rrhre - Mathbox for Thierry Arnoux

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Theorem rrhre 34356

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

**Assertion**
Ref	Expression
rrhre	⊢ (ℝHom‘ℝ_fld) = ( I ↾ ℝ)

Proof of Theorem rrhre Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniretop 24888	. . 3 ⊢ ℝ = ∪ (topGen‘ran (,))
2		rehaus 24925	. . . 4 ⊢ (topGen‘ran (,)) ∈ Haus
3	2	a1i 11	. . 3 ⊢ (⊤ → (topGen‘ran (,)) ∈ Haus)
4		rerrext 34344	. . . 4 ⊢ ℝ_fld ∈ ℝExt
5		eqid 2769	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
6		retopn 25507	. . . . 5 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
7	5, 6	rrhcne 34348	. . . 4 ⊢ (ℝ_fld ∈ ℝExt → (ℝHom‘ℝ_fld) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
8	4, 7	mp1i 14	. . 3 ⊢ (⊤ → (ℝHom‘ℝ_fld) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
9		retop 24887	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
10	1	toptopon 23043	. . . . . 6 ⊢ ((topGen‘ran (,)) ∈ Top ↔ (topGen‘ran (,)) ∈ (TopOn‘ℝ))
11	9, 10	mpbi 233	. . . . 5 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
12		idcn 23383	. . . . 5 ⊢ ((topGen‘ran (,)) ∈ (TopOn‘ℝ) → ( I ↾ ℝ) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
13	11, 12	ax-mp 5	. . . 4 ⊢ ( I ↾ ℝ) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,)))
14	13	a1i 11	. . 3 ⊢ (⊤ → ( I ↾ ℝ) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
15	9	a1i 11	. . . . . . 7 ⊢ (⊤ → (topGen‘ran (,)) ∈ Top)
16		f1oi 6860	. . . . . . . . . 10 ⊢ ( I ↾ ℚ):ℚ–1-1-onto→ℚ
17		f1of 6821	. . . . . . . . . 10 ⊢ (( I ↾ ℚ):ℚ–1-1-onto→ℚ → ( I ↾ ℚ):ℚ⟶ℚ)
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ ℚ):ℚ⟶ℚ
19		qssre 12983	. . . . . . . . 9 ⊢ ℚ ⊆ ℝ
20		fss 6723	. . . . . . . . 9 ⊢ ((( I ↾ ℚ):ℚ⟶ℚ ∧ ℚ ⊆ ℝ) → ( I ↾ ℚ):ℚ⟶ℝ)
21	18, 19, 20	mp2an 704	. . . . . . . 8 ⊢ ( I ↾ ℚ):ℚ⟶ℝ
22	21	a1i 11	. . . . . . 7 ⊢ (⊤ → ( I ↾ ℚ):ℚ⟶ℝ)
23	19	a1i 11	. . . . . . 7 ⊢ (⊤ → ℚ ⊆ ℝ)
24		qdensere 24895	. . . . . . . 8 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
25	24	a1i 11	. . . . . . 7 ⊢ (⊤ → ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ)
26	9	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → (topGen‘ran (,)) ∈ Top)
27		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ (topGen‘ran (,)))
28		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
29		opnneip 23245	. . . . . . . . . . . . . . . 16 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑎 ∈ (topGen‘ran (,)) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘(topGen‘ran (,)))‘{𝑥}))
30	26, 27, 28, 29	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘(topGen‘ran (,)))‘{𝑥}))
31		fvex 6895	. . . . . . . . . . . . . . . 16 ⊢ ((nei‘(topGen‘ran (,)))‘{𝑥}) ∈ V
32		qex 12985	. . . . . . . . . . . . . . . 16 ⊢ ℚ ∈ V
33		elrestr 17481	. . . . . . . . . . . . . . . 16 ⊢ ((((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 1477	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ((nei‘(topGen‘ran (,)))‘{𝑥}) → (𝑎 ∩ ℚ) ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))
35	30, 34	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → (𝑎 ∩ ℚ) ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))
36		inss2 4198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∩ ℚ) ⊆ ℚ
37		resiima 6079	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∩ ℚ) ⊆ ℚ → (( I ↾ ℚ) “ (𝑎 ∩ ℚ)) = (𝑎 ∩ ℚ))
38	36, 37	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (( I ↾ ℚ) “ (𝑎 ∩ ℚ)) = (𝑎 ∩ ℚ)
39		inss1 4197	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∩ ℚ) ⊆ 𝑎
40	38, 39	eqsstri 3991	. . . . . . . . . . . . . . 15 ⊢ (( I ↾ ℚ) “ (𝑎 ∩ ℚ)) ⊆ 𝑎
41	40	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → (( I ↾ ℚ) “ (𝑎 ∩ ℚ)) ⊆ 𝑎)
42		imaeq2 6059	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑎 ∩ ℚ) → (( I ↾ ℚ) “ 𝑏) = (( I ↾ ℚ) “ (𝑎 ∩ ℚ)))
43	42	sseq1d 3976	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∩ ℚ) → ((( I ↾ ℚ) “ 𝑏) ⊆ 𝑎 ↔ (( I ↾ ℚ) “ (𝑎 ∩ ℚ)) ⊆ 𝑎))
44	43	rspcev 3590	. . . . . . . . . . . . . 14 ⊢ (((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 595	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎)
46	45	ex 417	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) → (𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎))
47	46	ralrimiva 3163	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → ∀𝑎 ∈ (topGen‘ran (,))(𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎))
48	47	ancli 557	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ℝ ∧ ∀𝑎 ∈ (topGen‘ran (,))(𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎)))
49	24	eleq2i 2861	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((cls‘(topGen‘ran (,)))‘ℚ) ↔ 𝑥 ∈ ℝ)
