Theorem rrhre 24340

Description: The RRHom homomorphism for the real numbers structure is the identity. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Assertion

Ref	Expression
rrhre	|- (RRHom ` (ℂfld ↾s RR ) ) = ( _I |` RR )

Proof of Theorem rrhre

Dummy variables a b x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniretop 18749	. . 3 |- RR = U. ( topGen ` ran (,) )
2		rehaus 18783	. . . 4 |- ( topGen ` ran (,) ) e. Haus
3	2	a1i 11	. . 3 |- ( T. -> ( topGen ` ran (,) ) e. Haus )
4		reex 9037	. . . . . 6 |- RR e. _V
5		eqid 2404	. . . . . . 7 |- (ℂfld ↾s RR ) = (ℂfld ↾s RR )
6		cnfldds 16668	. . . . . . 7 |- ( abs o. - ) = ( dist ` ℂfld )
7	5, 6	ressds 13596	. . . . . 6 |- ( RR e. _V -> ( abs o. - ) = ( dist ` (ℂfld ↾s RR ) ) )
8	4, 7	ax-mp 8	. . . . 5 |- ( abs o. - ) = ( dist ` (ℂfld ↾s RR ) )
9	8	reseq1i 5101	. . . 4 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( dist ` (ℂfld ↾s RR ) ) |` ( RR X. RR ) )
10		eqid 2404	. . . 4 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
11	5	rebase 24222	. . . 4 |- RR = ( Base ` (ℂfld ↾s RR ) )
12		eqid 2404	. . . . 5 |- ( TopOpen ` (ℂfld ↾s RR ) ) = ( TopOpen ` (ℂfld ↾s RR ) )
13	12	tgioo3 18789	. . . 4 |- ( topGen ` ran (,) ) = ( TopOpen ` (ℂfld ↾s RR ) )
14	5	refld 24232	. . . . . . 7 |- (ℂfld ↾s RR ) e. Field
15		isfld 15799	. . . . . . 7 |- ( (ℂfld ↾s RR ) e. Field <-> ( (ℂfld ↾s RR ) e. DivRing /\ (ℂfld ↾s RR ) e. CRing ) )
16	14, 15	mpbi 200	. . . . . 6 |- ( (ℂfld ↾s RR ) e. DivRing /\ (ℂfld ↾s RR ) e. CRing )
17	16	simpli 445	. . . . 5 |- (ℂfld ↾s RR ) e. DivRing
18	17	a1i 11	. . . 4 |- ( T. -> (ℂfld ↾s RR ) e. DivRing )
19		cnnrg 18768	. . . . . 6 |- ℂfld e. NrmRing
20		resubdrg 16705	. . . . . . 7 |- ( RR e. (SubRing ` ℂfld ) /\ (ℂfld ↾s RR ) e. DivRing )
21	20	simpli 445	. . . . . 6 |- RR e. (SubRing ` ℂfld )
22	5	subrgnrg 18662	. . . . . 6 |- ( (ℂfld e. NrmRing /\ RR e. (SubRing ` ℂfld ) ) -> (ℂfld ↾s RR ) e. NrmRing )
23	19, 21, 22	mp2an 654	. . . . 5 |- (ℂfld ↾s RR ) e. NrmRing
24	23	a1i 11	. . . 4 |- ( T. -> (ℂfld ↾s RR ) e. NrmRing )
25	5	rezh 24308	. . . . 5 |- ( ZMod ` (ℂfld ↾s RR ) ) e. NrmMod
26	25	a1i 11	. . . 4 |- ( T. -> ( ZMod ` (ℂfld ↾s RR ) ) e. NrmMod )
27	5	reofld 24233	. . . . . 6 |- (ℂfld ↾s RR ) e. oField
28		ofldchr 24197	. . . . . 6 |- ( (ℂfld ↾s RR ) e. oField -> (chr ` (ℂfld ↾s RR ) ) = 0 )
29	27, 28	ax-mp 8	. . . . 5 |- (chr ` (ℂfld ↾s RR ) ) = 0
30	29	a1i 11	. . . 4 |- ( T. -> (chr ` (ℂfld ↾s RR ) ) = 0 )
31	5	recms 24296	. . . . . 6 |- (ℂfld ↾s RR ) e. CMetSp
32		cmsms 19254	. . . . . 6 |- ( (ℂfld ↾s RR ) e. CMetSp -> (ℂfld ↾s RR ) e. MetSp )
33		mstps 18438	. . . . . 6 |- ( (ℂfld ↾s RR ) e. MetSp -> (ℂfld ↾s RR ) e. TopSp )
34	31, 32, 33	mp2b 10	. . . . 5 |- (ℂfld ↾s RR ) e. TopSp
35	34	a1i 11	. . . 4 |- ( T. -> (ℂfld ↾s RR ) e. TopSp )
36	5	recusp 24298	. . . . 5 |- (ℂfld ↾s RR ) e. CUnifSp
37	36	a1i 11	. . . 4 |- ( T. -> (ℂfld ↾s RR ) e. CUnifSp )
38		ressuss 18246	. . . . . . 7 |- ( RR e. _V -> (UnifSt ` (ℂfld ↾s RR ) ) = ( (UnifSt ` ℂfld ) ↾t ( RR X. RR ) ) )
39	4, 38	ax-mp 8	. . . . . 6 |- (UnifSt ` (ℂfld ↾s RR ) ) = ( (UnifSt ` ℂfld ) ↾t ( RR X. RR ) )
40		eqid 2404	. . . . . . . 8 |- (UnifSt ` ℂfld ) = (UnifSt ` ℂfld )
41	40	cnflduss 19263	. . . . . . 7 |- (UnifSt ` ℂfld ) = (metUnif ` ( abs o. - ) )
