Metamath Proof Explorer


Theorem rrhre

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

Ref Expression
Assertion rrhre ( ℝHom ‘ ℝfld ) = ( I ↾ ℝ )

Proof

Step Hyp Ref Expression
1 uniretop ℝ = ( topGen ‘ ran (,) )
2 rehaus ( topGen ‘ ran (,) ) ∈ Haus
3 2 a1i ( ⊤ → ( topGen ‘ ran (,) ) ∈ Haus )
4 rerrext fld ∈ ℝExt
5 eqid ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
6 retopn ( topGen ‘ ran (,) ) = ( TopOpen ‘ ℝfld )
7 5 6 rrhcne ( ℝfld ∈ ℝExt → ( ℝHom ‘ ℝfld ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) )
8 4 7 mp1i ( ⊤ → ( ℝHom ‘ ℝfld ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) )
9 retop ( topGen ‘ ran (,) ) ∈ Top
10 1 toptopon ( ( topGen ‘ ran (,) ) ∈ Top ↔ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) )
11 9 10 mpbi ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
12 idcn ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) → ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) )
13 11 12 ax-mp ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) )
14 13 a1i ( ⊤ → ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) )
15 9 a1i ( ⊤ → ( topGen ‘ ran (,) ) ∈ Top )
16 f1oi ( I ↾ ℚ ) : ℚ –1-1-onto→ ℚ
17 f1of ( ( I ↾ ℚ ) : ℚ –1-1-onto→ ℚ → ( I ↾ ℚ ) : ℚ ⟶ ℚ )
18 16 17 ax-mp ( I ↾ ℚ ) : ℚ ⟶ ℚ
19 qssre ℚ ⊆ ℝ
20 fss ( ( ( I ↾ ℚ ) : ℚ ⟶ ℚ ∧ ℚ ⊆ ℝ ) → ( I ↾ ℚ ) : ℚ ⟶ ℝ )
21 18 19 20 mp2an ( I ↾ ℚ ) : ℚ ⟶ ℝ
22 21 a1i ( ⊤ → ( I ↾ ℚ ) : ℚ ⟶ ℝ )
23 19 a1i ( ⊤ → ℚ ⊆ ℝ )
24 qdensere ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) = ℝ
25 24 a1i ( ⊤ → ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) = ℝ )
26 9 a1i ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥𝑎 ) → ( topGen ‘ ran (,) ) ∈ Top )
27 simplr ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥𝑎 ) → 𝑎 ∈ ( topGen ‘ ran (,) ) )
28 simpr ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥𝑎 ) → 𝑥𝑎 )
29 opnneip ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ∧ 𝑥𝑎 ) → 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) )
30 26 27 28 29 syl3anc ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥𝑎 ) → 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) )
31 fvex ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ∈ V
32 qex ℚ ∈ V
33 elrestr ( ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ∈ V ∧ ℚ ∈ V ∧ 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ) → ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) )
34 31 32 33 mp3an12 ( 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) → ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) )
35 30 34 syl ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥𝑎 ) → ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) )
36 inss2 ( 𝑎 ∩ ℚ ) ⊆ ℚ
37 resiima ( ( 𝑎 ∩ ℚ ) ⊆ ℚ → ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) = ( 𝑎 ∩ ℚ ) )
38 36 37 ax-mp ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) = ( 𝑎 ∩ ℚ )
39 inss1 ( 𝑎 ∩ ℚ ) ⊆ 𝑎
40 38 39 eqsstri ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎
41 40 a1i ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥𝑎 ) → ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎 )
42 imaeq2 ( 𝑏 = ( 𝑎 ∩ ℚ ) → ( ( I ↾ ℚ ) “ 𝑏 ) = ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) )
43 42 sseq1d ( 𝑏 = ( 𝑎 ∩ ℚ ) → ( ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ↔ ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎 ) )
44 43 rspcev ( ( ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∧ ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎 ) → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 )
