Theorem qqhucn 30036

Description: The QQHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)

Hypotheses

Ref	Expression
qqhucn.b	|- B = ( Base ` R )
qqhucn.q	|- Q = (ℂfld ↾s QQ )
qqhucn.u	|- U = (UnifSt ` Q )
qqhucn.v	|- V = (metUnif ` ( ( dist ` R ) |` ( B X. B ) ) )
qqhucn.z	|- Z = ( ZMod ` R )
qqhucn.1	|- ( ph -> R e. NrmRing )
qqhucn.2	|- ( ph -> R e. DivRing )
qqhucn.3	|- ( ph -> Z e. NrmMod )
qqhucn.4	|- ( ph -> (chr ` R ) = 0 )

Assertion

Ref	Expression
qqhucn	|- ( ph -> (QQHom ` R ) e. ( U Cnu V ) )

Proof of Theorem qqhucn

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

Step	Hyp	Ref	Expression
1		qqhucn.2	. . . 4 |- ( ph -> R e. DivRing )
2		qqhucn.4	. . . 4 |- ( ph -> (chr ` R ) = 0 )
3		qqhucn.b	. . . . 5 |- B = ( Base ` R )
4		eqid 2622	. . . . 5 |- (/r ` R ) = (/r ` R )
5		eqid 2622	. . . . 5 |- ( ZRHom ` R ) = ( ZRHom ` R )
6	3, 4, 5	qqhf 30030	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
7	1, 2, 6	syl2anc 693	. . 3 |- ( ph -> (QQHom ` R ) : QQ --> B )
8		simpr 477	. . . . 5 |- ( ( ph /\ e e. RR+ ) -> e e. RR+ )
9		qqhucn.1	. . . . . . . . . . . . . . 15 |- ( ph -> R e. NrmRing )
10		nrgngp 22466	. . . . . . . . . . . . . . 15 |- ( R e. NrmRing -> R e. NrmGrp )
11	9, 10	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> R e. NrmGrp )
12	11	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> R e. NrmGrp )
13	7	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. QQ ) -> ( (QQHom ` R ) ` p ) e. B )
14	13	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( (QQHom ` R ) ` p ) e. B )
15	7	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. QQ ) -> (QQHom ` R ) : QQ --> B )
16	15	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( (QQHom ` R ) ` q ) e. B )
17		eqid 2622	. . . . . . . . . . . . . 14 |- ( norm ` R ) = ( norm ` R )
18		eqid 2622	. . . . . . . . . . . . . 14 |- ( -g ` R ) = ( -g ` R )
19		eqid 2622	. . . . . . . . . . . . . 14 |- ( dist ` R ) = ( dist ` R )
20	17, 3, 18, 19	ngpdsr 22409	. . . . . . . . . . . . 13 |- ( ( R e. NrmGrp /\ ( (QQHom ` R ) ` p ) e. B /\ ( (QQHom ` R ) ` q ) e. B ) -> ( ( (QQHom ` R ) ` p ) ( dist ` R ) ( (QQHom ` R ) ` q ) ) = ( ( norm ` R ) ` ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` p ) ) ) )
21	12, 14, 16, 20	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` p ) ( dist ` R ) ( (QQHom ` R ) ` q ) ) = ( ( norm ` R ) ` ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` p ) ) ) )
22		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> q e. QQ )
23		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> p e. QQ )
24		qsubdrg 19798	. . . . . . . . . . . . . . . . . . 19 |- ( QQ e. (SubRing ` ℂfld ) /\ (ℂfld ↾s QQ ) e. DivRing )
25	24	simpli 474	. . . . . . . . . . . . . . . . . 18 |- QQ e. (SubRing ` ℂfld )
26		subrgsubg 18786	. . . . . . . . . . . . . . . . . 18 |- ( QQ e. (SubRing ` ℂfld ) -> QQ e. (SubGrp ` ℂfld ) )
27	25, 26	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- QQ e. (SubGrp ` ℂfld )
28		cnfldsub 19774	. . . . . . . . . . . . . . . . . 18 |- - = ( -g ` ℂfld )
29		qqhucn.q	. . . . . . . . . . . . . . . . . 18 |- Q = (ℂfld ↾s QQ )
30		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( -g ` Q ) = ( -g ` Q )
31	28, 29, 30	subgsub 17606	. . . . . . . . . . . . . . . . 17 |- ( ( QQ e. (SubGrp ` ℂfld ) /\ q e. QQ /\ p e. QQ ) -> ( q - p ) = ( q ( -g ` Q ) p ) )
32	27, 31	mp3an1 1411	. . . . . . . . . . . . . . . 16 |- ( ( q e. QQ /\ p e. QQ ) -> ( q - p ) = ( q ( -g ` Q ) p ) )
33	22, 23, 32	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( q - p ) = ( q ( -g ` Q ) p ) )
