Theorem qqhcn 30035

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 3834	. . . . . . . 8 |- (NrmRing i^i DivRing ) C_ DivRing
2	1	sseli 3599	. . . . . . 7 |- ( R e. (NrmRing i^i DivRing ) -> R e. DivRing )
3	2	3ad2ant1 1082	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. DivRing )
4		simp3 1063	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (chr ` R ) = 0 )
5		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
6		eqid 2622	. . . . . . 7 |- (/r ` R ) = (/r ` R )
7		eqid 2622	. . . . . . 7 |- ( ZRHom ` R ) = ( ZRHom ` R )
8	5, 6, 7	qqhf 30030	. . . . . 6 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
9	3, 4, 8	syl2anc 693	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
10		simpr 477	. . . . . . 7 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) -> e e. RR+ )
11		qsscn 11799	. . . . . . . . . . . . . 14 |- QQ C_ CC
12		simpr 477	. . . . . . . . . . . . . 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 3601	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> q e. CC )
14		0cn 10032	. . . . . . . . . . . . . . 15 |- 0 e. CC
15		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( abs o. - ) = ( abs o. - )
16	15	cnmetdval 22574	. . . . . . . . . . . . . . 15 |- ( ( 0 e. CC /\ q e. CC ) -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
17	14, 16	mpan 706	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
18		df-neg 10269	. . . . . . . . . . . . . . . 16 |- -u q = ( 0 - q )
19	18	fveq2i 6194	. . . . . . . . . . . . . . 15 |- ( abs ` -u q ) = ( abs ` ( 0 - q ) )
20	19	a1i 11	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` ( 0 - q ) ) )
21		absneg 14017	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` q ) )
22	17, 20, 21	3eqtr2d 2662	. . . . . . . . . . . . 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 11795	. . . . . . . . . . . . . . 15 |- ZZ C_ QQ
25		0z 11388	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
26	24, 25	sselii 3600	. . . . . . . . . . . . . 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 6801	. . . . . . . . . . . 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 2622	. . . . . . . . . . . . . 14 |- ( norm ` R ) = ( norm ` R )
30		qqhcn.z	. . . . . . . . . . . . . 14 |- Z = ( ZMod ` R )
31	29, 30	qqhnm 30034	. . . . . . . . . . . . 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 751	. . . . . . . . . . . 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 2666	. . . . . . . . . . 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 762	. . . . . . . . . . . . . 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 6360	. . . . . . . . . . . . 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 6360	. . . . . . . . . . . . 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 6801	. . . . . . . . . . . 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 3833	. . . . . . . . . . . . . . . . 17 |- (NrmRing i^i DivRing ) C_ NrmRing
39	38	sseli 3599	. . . . . . . . . . . . . . . 16 |- ( R e. (NrmRing i^i DivRing ) -> R e. NrmRing )
40	39	3ad2ant1 1082	. . . . . . . . . . . . . . 15 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. NrmRing )
41	40	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. NrmRing )
42		nrgngp 22466	. . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . 14 |- ( -g ` R ) = ( -g ` R )
45		eqid 2622	. . . . . . . . . . . . . 14 |- ( dist ` R ) = ( dist ` R )
46	29, 5, 44, 45	ngpdsr 22409	. . . . . . . . . . . . 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 1326	. . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . 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 30028	. . . . . . . . . . . . . . . 16 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 0 ) = ( 0g ` R ) )
51	48, 49, 50	syl2anc 693	. . . . . . . . . . . . . . 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 6666	. . . . . . . . . . . . . 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 22403	. . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) = ( 0g ` R )
56	5, 55, 44	grpsubid1 17500	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . 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 6195	. . . . . . . . . . . 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 2660	. . . . . . . . . . 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 2659	. . . . . . . . . 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 4663	. . . . . . . . 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 219	. . . . . . . 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 2966	. . . . . . 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 4657	. . . . . . . . . 10 |- ( d = e -> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d <-> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e ) )