45 35 41 44 syl2anc ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥𝑎 ) → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 )
46 45 ex ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) → ( 𝑥𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) )
47 46 ralrimiva ( 𝑥 ∈ ℝ → ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) )
48 47 ancli ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) )
49 24 eleq2i ( 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) ↔ 𝑥 ∈ ℝ )
50 49 biimpri ( 𝑥 ∈ ℝ → 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) )
51 trnei ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ℚ ⊆ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) ↔ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) ) )
52 11 19 51 mp3an12 ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) ↔ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) ) )
53 50 52 mpbid ( 𝑥 ∈ ℝ → ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) )
54 isflf ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) ∧ ( I ↾ ℚ ) : ℚ ⟶ ℝ ) → ( 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ↔ ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) ) )
55 11 21 54 mp3an13 ( ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) → ( 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ↔ ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) ) )
56 53 55 syl ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ↔ ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) ) )
57 48 56 mpbird ( 𝑥 ∈ ℝ → 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) )
58 57 ne0d ( 𝑥 ∈ ℝ → ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ≠ ∅ )
59 58 adantl ( ( ⊤ ∧ 𝑥 ∈ ℝ ) → ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ≠ ∅ )
60 recusp fld ∈ CUnifSp
61 cuspusp ( ℝfld ∈ CUnifSp → ℝfld ∈ UnifSp )
62 60 61 ax-mp fld ∈ UnifSp
63 6 uspreg ( ( ℝfld ∈ UnifSp ∧ ( topGen ‘ ran (,) ) ∈ Haus ) → ( topGen ‘ ran (,) ) ∈ Reg )
64 62 2 63 mp2an ( topGen ‘ ran (,) ) ∈ Reg
65 64 a1i ( ⊤ → ( topGen ‘ ran (,) ) ∈ Reg )
66 resabs1 ( ℚ ⊆ ℝ → ( ( I ↾ ℝ ) ↾ ℚ ) = ( I ↾ ℚ ) )
67 19 66 ax-mp ( ( I ↾ ℝ ) ↾ ℚ ) = ( I ↾ ℚ )
68 1 cnrest ( ( ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) ∧ ℚ ⊆ ℝ ) → ( ( I ↾ ℝ ) ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) ) )
69 13 19 68 mp2an ( ( I ↾ ℝ ) ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) )
70 67 69 eqeltrri ( I ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) )
71 70 a1i ( ⊤ → ( I ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) ) )
72 1 1 15 3 22 23 25 59 65 71 cnextfres1 ( ⊤ → ( ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) ↾ ℚ ) = ( I ↾ ℚ ) )
73 72 mptru ( ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) ↾ ℚ ) = ( I ↾ ℚ )
74 recms fld ∈ CMetSp
75 74 elexi fld ∈ V
76 5 6 rrhval ( ℝfld ∈ V → ( ℝHom ‘ ℝfld ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( ℚHom ‘ ℝfld ) ) )
77 75 76 ax-mp ( ℝHom ‘ ℝfld ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( ℚHom ‘ ℝfld ) )
78 qqhre ( ℚHom ‘ ℝfld ) = ( I ↾ ℚ )
79 78 fveq2i ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( ℚHom ‘ ℝfld ) ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) )
80 77 79 eqtri ( ℝHom ‘ ℝfld ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) )
81 80 reseq1i ( ( ℝHom ‘ ℝfld ) ↾ ℚ ) = ( ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) ↾ ℚ )
82 73 81 67 3eqtr4i ( ( ℝHom ‘ ℝfld ) ↾ ℚ ) = ( ( I ↾ ℝ ) ↾ ℚ )
83 82 a1i ( ⊤ → ( ( ℝHom ‘ ℝfld ) ↾ ℚ ) = ( ( I ↾ ℝ ) ↾ ℚ ) )
84 1 3 8 14 83 23 25 hauseqcn ( ⊤ → ( ℝHom ‘ ℝfld ) = ( I ↾ ℝ ) )
85 84 mptru ( ℝHom ‘ ℝfld ) = ( I ↾ ℝ )