42	41	oveq1i 6050	. . . . . 6 |- ( (UnifSt ` ℂfld ) ↾t ( RR X. RR ) ) = ( (metUnif ` ( abs o. - ) ) ↾t ( RR X. RR ) )
43		0re 9047	. . . . . . . 8 |- 0 e. RR
44		ne0i 3594	. . . . . . . 8 |- ( 0 e. RR -> RR =/= (/) )
45	43, 44	ax-mp 8	. . . . . . 7 |- RR =/= (/)
46		cnxmet 18760	. . . . . . . 8 |- ( abs o. - ) e. ( * Met ` CC )
47		xmetpsmet 18331	. . . . . . . 8 |- ( ( abs o. - ) e. ( * Met ` CC ) -> ( abs o. - ) e. (PsMet ` CC ) )
48	46, 47	ax-mp 8	. . . . . . 7 |- ( abs o. - ) e. (PsMet ` CC )
49		ax-resscn 9003	. . . . . . 7 |- RR C_ CC
50		restmetu 18570	. . . . . . 7 |- ( ( RR =/= (/) /\ ( abs o. - ) e. (PsMet ` CC ) /\ RR C_ CC ) -> ( (metUnif ` ( abs o. - ) ) ↾t ( RR X. RR ) ) = (metUnif ` ( ( abs o. - ) |` ( RR X. RR ) ) ) )
51	45, 48, 49, 50	mp3an 1279	. . . . . 6 |- ( (metUnif ` ( abs o. - ) ) ↾t ( RR X. RR ) ) = (metUnif ` ( ( abs o. - ) |` ( RR X. RR ) ) )
52	39, 42, 51	3eqtri 2428	. . . . 5 |- (UnifSt ` (ℂfld ↾s RR ) ) = (metUnif ` ( ( abs o. - ) |` ( RR X. RR ) ) )
53	52	a1i 11	. . . 4 |- ( T. -> (UnifSt ` (ℂfld ↾s RR ) ) = (metUnif ` ( ( abs o. - ) |` ( RR X. RR ) ) ) )
54	9, 10, 11, 13, 18, 24, 26, 30, 35, 37, 3, 53	rrhcn 24333	. . 3 |- ( T. -> (RRHom ` (ℂfld ↾s RR ) ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
55		retop 18748	. . . . . 6 |- ( topGen ` ran (,) ) e. Top
56	1	toptopon 16953	. . . . . 6 |- ( ( topGen ` ran (,) ) e. Top <-> ( topGen ` ran (,) ) e. (TopOn ` RR ) )
57	55, 56	mpbi 200	. . . . 5 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
58		idcn 17275	. . . . 5 |- ( ( topGen ` ran (,) ) e. (TopOn ` RR ) -> ( _I |` RR ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
59	57, 58	ax-mp 8	. . . 4 |- ( _I |` RR ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) )
60	59	a1i 11	. . 3 |- ( T. -> ( _I |` RR ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
61	55	a1i 11	. . . . . . 7 |- ( T. -> ( topGen ` ran (,) ) e. Top )
62		f1oi 5672	. . . . . . . . . 10 |- ( _I |` QQ ) : QQ -1-1-onto-> QQ
63		f1of 5633	. . . . . . . . . 10 |- ( ( _I |` QQ ) : QQ -1-1-onto-> QQ -> ( _I |` QQ ) : QQ --> QQ )
64	62, 63	ax-mp 8	. . . . . . . . 9 |- ( _I |` QQ ) : QQ --> QQ
65		qssre 10540	. . . . . . . . 9 |- QQ C_ RR
66		fss 5558	. . . . . . . . 9 |- ( ( ( _I |` QQ ) : QQ --> QQ /\ QQ C_ RR ) -> ( _I |` QQ ) : QQ --> RR )
67	64, 65, 66	mp2an 654	. . . . . . . 8 |- ( _I |` QQ ) : QQ --> RR
68	67	a1i 11	. . . . . . 7 |- ( T. -> ( _I |` QQ ) : QQ --> RR )
69	65	a1i 11	. . . . . . 7 |- ( T. -> QQ C_ RR )
70		qdensere 18757	. . . . . . . 8 |- ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) = RR
71	70	a1i 11	. . . . . . 7 |- ( T. -> ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) = RR )
72	55	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) /\ x e. a ) -> ( topGen ` ran (,) ) e. Top )
73		simplr 732	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) /\ x e. a ) -> a e. ( topGen ` ran (,) ) )
74		simpr 448	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) /\ x e. a ) -> x e. a )
75		opnneip 17138	. . . . . . . . . . . . . . . 16 |- ( ( ( topGen ` ran (,) ) e. Top /\ a e. ( topGen ` ran (,) ) /\ x e. a ) -> a e. ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) )
