Theorem qqhcn 28844

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 3665	. . . . . . . 8 |- (NrmRing i^i DivRing ) C_ DivRing
2	1	sseli 3440	. . . . . . 7 |- ( R e. (NrmRing i^i DivRing ) -> R e. DivRing )
3	2	3ad2ant1 1035	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. DivRing )
4		simp3 1016	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (chr ` R ) = 0 )
5		eqid 2462	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
6		eqid 2462	. . . . . . 7 |- (/r ` R ) = (/r ` R )
7		eqid 2462	. . . . . . 7 |- ( ZRHom ` R ) = ( ZRHom ` R )
8	5, 6, 7	qqhf 28839	. . . . . 6 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
9	3, 4, 8	syl2anc 671	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
10		simpr 467	. . . . . . 7 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) -> e e. RR+ )
11		qsscn 11304	. . . . . . . . . . . . . 14 |- QQ C_ CC
12		simpr 467	. . . . . . . . . . . . . 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 3442	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> q e. CC )
14		0cn 9661	. . . . . . . . . . . . . . 15 |- 0 e. CC
15		eqid 2462	. . . . . . . . . . . . . . . 16 |- ( abs o. - ) = ( abs o. - )
16	15	cnmetdval 21840	. . . . . . . . . . . . . . 15 |- ( ( 0 e. CC /\ q e. CC ) -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
17	14, 16	mpan 681	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
18		df-neg 9889	. . . . . . . . . . . . . . . 16 |- -u q = ( 0 - q )
19	18	fveq2i 5891	. . . . . . . . . . . . . . 15 |- ( abs ` -u q ) = ( abs ` ( 0 - q ) )
20	19	a1i 11	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` ( 0 - q ) ) )
21		absneg 13389	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` q ) )
22	17, 20, 21	3eqtr2d 2502	. . . . . . . . . . . . 13 |- ( q e. CC -> ( 0 ( abs o. - ) q ) = ( abs ` q ) )
23	13, 22	syl 17	. . . . . . . . . . . 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 11300	. . . . . . . . . . . . . . 15 |- ZZ C_ QQ
25		0z 10977	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
26	24, 25	sselii 3441	. . . . . . . . . . . . . 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 6464	. . . . . . . . . . . 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 2462	. . . . . . . . . . . . . 14 |- ( norm ` R ) = ( norm ` R )
30		qqhcn.z	. . . . . . . . . . . . . 14 |- Z = ( ZMod ` R )
31	29, 30	qqhnm 28843	. . . . . . . . . . . . 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 726	. . . . . . . . . . . 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 2506	. . . . . . . . . . 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 737	. . . . . . . . . . . . . 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 6046	. . . . . . . . . . . . 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 6046	. . . . . . . . . . . . 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 6464	. . . . . . . . . . . 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 3664	. . . . . . . . . . . . . . . . 17 |- (NrmRing i^i DivRing ) C_ NrmRing
39	38	sseli 3440	. . . . . . . . . . . . . . . 16 |- ( R e. (NrmRing i^i DivRing ) -> R e. NrmRing )
40	39	3ad2ant1 1035	. . . . . . . . . . . . . . 15 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. NrmRing )
41	40	ad2antrr 737	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. NrmRing )
42		nrgngp 21714	. . . . . . . . . . . . . 14 |- ( R e. NrmRing -> R e. NrmGrp )
43	41, 42	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. NrmGrp )
44		eqid 2462	. . . . . . . . . . . . . 14 |- ( -g ` R ) = ( -g ` R )
45		eqid 2462	. . . . . . . . . . . . . 14 |- ( dist ` R ) = ( dist ` R )
46	29, 5, 44, 45	ngpdsr 21667	. . . . . . . . . . . . 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 1276	. . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . 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 28837	. . . . . . . . . . . . . . . 16 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 0 ) = ( 0g ` R ) )
