Theorem qqhcn 28133

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 3715	. . . . . . . 8 |- (NrmRing i^i DivRing ) C_ DivRing
2	1	sseli 3495	. . . . . . 7 |- ( R e. (NrmRing i^i DivRing ) -> R e. DivRing )
3	2	3ad2ant1 1017	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. DivRing )
4		simp3 998	. . . . . 6 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (chr ` R ) = 0 )
5		eqid 2457	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
6		eqid 2457	. . . . . . 7 |- (/r ` R ) = (/r ` R )
7		eqid 2457	. . . . . . 7 |- ( ZRHom ` R ) = ( ZRHom ` R )
8	5, 6, 7	qqhf 28128	. . . . . 6 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
9	3, 4, 8	syl2anc 661	. . . . 5 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> ( Base ` R ) )
10		simpr 461	. . . . . . 7 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) -> e e. RR+ )
11		qsscn 11218	. . . . . . . . . . . . . 14 |- QQ C_ CC
12		simpr 461	. . . . . . . . . . . . . 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 3497	. . . . . . . . . . . . 13 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> q e. CC )
14		0cn 9605	. . . . . . . . . . . . . . 15 |- 0 e. CC
15		eqid 2457	. . . . . . . . . . . . . . . 16 |- ( abs o. - ) = ( abs o. - )
16	15	cnmetdval 21404	. . . . . . . . . . . . . . 15 |- ( ( 0 e. CC /\ q e. CC ) -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
17	14, 16	mpan 670	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( 0 ( abs o. - ) q ) = ( abs ` ( 0 - q ) ) )
18		df-neg 9827	. . . . . . . . . . . . . . . 16 |- -u q = ( 0 - q )
19	18	fveq2i 5875	. . . . . . . . . . . . . . 15 |- ( abs ` -u q ) = ( abs ` ( 0 - q ) )
20	19	a1i 11	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` ( 0 - q ) ) )
21		absneg 13122	. . . . . . . . . . . . . 14 |- ( q e. CC -> ( abs ` -u q ) = ( abs ` q ) )
22	17, 20, 21	3eqtr2d 2504	. . . . . . . . . . . . 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 11214	. . . . . . . . . . . . . . 15 |- ZZ C_ QQ
25		0z 10896	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
26	24, 25	sselii 3496	. . . . . . . . . . . . . 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 6442	. . . . . . . . . . . 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 2457	. . . . . . . . . . . . . 14 |- ( norm ` R ) = ( norm ` R )
30		qqhcn.z	. . . . . . . . . . . . . 14 |- Z = ( ZMod ` R )
31	29, 30	qqhnm 28132	. . . . . . . . . . . . 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 714	. . . . . . . . . . . 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 2508	. . . . . . . . . . 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 725	. . . . . . . . . . . . . 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 6033	. . . . . . . . . . . . 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 6033	. . . . . . . . . . . . 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 6442	. . . . . . . . . . . 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 3714	. . . . . . . . . . . . . . . . 17 |- (NrmRing i^i DivRing ) C_ NrmRing
39	38	sseli 3495	. . . . . . . . . . . . . . . 16 |- ( R e. (NrmRing i^i DivRing ) -> R e. NrmRing )
40	39	3ad2ant1 1017	. . . . . . . . . . . . . . 15 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. NrmRing )
41	40	ad2antrr 725	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ e e. RR+ ) /\ q e. QQ ) -> R e. NrmRing )
42		nrgngp 21297	. . . . . . . . . . . . . 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 2457	. . . . . . . . . . . . . 14 |- ( -g ` R ) = ( -g ` R )
45		eqid 2457	. . . . . . . . . . . . . 14 |- ( dist ` R ) = ( dist ` R )
46	29, 5, 44, 45	ngpdsr 21250	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . 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 28126	. . . . . . . . . . . . . . . 16 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 0 ) = ( 0g ` R ) )
51	48, 49, 50	syl2anc 661	. . . . . . . . . . . . . . 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 6312	. . . . . . . . . . . . . 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 21245	. . . . . . . . . . . . . . . 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 2457	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) = ( 0g ` R )
56	5, 55, 44	grpsubid1 16250	. . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . 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 2498	. . . . . . . . . . . . 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 5876	. . . . . . . . . . . 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 2502	. . . . . . . . . . 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 2501	. . . . . . . . . 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 4466	. . . . . . . . 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 207	. . . . . . . 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 2871	. . . . . . 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 4460	. . . . . . . . . 10 |- ( d = e -> ( ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d <-> ( 0 ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e ) )