76	72, 73, 74, 75	syl3anc 1184	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) /\ x e. a ) -> a e. ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) )
77		fvex 5701	. . . . . . . . . . . . . . . 16 |- ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) e. _V
78		qex 10542	. . . . . . . . . . . . . . . 16 |- QQ e. _V
79		elrestr 13611	. . . . . . . . . . . . . . . 16 |- ( ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) e. _V /\ QQ e. _V /\ a e. ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ) -> ( a i^i QQ ) e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) )
80	77, 78, 79	mp3an12 1269	. . . . . . . . . . . . . . 15 |- ( a e. ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) -> ( a i^i QQ ) e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) )
81	76, 80	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) /\ x e. a ) -> ( a i^i QQ ) e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) )
82		inss2 3522	. . . . . . . . . . . . . . . . 17 |- ( a i^i QQ ) C_ QQ
83		resiima 5179	. . . . . . . . . . . . . . . . 17 |- ( ( a i^i QQ ) C_ QQ -> ( ( _I |` QQ ) " ( a i^i QQ ) ) = ( a i^i QQ ) )
84	82, 83	ax-mp 8	. . . . . . . . . . . . . . . 16 |- ( ( _I |` QQ ) " ( a i^i QQ ) ) = ( a i^i QQ )
85		inss1 3521	. . . . . . . . . . . . . . . 16 |- ( a i^i QQ ) C_ a
86	84, 85	eqsstri 3338	. . . . . . . . . . . . . . 15 |- ( ( _I |` QQ ) " ( a i^i QQ ) ) C_ a
87	86	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) /\ x e. a ) -> ( ( _I |` QQ ) " ( a i^i QQ ) ) C_ a )
88		imaeq2 5158	. . . . . . . . . . . . . . . 16 |- ( b = ( a i^i QQ ) -> ( ( _I |` QQ ) " b ) = ( ( _I |` QQ ) " ( a i^i QQ ) ) )
89	88	sseq1d 3335	. . . . . . . . . . . . . . 15 |- ( b = ( a i^i QQ ) -> ( ( ( _I |` QQ ) " b ) C_ a <-> ( ( _I |` QQ ) " ( a i^i QQ ) ) C_ a ) )
90	89	rspcev 3012	. . . . . . . . . . . . . 14 |- ( ( ( a i^i QQ ) e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) /\ ( ( _I |` QQ ) " ( a i^i QQ ) ) C_ a ) -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a )
91	81, 87, 90	syl2anc 643	. . . . . . . . . . . . 13 |- ( ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) /\ x e. a ) -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a )
92	91	ex 424	. . . . . . . . . . . 12 |- ( ( x e. RR /\ a e. ( topGen ` ran (,) ) ) -> ( x e. a -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a ) )
93	92	ralrimiva 2749	. . . . . . . . . . 11 |- ( x e. RR -> A. a e. ( topGen ` ran (,) ) ( x e. a -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a ) )
94	93	ancli 535	. . . . . . . . . 10 |- ( x e. RR -> ( x e. RR /\ A. a e. ( topGen ` ran (,) ) ( x e. a -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a ) ) )
95	70	eleq2i 2468	. . . . . . . . . . . . 13 |- ( x e. ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) <-> x e. RR )
96	95	biimpri 198	. . . . . . . . . . . 12 |- ( x e. RR -> x e. ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) )
97		trnei 17877	. . . . . . . . . . . . 13 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ QQ C_ RR /\ x e. RR ) -> ( x e. ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) <-> ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) e. ( Fil ` QQ ) ) )
