Theorem qqhcn 24377

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 3564	. . . . . . . 8 |- (NrmRing i^i DivRing ) C_ DivRing
2	1	sseli 3346	. . . . . . 7 |- ( R e. (NrmRing i^i DivRing ) -> R e. DivRing )
3	2	3ad2ant1 979	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. DivRing )
4		simp3 960	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (chr ` R ) = 0 )
5		eqid 2438	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
6		eqid 2438	. . . . . . 7 |- (/r ` R ) = (/r ` R )
7		eqid 2438	. . . . . . 7 |- ( ZRHom ` R ) = ( ZRHom ` R )
8	5, 6, 7	qqhf 24372	. . . . . 6 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
9	3, 4, 8	syl2anc 644	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
10		simpr 449	. . . . . . 7 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) -> e e. RR+ )
11		qsscn 10587	. . . . . . . . . . . . . 14 |- QQ C_ CC
12		simpr 449	. . . . . . . . . . . . . 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 3348	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> q e. CC )
14		0cn 9086	. . . . . . . . . . . . . . 15 |- 0 e. CC
15		eqid 2438	. . . . . . . . . . . . . . . 16 |- ( abs o. - ) = ( abs o. - )
16	15	cnmetdval 18807	. . . . . . . . . . . . . . 15 |- ( ( 0 e. CC /\ q e. CC ) -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
17	14, 16	mpan 653	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
18		df-neg 9296	. . . . . . . . . . . . . . . 16 |- -u q = ( 0 - q )
19	18	fveq2i 5733	. . . . . . . . . . . . . . 15 |- ( abs ` -u q ) = ( abs ` ( 0 - q ) )
20	19	a1i 11	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` ( 0 - q ) ) )
21		absneg 12084	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` q ) )
22	17, 20, 21	3eqtr2d 2476	. . . . . . . . . . . . 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 10583	. . . . . . . . . . . . . . 15 |- ZZ C_ QQ
25		0z 10295	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
26	24, 25	sselii 3347	. . . . . . . . . . . . . 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 6216	. . . . . . . . . . . 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 2438	. . . . . . . . . . . . . 14 |- ( norm ` R ) = ( norm ` R )
30		qqhcn.z	. . . . . . . . . . . . . 14 |- Z = ( ZMod ` R )
31	29, 30	qqhnm 24376	. . . . . . . . . . . . 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 697	. . . . . . . . . . . 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 2480	. . . . . . . . . . 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 708	. . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . 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 5873	. . . . . . . . . . . . 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 6216	. . . . . . . . . . . 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 3563	. . . . . . . . . . . . . . . . 17 |- (NrmRing i^i DivRing ) C_ NrmRing
39	38	sseli 3346	. . . . . . . . . . . . . . . 16 |- ( R e. (NrmRing i^i DivRing ) -> R e. NrmRing )
40	39	3ad2ant1 979	. . . . . . . . . . . . . . 15 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. NrmRing )
41	40	ad2antrr 708	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. NrmRing )
42		nrgngp 18700	. . . . . . . . . . . . . 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 2438	. . . . . . . . . . . . . 14 |- ( -g ` R ) = ( -g ` R )
45		eqid 2438	. . . . . . . . . . . . . 14 |- ( dist ` R ) = ( dist ` R )
46	29, 5, 44, 45	ngpdsr 18653	. . . . . . . . . . . . 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 1185	. . . . . . . . . . . 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 708	. . . . . . . . . . . . . . . 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 708	. . . . . . . . . . . . . . . 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 24370	. . . . . . . . . . . . . . . 16 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 0 ) = ( 0g ` R ) )
51	48, 49, 50	syl2anc 644	. . . . . . . . . . . . . . 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 6099	. . . . . . . . . . . . . 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 18648	. . . . . . . . . . . . . . . 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 2438	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) = ( 0g ` R )
56	5, 55, 44	grpsubid1 14876	. . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . 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 2470	. . . . . . . . . . . . 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 5734	. . . . . . . . . . . 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 2474	. . . . . . . . . . 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 2473	. . . . . . . . . 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 4224	. . . . . . . . 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 200	. . . . . . . 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 2791	. . . . . . 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 4218	. . . . . . . . . 10 |- ( d = e -> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d <-> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e ) )