50	49	biimpri 231	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ((cls‘(topGen‘ran (,)))‘ℚ))
51		trnei 24018	. . . . . . . . . . . . 13 ⊢ (((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 1477	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ((cls‘(topGen‘ran (,)))‘ℚ) ↔ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ) ∈ (Fil‘ℚ)))
53	50, 52	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ) ∈ (Fil‘ℚ))
54		isflf 24119	. . . . . . . . . . . 12 ⊢ (((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 (,))(𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎))))
55	11, 21, 54	mp3an13 1478	. . . . . . . . . . 11 ⊢ ((((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 (,))(𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎))))
56	53, 55	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (((topGen‘ran (,)) fLimf (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))‘( I ↾ ℚ)) ↔ (𝑥 ∈ ℝ ∧ ∀𝑎 ∈ (topGen‘ran (,))(𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎))))
57	48, 56	mpbird 260	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ (((topGen‘ran (,)) fLimf (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))‘( I ↾ ℚ)))
58	57	ne0d 4303	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (((topGen‘ran (,)) fLimf (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))‘( I ↾ ℚ)) ≠ ∅)
59	58	adantl 486	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (((topGen‘ran (,)) fLimf (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))‘( I ↾ ℚ)) ≠ ∅)
60		recusp 25510	. . . . . . . . . 10 ⊢ ℝ_fld ∈ CUnifSp
61		cuspusp 24425	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ CUnifSp → ℝ_fld ∈ UnifSp)
62	60, 61	ax-mp 5	. . . . . . . . 9 ⊢ ℝ_fld ∈ UnifSp
63	6	uspreg 24399	. . . . . . . . 9 ⊢ ((ℝ_fld ∈ UnifSp ∧ (topGen‘ran (,)) ∈ Haus) → (topGen‘ran (,)) ∈ Reg)
64	62, 2, 63	mp2an 704	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ Reg
65	64	a1i 11	. . . . . . 7 ⊢ (⊤ → (topGen‘ran (,)) ∈ Reg)
66		resabs1 6006	. . . . . . . . . 10 ⊢ (ℚ ⊆ ℝ → (( I ↾ ℝ) ↾ ℚ) = ( I ↾ ℚ))
67	19, 66	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ ℝ) ↾ ℚ) = ( I ↾ ℚ)
68	1	cnrest 23411	. . . . . . . . . 10 ⊢ ((( I ↾ ℝ) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ∧ ℚ ⊆ ℝ) → (( I ↾ ℝ) ↾ ℚ) ∈ (((topGen‘ran (,)) ↾_t ℚ) Cn (topGen‘ran (,))))
69	13, 19, 68	mp2an 704	. . . . . . . . 9 ⊢ (( I ↾ ℝ) ↾ ℚ) ∈ (((topGen‘ran (,)) ↾_t ℚ) Cn (topGen‘ran (,)))
70	67, 69	eqeltrri 2866	. . . . . . . 8 ⊢ ( I ↾ ℚ) ∈ (((topGen‘ran (,)) ↾_t ℚ) Cn (topGen‘ran (,)))
71	70	a1i 11	. . . . . . 7 ⊢ (⊤ → ( I ↾ ℚ) ∈ (((topGen‘ran (,)) ↾_t ℚ) Cn (topGen‘ran (,))))
72	1, 1, 15, 3, 22, 23, 25, 59, 65, 71	cnextfres1 24194	. . . . . 6 ⊢ (⊤ → ((((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ)) ↾ ℚ) = ( I ↾ ℚ))
73	72	mptru 1574	. . . . 5 ⊢ ((((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ)) ↾ ℚ) = ( I ↾ ℚ)
74		recms 25508	. . . . . . . . 9 ⊢ ℝ_fld ∈ CMetSp
75	74	elexi 3485	. . . . . . . 8 ⊢ ℝ_fld ∈ V
76	5, 6	rrhval 34331	. . . . . . . 8 ⊢ (ℝ_fld ∈ V → (ℝHom‘ℝ_fld) = (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘(ℚHom‘ℝ_fld)))
77	75, 76	ax-mp 5	. . . . . . 7 ⊢ (ℝHom‘ℝ_fld) = (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘(ℚHom‘ℝ_fld))
78		qqhre 34355	. . . . . . . 8 ⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)
79	78	fveq2i 6885	. . . . . . 7 ⊢ (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘(ℚHom‘ℝ_fld)) = (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ))
80	77, 79	eqtri 2792	. . . . . 6 ⊢ (ℝHom‘ℝ_fld) = (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ))
81	80	reseq1i 5975	. . . . 5 ⊢ ((ℝHom‘ℝ_fld) ↾ ℚ) = ((((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ)) ↾ ℚ)
82	73, 81, 67	3eqtr4i 2802	. . . 4 ⊢ ((ℝHom‘ℝ_fld) ↾ ℚ) = (( I ↾ ℝ) ↾ ℚ)
83	82	a1i 11	. . 3 ⊢ (⊤ → ((ℝHom‘ℝ_fld) ↾ ℚ) = (( I ↾ ℝ) ↾ ℚ))
84	1, 3, 8, 14, 83, 23, 25	hauseqcn 34233	. 2 ⊢ (⊤ → (ℝHom‘ℝ_fld) = ( I ↾ ℝ))
85	84	mptru 1574	1 ⊢ (ℝHom‘ℝ_fld) = ( I ↾ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ⊤wtru 1568 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 Vcvv 3463 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 {csn 4594 I cid 5556 ran crn 5663 ↾ cres 5664 “ cima 5665 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 ℝcr 11099 ℚcq 12972 (,)cioo 13372 ↾_t crest 17473 topGenctg 17490 ℝ_fldcrefld 21723 Topctop 23019 TopOnctopon 23036 clsccl 23144 neicnei 23223 Cn ccn 23350 Hauscha 23434 Regcreg 23435 Filcfil 23971 fLimf cflf 24061 CnExtccnext 24185 UnifSpcusp 24380 CUnifSpccusp 24422 CMetSpccms 25460 ℚHomcqqh 34305 ℝHomcrrh 34328 ℝExt crrext 34329