98	57, 65, 97	mp3an12 1269	. . . . . . . . . . . 12 |- ( x e. RR -> ( x e. ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) <-> ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) e. ( Fil ` QQ ) ) )
99	96, 98	mpbid 202	. . . . . . . . . . 11 |- ( x e. RR -> ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) e. ( Fil ` QQ ) )
100		isflf 17978	. . . . . . . . . . . 12 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) e. ( Fil ` QQ ) /\ ( _I |` QQ ) : QQ --> RR ) -> ( x e. ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) <-> ( x e. RR /\ A. a e. ( topGen ` ran (,) ) ( x e. a -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a ) ) ) )
101	57, 67, 100	mp3an13 1270	. . . . . . . . . . 11 |- ( ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) e. ( Fil ` QQ ) -> ( x e. ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) <-> ( x e. RR /\ A. a e. ( topGen ` ran (,) ) ( x e. a -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a ) ) ) )
102	99, 101	syl 16	. . . . . . . . . 10 |- ( x e. RR -> ( x e. ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) <-> ( x e. RR /\ A. a e. ( topGen ` ran (,) ) ( x e. a -> E. b e. ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ( ( _I |` QQ ) " b ) C_ a ) ) ) )
103	94, 102	mpbird 224	. . . . . . . . 9 |- ( x e. RR -> x e. ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) )
104		ne0i 3594	. . . . . . . . 9 |- ( x e. ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) -> ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) =/= (/) )
105	103, 104	syl 16	. . . . . . . 8 |- ( x e. RR -> ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) =/= (/) )
106	105	adantl 453	. . . . . . 7 |- ( ( T. /\ x e. RR ) -> ( ( ( topGen ` ran (,) ) fLimf ( ( ( nei ` ( topGen ` ran (,) ) ) ` { x } ) ↾t QQ ) ) ` ( _I |` QQ ) ) =/= (/) )
107		cuspusp 18283	. . . . . . . . . 10 |- ( (ℂfld ↾s RR ) e. CUnifSp -> (ℂfld ↾s RR ) e. UnifSp )
108	36, 107	ax-mp 8	. . . . . . . . 9 |- (ℂfld ↾s RR ) e. UnifSp
109	13	uspreg 18257	. . . . . . . . 9 |- ( ( (ℂfld ↾s RR ) e. UnifSp /\ ( topGen ` ran (,) ) e. Haus ) -> ( topGen ` ran (,) ) e. Reg )
110	108, 2, 109	mp2an 654	. . . . . . . 8 |- ( topGen ` ran (,) ) e. Reg
111	110	a1i 11	. . . . . . 7 |- ( T. -> ( topGen ` ran (,) ) e. Reg )
112		resabs1 5134	. . . . . . . . . 10 |- ( QQ C_ RR -> ( ( _I |` RR ) |` QQ ) = ( _I |` QQ ) )
113	65, 112	ax-mp 8	. . . . . . . . 9 |- ( ( _I |` RR ) |` QQ ) = ( _I |` QQ )
114	1	cnrest 17303	. . . . . . . . . 10 |- ( ( ( _I |` RR ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) /\ QQ C_ RR ) -> ( ( _I |` RR ) |` QQ ) e. ( ( ( topGen ` ran (,) ) ↾t QQ ) Cn ( topGen ` ran (,) ) ) )
115	59, 65, 114	mp2an 654	. . . . . . . . 9 |- ( ( _I |` RR ) |` QQ ) e. ( ( ( topGen ` ran (,) ) ↾t QQ ) Cn ( topGen ` ran (,) ) )
116	113, 115	eqeltrri 2475	. . . . . . . 8 |- ( _I |` QQ ) e. ( ( ( topGen ` ran (,) ) ↾t QQ ) Cn ( topGen ` ran (,) ) )
117	116	a1i 11	. . . . . . 7 |- ( T. -> ( _I |` QQ ) e. ( ( ( topGen ` ran (,) ) ↾t QQ ) Cn ( topGen ` ran (,) ) ) )
118	1, 1, 61, 3, 68, 69, 71, 106, 111, 117	cnextfres 18052	. . . . . 6 |- ( T. -> ( ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` ( _I |` QQ ) ) |` QQ ) = ( _I |` QQ ) )
119	118	trud 1329	. . . . 5 |- ( ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` ( _I |` QQ ) ) |` QQ ) = ( _I |` QQ )