51	48, 49, 50	syl2anc 671	. . . . . . . . . . . . . . 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 6331	. . . . . . . . . . . . . 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 21662	. . . . . . . . . . . . . . . 16 |- ( R e. NrmGrp -> R e. Grp )
54	43, 53	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. Grp )
55		eqid 2462	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) = ( 0g ` R )
56	5, 55, 44	grpsubid1 16788	. . . . . . . . . . . . . . 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 671	. . . . . . . . . . . . . 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 2496	. . . . . . . . . . . . 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 5892	. . . . . . . . . . . 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 2500	. . . . . . . . . . 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 2499	. . . . . . . . . 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 4426	. . . . . . . . 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 212	. . . . . . . 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 2814	. . . . . . 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 4420	. . . . . . . . . 10 |- ( d = e -> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d <-> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e ) )
66	65	imbi1d 323	. . . . . . . . 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 2839	. . . . . . . 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 3162	. . . . . . 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 671	. . . . . 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 2814	. . . . 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 21846	. . . . . . . . 9 |- ℂfld e. *MetSp
73		qex 11305	. . . . . . . . 9 |- QQ e. _V
74		ressxms 21589	. . . . . . . . 9 |- ( (ℂfld e. *MetSp /\ QQ e. _V ) -> (ℂfld ↾s QQ ) e. *MetSp )
75	72, 73, 74	mp2an 683	. . . . . . . 8 |- (ℂfld ↾s QQ ) e. *MetSp
76	71, 75	eqeltri 2536	. . . . . . 7 |- Q e. *MetSp
77	71	qrngbas 24506	. . . . . . . 8 |- QQ = ( Base ` Q )
78		cnfldds 19029	. . . . . . . . . 10 |- ( abs o. - ) = ( dist ` ℂfld )
79	71, 78	ressds 15360	. . . . . . . . 9 |- ( QQ e. _V -> ( abs o. - ) = ( dist ` Q ) )
80	73, 79	ax-mp 5	. . . . . . . 8 |- ( abs o. - ) = ( dist ` Q )
81	77, 80	xmsxmet2 21523	. . . . . . 7 |- ( Q e. *MetSp -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( *Met ` QQ ) )
82	76, 81	mp1i 13	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( *Met ` QQ ) )
83		ngpxms 21664	. . . . . . . . 9 |- ( R e. NrmGrp -> R e. *MetSp )
84	39, 42, 83	3syl 18	. . . . . . . 8 |- ( R e. (NrmRing i^i DivRing ) -> R e. *MetSp )
85	84	3ad2ant1 1035	. . . . . . 7 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. *MetSp )
86	5, 45	xmsxmet2 21523	. . . . . . 7 |- ( R e. *MetSp -> ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( *Met ` ( Base ` R ) ) )
87	85, 86	syl 17	. . . . . 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 5120	. . . . . . . . 9 |- ( ( abs o. - ) |` ( QQ X. QQ ) ) = ( ( dist ` Q ) |` ( QQ X. QQ ) )
91	89, 77, 90	xmstopn 21515	. . . . . . . 8 |- ( Q e. *MetSp -> J = ( MetOpen ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) )
92	76, 91	ax-mp 5	. . . . . . 7 |- J = ( MetOpen ` ( ( abs o. - ) |` ( QQ X. QQ ) ) )
93		eqid 2462	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
94	92, 93	metcnp 21605	. . . . . 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 1276	. . . . 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 938	. . . 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 2462	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
99	97, 5, 98	xmstopn 21515	. . . . . . 7 |- ( R e. *MetSp -> K = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
100	85, 99	syl 17	. . . . . 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 6331	. . . . 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 5890	. . . 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 2543	. . 3 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( ( J CnP K ) ` 0 ) )