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179 ax-mulf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-tpos 8222 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16311 df-gcd 16553 df-numer 16794 df-denom 16795 df-gz 16990 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-proset 18350 df-poset 18369 df-plt 18384 df-toset 18471 df-ps 18622 df-tsr 18623 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-mhm 18841 df-submnd 18842 df-grp 19003 df-minusg 19004 df-sbg 19005 df-mulg 19134 df-subg 19189 df-ghm 19284 df-cntz 19387 df-od 19598 df-cmn 19852 df-abl 19853 df-omnd 20191 df-ogrp 20192 df-mgp 20217 df-rng 20231 df-ur 20264 df-ring 20317 df-cring 20318 df-oppr 20419 df-dvdsr 20439 df-unit 20440 df-invr 20470 df-dvr 20483 df-rhm 20554 df-nzr 20596 df-subrng 20631 df-subrg 20655 df-drng 20815 df-field 20816 df-abv 20890 df-orng 20940 df-ofld 20941 df-lmod 20961 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-metu 21490 df-cnfld 21492 df-zring 21566 df-zrh 21622 df-zlm 21623 df-chr 21624 df-refld 21724 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-cn 23353 df-cnp 23354 df-haus 23441 df-reg 23442 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-fcls 24067 df-cnext 24186 df-ust 24327 df-utop 24357 df-uss 24382 df-usp 24383 df-ucn 24401 df-cfilu 24412 df-cusp 24423 df-xms 24446 df-ms 24447 df-tms 24448 df-nm 24708 df-ngp 24709 df-nrg 24711 df-nlm 24712 df-cncf 25006 df-cfil 25383 df-cmet 25385 df-cms 25463 df-qqh 34306 df-rrh 34330 df-rrext 34334

This theorem is referenced by: sitmcl 34686

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