66	65	imbi1d 310	. . . . . . . . 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 2727	. . . . . . . 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 3054	. . . . . . 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 644	. . . . . 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 2791	. . . . 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 18813	. . . . . . . . 9 |- ℂfld e. * MetSp
73		qex 10588	. . . . . . . . 9 |- QQ e. _V
74		ressxms 18557	. . . . . . . . 9 |- ( (ℂfld e. * MetSp /\ QQ e. _V ) -> (ℂfld ↾s QQ ) e. * MetSp )
75	72, 73, 74	mp2an 655	. . . . . . . 8 |- (ℂfld ↾s QQ ) e. * MetSp
76	71, 75	eqeltri 2508	. . . . . . 7 |- Q e. * MetSp
77	71	qrngbas 21315	. . . . . . . 8 |- QQ = ( Base ` Q )
78		cnfldds 16715	. . . . . . . . . 10 |- ( abs o. - ) = ( dist ` ℂfld )
79	71, 78	ressds 13643	. . . . . . . . 9 |- ( QQ e. _V -> ( abs o. - ) = ( dist ` Q ) )
80	73, 79	ax-mp 8	. . . . . . . 8 |- ( abs o. - ) = ( dist ` Q )
81	77, 80	xmsxmet2 18491	. . . . . . 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 18650	. . . . . . . . 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 979	. . . . . . 7 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. * MetSp )
86	5, 45	xmsxmet2 18491	. . . . . . 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 5144	. . . . . . . . 9 |- ( ( abs o. - ) |` ( QQ X. QQ ) ) = ( ( dist ` Q ) |` ( QQ X. QQ ) )
91	89, 77, 90	xmstopn 18483	. . . . . . . 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 2438	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
94	92, 93	metcnp 18573	. . . . . 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 1185	. . . . 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 890	. . . 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 2438	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
99	97, 5, 98	xmstopn 18483	. . . . . . 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 6099	. . . . 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 5732	. . . 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 2515	. . 3 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( ( J CnP K ) ` 0 ) )
104		cnfldtgp 18901	. . . . . 6 |- ℂfld e. TopGrp
105		qsubdrg 16753	. . . . . . . 8 |- ( QQ e. (SubRing ` ℂfld ) /\ (ℂfld ↾s QQ ) e. DivRing )
106	105	simpli 446	. . . . . . 7 |- QQ e. (SubRing ` ℂfld )
107		subrgsubg 15876	. . . . . . 7 |- ( QQ e. (SubRing ` ℂfld ) -> QQ e. (SubGrp ` ℂfld ) )
108	106, 107	ax-mp 8	. . . . . 6 |- QQ e. (SubGrp ` ℂfld )
109	71	subgtgp 18137	. . . . . 6 |- ( (ℂfld e. TopGrp /\ QQ e. (SubGrp ` ℂfld ) ) -> Q e. TopGrp )
110	104, 108, 109	mp2an 655	. . . . 5 |- Q e. TopGrp
111		tgptmd 18111	. . . . 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 18727	. . . . 5 |- ( R e. NrmRing -> R e. TopRing )
114		trgtmd2 18200	. . . . 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 24374	. . . . 5 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
117	3, 4, 116	syl2anc 644	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
118	77, 89, 97	ghmcnp 18146	. . . 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 1185	. . 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 203	. 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 451	1 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 178   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958   i^i cin 3321   class class class wbr 4214   X. cxp 4878   |` cres 4882   o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   < clt 9122   - cmin 9293   -ucneg 9294   ZZcz 10284   QQcq 10576   RR+crp 10614   abscabs 12041   Basecbs 13471   ↾_s cress 13472   distcds 13540   TopOpenctopn 13651   0gc0g 13725   Grpcgrp 14687   -gcsg 14690  SubGrpcsubg 14940   GrpHom cghm 15005  /_rcdvr 15789   DivRingcdr 15837  SubRingcsubrg 15866   * Metcxmt 16688   MetOpencmopn 16693  ℂ_fldccnfld 16705   ZRHomczrh 16780   ZModczlm 16781  chrcchr 16782   Cn ccn 17290   CnP ccnp 17291  TopMndctmd 18102   TopGrpctgp 18103   TopRingctrg 18187   * MetSpcxme 18349   normcnm 18626  NrmGrpcngp 18627  NrmRingcnrg 18629  NrmModcnlm 18630  QQHomcqqh 24358

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-numer 13129  df-denom 13130  df-gz 13300  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-plusf 14693  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-ghm 15006  df-cntz 15118  df-od 15169  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-rnghom 15821  df-drng 15839  df-subrg 15868  df-abv 15907  df-lmod 15954  df-scaf 15955  df-sra 16246  df-rgmod 16247  df-nzr 16331  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-zrh 16784  df-zlm 16785  df-chr 16786  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cn 17293  df-cnp 17294  df-tx 17596  df-hmeo 17789  df-tmd 18104  df-tgp 18105  df-trg 18191  df-xms 18352  df-ms 18353  df-tms 18354  df-nm 18632  df-ngp 18633  df-nrg 18635  df-nlm 18636  df-qqh 24359