34	33	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( (QQHom ` R ) ` ( q - p ) ) = ( (QQHom ` R ) ` ( q ( -g ` Q ) p ) ) )
35	3, 4, 5, 29	qqhghm 30032	. . . . . . . . . . . . . . . . 17 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
36	1, 2, 35	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> (QQHom ` R ) e. ( Q GrpHom R ) )
37	36	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> (QQHom ` R ) e. ( Q GrpHom R ) )
38	29	qrngbas 25308	. . . . . . . . . . . . . . . 16 |- QQ = ( Base ` Q )
39	38, 30, 18	ghmsub 17668	. . . . . . . . . . . . . . 15 |- ( ( (QQHom ` R ) e. ( Q GrpHom R ) /\ q e. QQ /\ p e. QQ ) -> ( (QQHom ` R ) ` ( q ( -g ` Q ) p ) ) = ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` p ) ) )
40	37, 22, 23, 39	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( (QQHom ` R ) ` ( q ( -g ` Q ) p ) ) = ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` p ) ) )
41	34, 40	eqtr2d 2657	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` p ) ) = ( (QQHom ` R ) ` ( q - p ) ) )
42	41	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( norm ` R ) ` ( ( (QQHom ` R ) ` q ) ( -g ` R ) ( (QQHom ` R ) ` p ) ) ) = ( ( norm ` R ) ` ( (QQHom ` R ) ` ( q - p ) ) ) )
43	9, 1	elind 3798	. . . . . . . . . . . . . 14 |- ( ph -> R e. (NrmRing i^i DivRing ) )
44	43	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> R e. (NrmRing i^i DivRing ) )
45		qqhucn.3	. . . . . . . . . . . . . 14 |- ( ph -> Z e. NrmMod )
46	45	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> Z e. NrmMod )
47	2	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> (chr ` R ) = 0 )
48		qsubcl 11807	. . . . . . . . . . . . . 14 |- ( ( q e. QQ /\ p e. QQ ) -> ( q - p ) e. QQ )
49	22, 23, 48	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( q - p ) e. QQ )
50		qqhucn.z	. . . . . . . . . . . . . 14 |- Z = ( ZMod ` R )
51	17, 50	qqhnm 30034	. . . . . . . . . . . . 13 |- ( ( ( R e. (NrmRing i^i DivRing ) /\ Z e. NrmMod /\ (chr ` R ) = 0 ) /\ ( q - p ) e. QQ ) -> ( ( norm ` R ) ` ( (QQHom ` R ) ` ( q - p ) ) ) = ( abs ` ( q - p ) ) )
52	44, 46, 47, 49, 51	syl31anc 1329	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( norm ` R ) ` ( (QQHom ` R ) ` ( q - p ) ) ) = ( abs ` ( q - p ) ) )
53	21, 42, 52	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` p ) ( dist ` R ) ( (QQHom ` R ) ` q ) ) = ( abs ` ( q - p ) ) )
54	14, 16	ovresd 6801	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) = ( ( (QQHom ` R ) ` p ) ( dist ` R ) ( (QQHom ` R ) ` q ) ) )
55		qsscn 11799	. . . . . . . . . . . . . 14 |- QQ C_ CC
56	55, 23	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> p e. CC )
57	55, 22	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> q e. CC )
58		eqid 2622	. . . . . . . . . . . . . 14 |- ( abs o. - ) = ( abs o. - )
59	58	cnmetdval 22574	. . . . . . . . . . . . 13 |- ( ( p e. CC /\ q e. CC ) -> ( p ( abs o. - ) q ) = ( abs ` ( p - q ) ) )
60	56, 57, 59	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( p ( abs o. - ) q ) = ( abs ` ( p - q ) ) )
61	23, 22	ovresd 6801	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) = ( p ( abs o. - ) q ) )
62	57, 56	abssubd 14192	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( abs ` ( q - p ) ) = ( abs ` ( p - q ) ) )
63	60, 61, 62	3eqtr4d 2666	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) = ( abs ` ( q - p ) ) )
64	53, 54, 63	3eqtr4rd 2667	. . . . . . . . . 10 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) = ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) )
