rrhre - Mathbox for Thierry Arnoux

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Theorem rrhre 33967

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 24804	. . 3 ⊢ ℝ = ∪ (topGen‘ran (,))
2		rehaus 24840	. . . 4 ⊢ (topGen‘ran (,)) ∈ Haus
3	2	a1i 11	. . 3 ⊢ (⊤ → (topGen‘ran (,)) ∈ Haus)
4		rerrext 33955	. . . 4 ⊢ ℝ_fld ∈ ℝExt
5		eqid 2740	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
6		retopn 25432	. . . . 5 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
7	5, 6	rrhcne 33959	. . . 4 ⊢ (ℝ_fld ∈ ℝExt → (ℝHom‘ℝ_fld) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
8	4, 7	mp1i 13	. . 3 ⊢ (⊤ → (ℝHom‘ℝ_fld) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
9		retop 24803	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
10	1	toptopon 22944	. . . . . 6 ⊢ ((topGen‘ran (,)) ∈ Top ↔ (topGen‘ran (,)) ∈ (TopOn‘ℝ))
11	9, 10	mpbi 230	. . . . 5 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
12		idcn 23286	. . . . 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 6900	. . . . . . . . . 10 ⊢ ( I ↾ ℚ):ℚ–1-1-onto→ℚ
17		f1of 6862	. . . . . . . . . 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 13024	. . . . . . . . 9 ⊢ ℚ ⊆ ℝ
20		fss 6763	. . . . . . . . 9 ⊢ ((( I ↾ ℚ):ℚ⟶ℚ ∧ ℚ ⊆ ℝ) → ( I ↾ ℚ):ℚ⟶ℝ)
21	18, 19, 20	mp2an 691	. . . . . . . 8 ⊢ ( I ↾ ℚ):ℚ⟶ℝ
22	21	a1i 11	. . . . . . 7 ⊢ (⊤ → ( I ↾ ℚ):ℚ⟶ℝ)
23	19	a1i 11	. . . . . . 7 ⊢ (⊤ → ℚ ⊆ ℝ)
24		qdensere 24811	. . . . . . . 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 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ (topGen‘ran (,)))
28		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
29		opnneip 23148	. . . . . . . . . . . . . . . 16 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑎 ∈ (topGen‘ran (,)) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘(topGen‘ran (,)))‘{𝑥}))
30	26, 27, 28, 29	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘(topGen‘ran (,)))‘{𝑥}))
31		fvex 6933	. . . . . . . . . . . . . . . 16 ⊢ ((nei‘(topGen‘ran (,)))‘{𝑥}) ∈ V
32		qex 13026	. . . . . . . . . . . . . . . 16 ⊢ ℚ ∈ V
33		elrestr 17488	. . . . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ((nei‘(topGen‘ran (,)))‘{𝑥}) → (𝑎 ∩ ℚ) ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))
35	30, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → (𝑎 ∩ ℚ) ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))
36		inss2 4259	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∩ ℚ) ⊆ ℚ
37		resiima 6105	. . . . . . . . . . . . . . . . 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 4258	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∩ ℚ) ⊆ 𝑎
40	38, 39	eqsstri 4043	. . . . . . . . . . . . . . 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 6085	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑎 ∩ ℚ) → (( I ↾ ℚ) “ 𝑏) = (( I ↾ ℚ) “ (𝑎 ∩ ℚ)))
43	42	sseq1d 4040	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∩ ℚ) → ((( I ↾ ℚ) “ 𝑏) ⊆ 𝑎 ↔ (( I ↾ ℚ) “ (𝑎 ∩ ℚ)) ⊆ 𝑎))
44	43	rspcev 3635	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎)
46	45	ex 412	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑎 ∈ (topGen‘ran (,))) → (𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎))
47	46	ralrimiva 3152	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → ∀𝑎 ∈ (topGen‘ran (,))(𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎))
48	47	ancli 548	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ℝ ∧ ∀𝑎 ∈ (topGen‘ran (,))(𝑥 ∈ 𝑎 → ∃𝑏 ∈ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ)(( I ↾ ℚ) “ 𝑏) ⊆ 𝑎)))
49	24	eleq2i 2836	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((cls‘(topGen‘ran (,)))‘ℚ) ↔ 𝑥 ∈ ℝ)
50	49	biimpri 228	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ((cls‘(topGen‘ran (,)))‘ℚ))
51		trnei 23921	. . . . . . . . . . . . 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 1451	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ((cls‘(topGen‘ran (,)))‘ℚ) ↔ (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ) ∈ (Fil‘ℚ)))
53	50, 52	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ) ∈ (Fil‘ℚ))