120	31	elexi 2925	. . . . . . . 8 |- (ℂfld ↾s RR ) e. _V
121	10, 13	rrhval 24332	. . . . . . . 8 |- ( (ℂfld ↾s RR ) e. _V -> (RRHom ` (ℂfld ↾s RR ) ) = ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` (QQHom ` (ℂfld ↾s RR ) ) ) )
122	120, 121	ax-mp 8	. . . . . . 7 |- (RRHom ` (ℂfld ↾s RR ) ) = ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` (QQHom ` (ℂfld ↾s RR ) ) )
123		qqhre 24339	. . . . . . . 8 |- (QQHom ` (ℂfld ↾s RR ) ) = ( _I |` QQ )
124	123	fveq2i 5690	. . . . . . 7 |- ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` (QQHom ` (ℂfld ↾s RR ) ) ) = ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` ( _I |` QQ ) )
125	122, 124	eqtri 2424	. . . . . 6 |- (RRHom ` (ℂfld ↾s RR ) ) = ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` ( _I |` QQ ) )
126	125	reseq1i 5101	. . . . 5 |- ( (RRHom ` (ℂfld ↾s RR ) ) |` QQ ) = ( ( ( ( topGen ` ran (,) )CnExt ( topGen ` ran (,) ) ) ` ( _I |` QQ ) ) |` QQ )
127	119, 126, 113	3eqtr4i 2434	. . . 4 |- ( (RRHom ` (ℂfld ↾s RR ) ) |` QQ ) = ( ( _I |` RR ) |` QQ )
128	127	a1i 11	. . 3 |- ( T. -> ( (RRHom ` (ℂfld ↾s RR ) ) |` QQ ) = ( ( _I |` RR ) |` QQ ) )
129	1, 3, 54, 60, 128, 69, 71	hauseqcn 24246	. 2 |- ( T. -> (RRHom ` (ℂfld ↾s RR ) ) = ( _I |` RR ) )
130	129	trud 1329	1 |- (RRHom ` (ℂfld ↾s RR ) ) = ( _I |` RR )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   T. wtru 1322   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916   i^i cin 3279   C_ wss 3280   (/)c0 3588   {csn 3774   _I cid 4453   X. cxp 4835   ran crn 4838   |` cres 4839   "cima 4840   o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   - cmin 9247   QQcq 10530   (,)cioo 10872   abscabs 11994   ↾_s cress 13425   distcds 13493   ↾_t crest 13603   TopOpenctopn 13604   topGenctg 13620   CRingccrg 15616   DivRingcdr 15790  Fieldcfield 15791  SubRingcsubrg 15819  PsMetcpsmet 16640   * Metcxmt 16641  metUnifcmetu 16648  ℂ_fldccnfld 16658   ZModczlm 16734  chrcchr 16735   Topctop 16913  TopOnctopon 16914   TopSpctps 16916   clsccl 17037   neicnei 17116   Cn ccn 17242   Hauscha 17326   Regcreg 17327   Filcfil 17830   fLimf cflf 17920  CnExtccnext 18043  UnifStcuss 18236  UnifSpcusp 18237  CUnifSpccusp 18280   MetSpcmt 18301  NrmRingcnrg 18580  NrmModcnlm 18581  CMetSpccms 19238  oFieldcofld 24186  QQHomcqqh 24309  RRHomcrrh 24330

This theorem is referenced by:  sitmcl  24616

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-toset 14418  df-ps 14584  df-tsr 14585  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-od 15122  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-field 15793  df-subrg 15821  df-abv 15860  df-lmod 15907  df-nzr 16284  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-metu 16657  df-cnfld 16659  df-zrh 16737  df-zlm 16738  df-chr 16739  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-haus 17333  df-reg 17334  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-fcls 17926  df-cnext 18044  df-ust 18183  df-utop 18214  df-uss 18239  df-usp 18240  df-ucn 18259  df-cfilu 18270  df-cusp 18281  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-cncf 18861  df-cfil 19161  df-cmet 19163  df-cms 19241  df-ofld 24187  df-qqh 24310  df-rrh 24331