65	64	breq1d 4663	. . . . . . . . 9 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e <-> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
66	65	biimpd 219	. . . . . . . 8 |- ( ( ( ph /\ p e. QQ ) /\ q e. QQ ) -> ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
67	66	ralrimiva 2966	. . . . . . 7 |- ( ( ph /\ p e. QQ ) -> A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
68	67	ralrimiva 2966	. . . . . 6 |- ( ph -> A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
69	68	adantr 481	. . . . 5 |- ( ( ph /\ e e. RR+ ) -> A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
70		breq2 4657	. . . . . . . 8 |- ( d = e -> ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d <-> ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e ) )
71	70	imbi1d 331	. . . . . . 7 |- ( d = e -> ( ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) <-> ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) ) )
72	71	2ralbidv 2989	. . . . . 6 |- ( d = e -> ( A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) <-> A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) ) )
73	72	rspcev 3309	. . . . 5 |- ( ( e e. RR+ /\ A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < e -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) ) -> E. d e. RR+ A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
74	8, 69, 73	syl2anc 693	. . . 4 |- ( ( ph /\ e e. RR+ ) -> E. d e. RR+ A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
75	74	ralrimiva 2966	. . 3 |- ( ph -> A. e e. RR+ E. d e. RR+ A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) )
76		eqid 2622	. . . 4 |- (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) = (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) )
77		qqhucn.v	. . . 4 |- V = (metUnif ` ( ( dist ` R ) |` ( B X. B ) ) )
78		0z 11388	. . . . . 6 |- 0 e. ZZ
79		zq 11794	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
80		ne0i 3921	. . . . . 6 |- ( 0 e. QQ -> QQ =/= (/) )
81	78, 79, 80	mp2b 10	. . . . 5 |- QQ =/= (/)
82	81	a1i 11	. . . 4 |- ( ph -> QQ =/= (/) )
83		drngring 18754	. . . . 5 |- ( R e. DivRing -> R e. Ring )
84		eqid 2622	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
85	3, 84	ringidcl 18568	. . . . 5 |- ( R e. Ring -> ( 1r ` R ) e. B )
86		ne0i 3921	. . . . 5 |- ( ( 1r ` R ) e. B -> B =/= (/) )
87	1, 83, 85, 86	4syl 19	. . . 4 |- ( ph -> B =/= (/) )
88		cnfldxms 22580	. . . . . . . 8 |- ℂfld e. *MetSp
89		qex 11800	. . . . . . . 8 |- QQ e. _V
90		ressxms 22330	. . . . . . . 8 |- ( (ℂfld e. *MetSp /\ QQ e. _V ) -> (ℂfld ↾s QQ ) e. *MetSp )
91	88, 89, 90	mp2an 708	. . . . . . 7 |- (ℂfld ↾s QQ ) e. *MetSp
92	29, 91	eqeltri 2697	. . . . . 6 |- Q e. *MetSp
93		cnfldds 19756	. . . . . . . . 9 |- ( abs o. - ) = ( dist ` ℂfld )
94	29, 93	ressds 16073	. . . . . . . 8 |- ( QQ e. _V -> ( abs o. - ) = ( dist ` Q ) )
95	89, 94	ax-mp 5	. . . . . . 7 |- ( abs o. - ) = ( dist ` Q )
96	38, 95	xmsxmet2 22264	. . . . . 6 |- ( Q e. *MetSp -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( *Met ` QQ ) )
97	92, 96	mp1i 13	. . . . 5 |- ( ph -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( *Met ` QQ ) )
98		xmetpsmet 22153	. . . . 5 |- ( ( ( abs o. - ) |` ( QQ X. QQ ) ) e. ( *Met ` QQ ) -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. (PsMet ` QQ ) )
99	97, 98	syl 17	. . . 4 |- ( ph -> ( ( abs o. - ) |` ( QQ X. QQ ) ) e. (PsMet ` QQ ) )
100		ngpxms 22405	. . . . . 6 |- ( R e. NrmGrp -> R e. *MetSp )
101	3, 19	xmsxmet2 22264	. . . . . 6 |- ( R e. *MetSp -> ( ( dist ` R ) |` ( B X. B ) ) e. ( *Met ` B ) )