54		isflf 24022	. . . . . . . . . . . 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 1452	. . . . . . . . . . 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 17	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ (((topGen‘ran (,)) fLimf (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))‘( I ↾ ℚ)))
58	57	ne0d 4365	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (((topGen‘ran (,)) fLimf (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))‘( I ↾ ℚ)) ≠ ∅)
59	58	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (((topGen‘ran (,)) fLimf (((nei‘(topGen‘ran (,)))‘{𝑥}) ↾_t ℚ))‘( I ↾ ℚ)) ≠ ∅)
60		recusp 25435	. . . . . . . . . 10 ⊢ ℝ_fld ∈ CUnifSp
61		cuspusp 24330	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ CUnifSp → ℝ_fld ∈ UnifSp)
62	60, 61	ax-mp 5	. . . . . . . . 9 ⊢ ℝ_fld ∈ UnifSp
63	6	uspreg 24304	. . . . . . . . 9 ⊢ ((ℝ_fld ∈ UnifSp ∧ (topGen‘ran (,)) ∈ Haus) → (topGen‘ran (,)) ∈ Reg)
64	62, 2, 63	mp2an 691	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ Reg
65	64	a1i 11	. . . . . . 7 ⊢ (⊤ → (topGen‘ran (,)) ∈ Reg)
66		resabs1 6036	. . . . . . . . . 10 ⊢ (ℚ ⊆ ℝ → (( I ↾ ℝ) ↾ ℚ) = ( I ↾ ℚ))
67	19, 66	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ ℝ) ↾ ℚ) = ( I ↾ ℚ)
68	1	cnrest 23314	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ (( I ↾ ℝ) ↾ ℚ) ∈ (((topGen‘ran (,)) ↾_t ℚ) Cn (topGen‘ran (,)))
70	67, 69	eqeltrri 2841	. . . . . . . 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 24097	. . . . . 6 ⊢ (⊤ → ((((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ)) ↾ ℚ) = ( I ↾ ℚ))
73	72	mptru 1544	. . . . 5 ⊢ ((((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ)) ↾ ℚ) = ( I ↾ ℚ)
74		recms 25433	. . . . . . . . 9 ⊢ ℝ_fld ∈ CMetSp
75	74	elexi 3511	. . . . . . . 8 ⊢ ℝ_fld ∈ V
76	5, 6	rrhval 33942	. . . . . . . 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 33966	. . . . . . . 8 ⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)
79	78	fveq2i 6923	. . . . . . 7 ⊢ (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘(ℚHom‘ℝ_fld)) = (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ))
80	77, 79	eqtri 2768	. . . . . 6 ⊢ (ℝHom‘ℝ_fld) = (((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ))
81	80	reseq1i 6005	. . . . 5 ⊢ ((ℝHom‘ℝ_fld) ↾ ℚ) = ((((topGen‘ran (,))CnExt(topGen‘ran (,)))‘( I ↾ ℚ)) ↾ ℚ)
82	73, 81, 67	3eqtr4i 2778	. . . 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 33844	. 2 ⊢ (⊤ → (ℝHom‘ℝ_fld) = ( I ↾ ℝ))
85	84	mptru 1544	1 ⊢ (ℝHom‘ℝ_fld) = ( I ↾ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 {csn 4648 I cid 5592 ran crn 5701 ↾ cres 5702 “ cima 5703 ⟶wf 6569 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 ℚcq 13013 (,)cioo 13407 ↾_t crest 17480 topGenctg 17497 ℝ_fldcrefld 21645 Topctop 22920 TopOnctopon 22937 clsccl 23047 neicnei 23126 Cn ccn 23253 Hauscha 23337 Regcreg 23338 Filcfil 23874 fLimf cflf 23964 CnExtccnext 24088 UnifSpcusp 24284 CUnifSpccusp 24327 CMetSpccms 25385 ℚHomcqqh 33918 ℝHomcrrh 33939 ℝExt crrext 33940

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-numer 16782 df-denom 16783 df-gz 16977 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-proset 18365 df-poset 18383 df-plt 18400 df-toset 18487 df-ps 18636 df-tsr 18637 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-od 19570 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-nzr 20539 df-subrng 20572 df-subrg 20597 df-drng 20753 df-field 20754 df-abv 20832 df-lmod 20882 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-metu 21386 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-zlm 21538 df-chr 21539 df-refld 21646 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-cn 23256 df-cnp 23257 df-haus 23344 df-reg 23345 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-fcls 23970 df-cnext 24089 df-ust 24230 df-utop 24261 df-uss 24286 df-usp 24287 df-ucn 24306 df-cfilu 24317 df-cusp 24328 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-cncf 24923 df-cfil 25308 df-cmet 25310 df-cms 25388 df-omnd 33049 df-ogrp 33050 df-orng 33292 df-ofld 33293 df-qqh 33919 df-rrh 33941 df-rrext 33945

This theorem is referenced by: sitmcl 34316

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