104		cnfldtgp 21950	. . . . . 6 |- ℂfld e. TopGrp
105		qsubdrg 19069	. . . . . . . 8 |- ( QQ e. (SubRing ` ℂfld ) /\ (ℂfld ↾s QQ ) e. DivRing )
106	105	simpli 464	. . . . . . 7 |- QQ e. (SubRing ` ℂfld )
107		subrgsubg 18063	. . . . . . 7 |- ( QQ e. (SubRing ` ℂfld ) -> QQ e. (SubGrp ` ℂfld ) )
108	106, 107	ax-mp 5	. . . . . 6 |- QQ e. (SubGrp ` ℂfld )
109	71	subgtgp 21169	. . . . . 6 |- ( (ℂfld e. TopGrp /\ QQ e. (SubGrp ` ℂfld ) ) -> Q e. TopGrp )
110	104, 108, 109	mp2an 683	. . . . 5 |- Q e. TopGrp
111		tgptmd 21143	. . . . 5 |- ( Q e. TopGrp -> Q e. TopMnd )
112	110, 111	mp1i 13	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> Q e. TopMnd )
113		nrgtrg 21741	. . . . 5 |- ( R e. NrmRing -> R e. TopRing )
114		trgtmd2 21232	. . . . 5 |- ( R e. TopRing -> R e. TopMnd )
115	40, 113, 114	3syl 18	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. TopMnd )
116	5, 6, 7, 71	qqhghm 28841	. . . . 5 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
117	3, 4, 116	syl2anc 671	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
118	77, 89, 97	ghmcnp 21178	. . . 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 1276	. . 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 215	. 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 469	1 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057   i^i cin 3415   class class class wbr 4416   X. cxp 4851   |` cres 4855   o. ccom 4857   -->wf 5597   ` cfv 5601  (class class class)co 6315   CCcc 9563   0cc0 9565   < clt 9701   - cmin 9886   -ucneg 9887   ZZcz 10966   QQcq 11293   RR+crp 11331   abscabs 13346   Basecbs 15170   ↾_s cress 15171   distcds 15248   TopOpenctopn 15369   0gc0g 15387   Grpcgrp 16718   -gcsg 16720  SubGrpcsubg 16860   GrpHom cghm 16929  /_rcdvr 17959   DivRingcdr 18024  SubRingcsubrg 18053   *Metcxmt 19004   MetOpencmopn 19009  ℂ_fldccnfld 19019   ZRHomczrh 19120   ZModczlm 19121  chrcchr 19122   Cn ccn 20289   CnP ccnp 20290  TopMndctmd 21134   TopGrpctgp 21135   TopRingctrg 21219   *MetSpcxme 21381   normcnm 21640  NrmGrpcngp 21641  NrmRingcnrg 21643  NrmModcnlm 21644  QQHomcqqh 28825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644  ax-mulf 9645

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-tpos 6999  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-map 7500  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-fi 7951  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-mod 12129  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-dvds 14355  df-gcd 14518  df-numer 14733  df-denom 14734  df-gz 14923  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-starv 15254  df-sca 15255  df-vsca 15256  df-ip 15257  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-hom 15263  df-cco 15264  df-rest 15370  df-topn 15371  df-0g 15389  df-gsum 15390  df-topgen 15391  df-pt 15392  df-prds 15395  df-xrs 15449  df-qtop 15455  df-imas 15456  df-xps 15459  df-mre 15541  df-mrc 15542  df-acs 15544  df-plusf 16536  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-mhm 16631  df-submnd 16632  df-grp 16722  df-minusg 16723  df-sbg 16724  df-mulg 16725  df-subg 16863  df-ghm 16930  df-cntz 17020  df-od 17221  df-cmn 17481  df-abl 17482  df-mgp 17773  df-ur 17785  df-ring 17831  df-cring 17832  df-oppr 17900  df-dvdsr 17918  df-unit 17919  df-invr 17949  df-dvr 17960  df-rnghom 17992  df-drng 18026  df-subrg 18055  df-abv 18094  df-lmod 18142  df-scaf 18143  df-sra 18444  df-rgmod 18445  df-nzr 18531  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-cnfld 19020  df-zring 19089  df-zrh 19124  df-zlm 19125  df-chr 19126  df-top 19970  df-bases 19971  df-topon 19972  df-topsp 19973  df-cn 20292  df-cnp 20293  df-tx 20626  df-hmeo 20819  df-tmd 21136  df-tgp 21137  df-trg 21223  df-xms 21384  df-ms 21385  df-tms 21386  df-nm 21646  df-ngp 21647  df-nrg 21649  df-nlm 21650  df-qqh 28826

This theorem is referenced by:  rrhqima  28867