102	9, 10, 100, 101	4syl 19	. . . . 5 |- ( ph -> ( ( dist ` R ) |` ( B X. B ) ) e. ( *Met ` B ) )
103		xmetpsmet 22153	. . . . 5 |- ( ( ( dist ` R ) |` ( B X. B ) ) e. ( *Met ` B ) -> ( ( dist ` R ) |` ( B X. B ) ) e. (PsMet ` B ) )
104	102, 103	syl 17	. . . 4 |- ( ph -> ( ( dist ` R ) |` ( B X. B ) ) e. (PsMet ` B ) )
105	76, 77, 82, 87, 99, 104	metucn 22376	. . 3 |- ( ph -> ( (QQHom ` R ) e. ( (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) Cnu V ) <-> ( (QQHom ` R ) : QQ --> B /\ A. e e. RR+ E. d e. RR+ A. p e. QQ A. q e. QQ ( ( p ( ( abs o. - ) |` ( QQ X. QQ ) ) q ) < d -> ( ( (QQHom ` R ) ` p ) ( ( dist ` R ) |` ( B X. B ) ) ( (QQHom ` R ) ` q ) ) < e ) ) ) )
106	7, 75, 105	mpbir2and 957	. 2 |- ( ph -> (QQHom ` R ) e. ( (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) Cnu V ) )
107		qqhucn.u	. . . . . 6 |- U = (UnifSt ` Q )
108	29	fveq2i 6194	. . . . . 6 |- (UnifSt ` Q ) = (UnifSt ` (ℂfld ↾s QQ ) )
109		ressuss 22067	. . . . . . 7 |- ( QQ e. _V -> (UnifSt ` (ℂfld ↾s QQ ) ) = ( (UnifSt ` ℂfld ) ↾t ( QQ X. QQ ) ) )
110	89, 109	ax-mp 5	. . . . . 6 |- (UnifSt ` (ℂfld ↾s QQ ) ) = ( (UnifSt ` ℂfld ) ↾t ( QQ X. QQ ) )
111	107, 108, 110	3eqtri 2648	. . . . 5 |- U = ( (UnifSt ` ℂfld ) ↾t ( QQ X. QQ ) )
112		eqid 2622	. . . . . . 7 |- (UnifSt ` ℂfld ) = (UnifSt ` ℂfld )
113	112	cnflduss 23152	. . . . . 6 |- (UnifSt ` ℂfld ) = (metUnif ` ( abs o. - ) )
114	113	oveq1i 6660	. . . . 5 |- ( (UnifSt ` ℂfld ) ↾t ( QQ X. QQ ) ) = ( (metUnif ` ( abs o. - ) ) ↾t ( QQ X. QQ ) )
115		cnxmet 22576	. . . . . . 7 |- ( abs o. - ) e. ( *Met ` CC )
116		xmetpsmet 22153	. . . . . . 7 |- ( ( abs o. - ) e. ( *Met ` CC ) -> ( abs o. - ) e. (PsMet ` CC ) )
117	115, 116	ax-mp 5	. . . . . 6 |- ( abs o. - ) e. (PsMet ` CC )
118		restmetu 22375	. . . . . 6 |- ( ( QQ =/= (/) /\ ( abs o. - ) e. (PsMet ` CC ) /\ QQ C_ CC ) -> ( (metUnif ` ( abs o. - ) ) ↾t ( QQ X. QQ ) ) = (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) )
119	81, 117, 55, 118	mp3an 1424	. . . . 5 |- ( (metUnif ` ( abs o. - ) ) ↾t ( QQ X. QQ ) ) = (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) )
120	111, 114, 119	3eqtri 2648	. . . 4 |- U = (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) )
121	120	a1i 11	. . 3 |- ( ph -> U = (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) )
122	121	oveq1d 6665	. 2 |- ( ph -> ( U Cnu V ) = ( (metUnif ` ( ( abs o. - ) |` ( QQ X. QQ ) ) ) Cnu V ) )
123	106, 122	eleqtrrd 2704	1 |- ( ph -> (QQHom ` R ) e. ( U Cnu V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   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   ZZcz 11377   QQcq 11788   RR+crp 11832   abscabs 13974   Basecbs 15857   ↾_s cress 15858   distcds 15950   ↾_t crest 16081   -gcsg 17424  SubGrpcsubg 17588   GrpHom cghm 17657   1rcur 18501   Ringcrg 18547  /_rcdvr 18682   DivRingcdr 18747  SubRingcsubrg 18776  PsMetcpsmet 19730   *Metcxmt 19731  metUnifcmetu 19737  ℂ_fldccnfld 19746   ZRHomczrh 19848   ZModczlm 19849  chrcchr 19850  UnifStcuss 22057   Cn_ucucn 22079   *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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  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-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  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-nzr 19258  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-metu 19745  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-fil 21650  df-ust 22004  df-uss 22060  df-ucn 22080  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-qqh 30017

This theorem is referenced by:  rrhcn  30041