Q e. * MetSp
77	71	qrngbas 21266	. . . . . . . 8 |- QQ = ( Base ` Q )
78		cnfldds 16668	. . . . . . . . . 10 |- ( abs o. - ) = ( dist ` ℂfld )
79	71, 78	ressds 13596	. . . . . . . . 9 |- ( QQ e. _V -> ( abs o. - ) = ( dist ` Q ) )
80	73, 79	ax-mp 8	. . . . . . . 8 |- ( abs o. - ) = ( dist ` Q )
81	77, 80	xmsxmet2 18442	. . . . . . 7 |- ( Q e. * MetSp -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( * Met ` QQ ) )
82	76, 81	mp1i 12	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( * Met ` QQ ) )
83		ngpxms 18601	. . . . . . . . 9 |- ( R e. NrmGrp -> R e. * MetSp )
84	39, 42, 83	3syl 19	. . . . . . . 8 |- ( R e. (NrmRing i^i DivRing ) -> R e. * MetSp )
85	84	3ad2ant1 978	. . . . . . 7 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. * MetSp )
86	5, 45	xmsxmet2 18442	. . . . . . 7 |- ( R e. * MetSp -> ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( * Met ` ( Base ` R ) ) )
87	85, 86	syl 16	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( * Met ` ( Base ` R ) ) )
88	26	a1i 11	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> 0 e. QQ )
89		qqhcn.j	. . . . . . . . 9 |- J = ( TopOpen ` Q )
90	80	reseq1i 5101	. . . . . . . . 9 |- ( ( abs o. - ) |` ( QQ X. QQ ) ) = ( ( dist ` Q ) |` ( QQ X. QQ ) )
91	89, 77, 90	xmstopn 18434	. . . . . . . 8 |- ( Q e. * MetSp -> J = ( MetOpen ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) )
92	76, 91	ax-mp 8	. . . . . . 7 |- J = ( MetOpen ` ( ( abs o. - ) |` ( QQ X. QQ ) ) )
93		eqid 2404	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
94	92, 93	metcnp 18524	. . . . . 6 |- ( ( ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( * Met ` QQ ) /\ ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( * Met ` ( Base ` R ) ) /\ 0 e. QQ ) -> ( (QQHom ` R ) e. ( ( J CnP ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) ) ` 0 ) <-> ( (QQHom ` R ) : QQ --> ( Base ` R ) /\ A. e e. RR+ E. d e. RR+ A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) ) ) )
95	82, 87, 88, 94	syl3anc 1184	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) e. ( ( J CnP ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) ) ` 0 ) <-> ( (QQHom ` R ) : QQ --> ( Base ` R ) /\ A. e e. RR+ E. d e. RR+ A. q e. QQ ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` 0 ) ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ( (QQHom ` R ) ` q ) ) < e ) ) ) )
96	9, 70, 95	mpbir2and 889	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( ( J CnP ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) ) ` 0 ) )
97		qqhcn.k	. . . . . . . 8 |- K = ( TopOpen ` R )
98		eqid 2404	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
99	97, 5, 98	xmstopn 18434	. . . . . . 7 |- ( R e. * MetSp -> K = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
100	85, 99	syl 16	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> K = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
101	100	oveq2d 6056	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( J CnP K ) = ( J CnP ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) ) )
102	101	fveq1d 5689	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( ( J CnP K ) ` 0 ) = ( ( J CnP ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) ) ` 0 ) )
103	96, 102	eleqtrrd 2481	. . 3 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( ( J CnP K ) ` 0 ) )
104		cnfldtgp 18852	. . . . . 6 |- ℂfld e. TopGrp
105		qsubdrg 16706	. . . . . . . 8 |- ( QQ e. (SubRing ` ℂfld ) /\ (ℂfld ↾s QQ ) e. DivRing )
106	105	simpli 445	. . . . . . 7 |- QQ e. (SubRing ` ℂfld )
107		subrgsubg 15829	. . . . . . 7 |- ( QQ e. (SubRing ` ℂfld ) -> QQ e. (SubGrp ` ℂfld ) )
108	106, 107	ax-mp 8	. . . . . 6 |- QQ e. (SubGrp ` ℂfld )
109	71	subgtgp 18088	. . . . . 6 |- ( (ℂfld e. TopGrp /\ QQ e. (SubGrp ` ℂfld ) ) -> Q e. TopGrp )
110	104, 108, 109	mp2an 654	. . . . 5 |- Q e. TopGrp
111		tgptmd 18062	. . . . 5 |- ( Q e. TopGrp -> Q e. TopMnd )
112	110, 111	mp1i 12	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> Q e. TopMnd )
113		nrgtrg 18678	. . . . 5 |- ( R e. NrmRing -> R e. TopRing )
114		trgtmd2 18151	. . . . 5 |- ( R e. TopRing -> R e. TopMnd )
115	40, 113, 114	3syl 19	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. TopMnd )
116	5, 6, 7, 71	qqhghm 24325	. . . . 5 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
117	3, 4, 116	syl2anc 643	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
118	77, 89, 97	ghmcnp 18097	. . . 4 |- ( ( Q e. TopMnd /\ R e. TopMnd /\ (QQHom ` R ) e. ( Q GrpHom R ) ) -> ( (QQHom ` R ) e. ( ( J CnP K ) ` 0 ) <-> ( 0 e. QQ /\ (QQHom ` R ) e. ( J Cn K ) ) ) )
119	112, 115, 117, 118	syl3anc 1184	. . 3 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) e. ( ( J CnP K ) ` 0 ) <-> ( 0 e. QQ /\ (QQHom ` R ) e. ( J Cn K ) ) ) )
120	103, 119	mpbid 202	. 2 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( 0 e. QQ /\ (QQHom ` R ) e. ( J Cn K ) ) )
121	120	simprd 450	1 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   i^i cin 3279   class class class wbr 4172   X. cxp 4835   |` cres 4839   o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   < clt 9076   - cmin 9247   -ucneg 9248   ZZcz 10238   QQcq 10530   RR+crp 10568   abscabs 11994   Basecbs 13424   ↾_s cress 13425   distcds 13493   TopOpenctopn 13604   0gc0g 13678   Grpcgrp 14640   -gcsg 14643  SubGrpcsubg 14893   GrpHom cghm 14958  /_rcdvr 15742   DivRingcdr 15790  SubRingcsubrg 15819   * Metcxmt 16641   MetOpencmopn 16646  ℂ_fldccnfld 16658   ZRHomczrh 16733   ZModczlm 16734  chrcchr 16735   Cn ccn 17242   CnP ccnp 17243  TopMndctmd 18053   TopGrpctgp 18054   TopRingctrg 18138   * MetSpcxme 18300   normcnm 18577  NrmGrpcngp 18578  NrmRingcnrg 18580  NrmModcnlm 18581  QQHomcqqh 24309

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-plusf 14646  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-od 15122  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  df-nzr 16284  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-zrh 16737  df-zlm 16738  df-chr 16739  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-tmd 18055  df-tgp 18056  df-trg 18142  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-qqh 24310