66	65	imbi1d 317	. . . . . . . . 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 2896	. . . . . . . 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 3210	. . . . . . 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 661	. . . . . 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 2871	. . . . 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 21410	. . . . . . . . 9 |- ℂfld e. *MetSp
73		qex 11219	. . . . . . . . 9 |- QQ e. _V
74		ressxms 21154	. . . . . . . . 9 |- ( (ℂfld e. *MetSp /\ QQ e. _V ) -> (ℂfld ↾s QQ ) e. *MetSp )
75	72, 73, 74	mp2an 672	. . . . . . . 8 |- (ℂfld ↾s QQ ) e. *MetSp
76	71, 75	eqeltri 2541	. . . . . . 7 |- Q e. *MetSp
77	71	qrngbas 23930	. . . . . . . 8 |- QQ = ( Base ` Q )
78		cnfldds 18557	. . . . . . . . . 10 |- ( abs o. - ) = ( dist ` ℂfld )
79	71, 78	ressds 14830	. . . . . . . . 9 |- ( QQ e. _V -> ( abs o. - ) = ( dist ` Q ) )
80	73, 79	ax-mp 5	. . . . . . . 8 |- ( abs o. - ) = ( dist ` Q )
81	77, 80	xmsxmet2 21088	. . . . . . 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 21247	. . . . . . . . 9 |- ( R e. NrmGrp -> R e. *MetSp )
84	39, 42, 83	3syl 20	. . . . . . . 8 |- ( R e. (NrmRing i^i DivRing ) -> R e. *MetSp )
85	84	3ad2ant1 1017	. . . . . . 7 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. *MetSp )
86	5, 45	xmsxmet2 21088	. . . . . . 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 5279	. . . . . . . . 9 |- ( ( abs o. - ) |` ( QQ X. QQ ) ) = ( ( dist ` Q ) |` ( QQ X. QQ ) )
91	89, 77, 90	xmstopn 21080	. . . . . . . 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 2457	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
94	92, 93	metcnp 21170	. . . . . 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 1228	. . . . 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 922	. . . 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 2457	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
99	97, 5, 98	xmstopn 21080	. . . . . . 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 6312	. . . . 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 5874	. . . 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 2548	. . 3 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( ( J CnP K ) ` 0 ) )
104		cnfldtgp 21499	. . . . . 6 |- ℂfld e. TopGrp
105		qsubdrg 18597	. . . . . . . 8 |- ( QQ e. (SubRing ` ℂfld ) /\ (ℂfld ↾s QQ ) e. DivRing )
106	105	simpli 458	. . . . . . 7 |- QQ e. (SubRing ` ℂfld )
107		subrgsubg 17562	. . . . . . 7 |- ( QQ e. (SubRing ` ℂfld ) -> QQ e. (SubGrp ` ℂfld ) )
108	106, 107	ax-mp 5	. . . . . 6 |- QQ e. (SubGrp ` ℂfld )
109	71	subgtgp 20730	. . . . . 6 |- ( (ℂfld e. TopGrp /\ QQ e. (SubGrp ` ℂfld ) ) -> Q e. TopGrp )
110	104, 108, 109	mp2an 672	. . . . 5 |- Q e. TopGrp
111		tgptmd 20704	. . . . 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 21324	. . . . 5 |- ( R e. NrmRing -> R e. TopRing )
114		trgtmd2 20797	. . . . 5 |- ( R e. TopRing -> R e. TopMnd )
115	40, 113, 114	3syl 20	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> R e. TopMnd )
116	5, 6, 7, 71	qqhghm 28130	. . . . 5 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
117	3, 4, 116	syl2anc 661	. . . 4 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
118	77, 89, 97	ghmcnp 20739	. . . 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 1228	. . 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 210	. 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 463	1 |- ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109   i^i cin 3470   class class class wbr 4456   X. cxp 5006   |` cres 5010   o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   < clt 9645   - cmin 9824   -ucneg 9825   ZZcz 10885   QQcq 11207   RR+crp 11245   abscabs 13079   Basecbs 14644   ↾_s cress 14645   distcds 14721   TopOpenctopn 14839   0gc0g 14857   Grpcgrp 16180   -gcsg 16182  SubGrpcsubg 16322   GrpHom cghm 16391  /_rcdvr 17458   DivRingcdr 17523  SubRingcsubrg 17552   *Metcxmt 18530   MetOpencmopn 18535  ℂ_fldccnfld 18547   ZRHomczrh 18664   ZModczlm 18665  chrcchr 18666   Cn ccn 19852   CnP ccnp 19853  TopMndctmd 20695   TopGrpctgp 20696   TopRingctrg 20784   *MetSpcxme 20946   normcnm 21223  NrmGrpcngp 21224  NrmRingcnrg 21226  NrmModcnlm 21227  QQHomcqqh 28114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-numer 14280  df-denom 14281  df-gz 14460  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-plusf 15998  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-cntz 16482  df-od 16680  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-rnghom 17491  df-drng 17525  df-subrg 17554  df-abv 17593  df-lmod 17641  df-scaf 17642  df-sra 17945  df-rgmod 17946  df-nzr 18033  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-zring 18616  df-zrh 18668  df-zlm 18669  df-chr 18670  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cn 19855  df-cnp 19856  df-tx 20189  df-hmeo 20382  df-tmd 20697  df-tgp 20698  df-trg 20788  df-xms 20949  df-ms 20950  df-tms 20951  df-nm 21229  df-ngp 21230  df-nrg 21232  df-nlm 21233  df-qqh 28115

This theorem is referenced by: (None)