66	65	imbi1d 331	. . . . . . . . 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 2986	. . . . . . . 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 3309	. . . . . . 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 693	. . . . . 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 2966	. . . . 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 22580	. . . . . . . . 9 |- ℂfld e. *MetSp
73		qex 11800	. . . . . . . . 9 |- QQ e. _V
74		ressxms 22330	. . . . . . . . 9 |- ( (ℂfld e. *MetSp /\ QQ e. _V ) -> (ℂfld ↾s QQ ) e. *MetSp )
75	72, 73, 74	mp2an 708	. . . . . . . 8 |- (ℂfld ↾s QQ ) e. *MetSp
76	71, 75	eqeltri 2697	. . . . . . 7 |- Q e. *MetSp
77	71	qrngbas 25308	. . . . . . . 8 |- QQ = ( Base ` Q )
78		cnfldds 19756	. . . . . . . . . 10 |- ( abs o. - ) = ( dist ` ℂfld )
79	71, 78	ressds 16073	. . . . . . . . 9 |- ( QQ e. _V -> ( abs o. - ) = ( dist ` Q ) )
80	73, 79	ax-mp 5	. . . . . . . 8 |- ( abs o. - ) = ( dist ` Q )
81	77, 80	xmsxmet2 22264	. . . . . . 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 22405	. . . . . . . . 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 1082	. . . . . . 7 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. *MetSp )
86	5, 45	xmsxmet2 22264	. . . . . . 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 5392	. . . . . . . . 9 |- ( ( abs o. - ) |` ( QQ X. QQ ) ) = ( ( dist ` Q ) |` ( QQ X. QQ ) )
91	89, 77, 90	xmstopn 22256	. . . . . . . 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 2622	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
94	92, 93	metcnp 22346	. . . . . 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 1326	. . . . 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 957	. . . 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 2622	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
99	97, 5, 98	xmstopn 22256	. . . . . . 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 6666	. . . . 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 6193	. . . 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 2704	. . 3 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( ( J CnP K ) ` 0 ) )
104		cnfldtgp 22672	. . . . . 6 |- ℂfld e. TopGrp
105		qsubdrg 19798	. . . . . . . 8 |- ( QQ e. (SubRing ` ℂfld ) /\ (ℂfld ↾s QQ ) e. DivRing )
106	105	simpli 474	. . . . . . 7 |- QQ e. (SubRing ` ℂfld )
107		subrgsubg 18786	. . . . . . 7 |- ( QQ e. (SubRing ` ℂfld ) -> QQ e. (SubGrp ` ℂfld ) )
108	106, 107	ax-mp 5	. . . . . 6 |- QQ e. (SubGrp ` ℂfld )
109	71	subgtgp 21909	. . . . . 6 |- ( (ℂfld e. TopGrp /\ QQ e. (SubGrp ` ℂfld ) ) -> Q e. TopGrp )
110	104, 108, 109	mp2an 708	. . . . 5 |- Q e. TopGrp
111		tgptmd 21883	. . . . 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 22494	. . . . 5 |- ( R e. NrmRing -> R e. TopRing )
114		trgtmd2 21972	. . . . 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 30032	. . . . 5 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
117	3, 4, 116	syl2anc 693	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
118	77, 89, 97	ghmcnp 21918	. . . 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 1326	. . 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 222	. 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 479	1 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   class class class wbr 4653   X. cxp 5112   |` cres 5116   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   < clt 10074   - cmin 10266   -ucneg 10267   ZZcz 11377   QQcq 11788   RR+crp 11832   abscabs 13974   Basecbs 15857   ↾_s cress 15858   distcds 15950   TopOpenctopn 16082   0gc0g 16100   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588   GrpHom cghm 17657  /_rcdvr 18682   DivRingcdr 18747  SubRingcsubrg 18776   *Metcxmt 19731   MetOpencmopn 19736  ℂ_fldccnfld 19746   ZRHomczrh 19848   ZModczlm 19849  chrcchr 19850   Cn ccn 21028   CnP ccnp 21029  TopMndctmd 21874   TopGrpctgp 21875   TopRingctrg 21959   *MetSpcxme 22122   normcnm 22381  NrmGrpcngp 22382  NrmRingcnrg 22384  NrmModcnlm 22385  QQHomcqqh 30016

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-inf2 8538  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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-od 17948  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-nzr 19258  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zlm 19853  df-chr 19854  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-tmd 21876  df-tgp 21877  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-qqh 30017

This theorem is referenced by:  rrhqima  30058