qqhucn - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > qqhucn		Structured version Visualization version GIF version

Theorem qqhucn 34028

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

**Hypotheses**
Ref	Expression
qqhucn.b	⊢ 𝐵 = (Base‘𝑅)
qqhucn.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qqhucn.u	⊢ 𝑈 = (UnifSt‘𝑄)
qqhucn.v	⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))
qqhucn.z	⊢ 𝑍 = (ℤMod‘𝑅)
qqhucn.1	⊢ (𝜑 → 𝑅 ∈ NrmRing)
qqhucn.2	⊢ (𝜑 → 𝑅 ∈ DivRing)
qqhucn.3	⊢ (𝜑 → 𝑍 ∈ NrmMod)
qqhucn.4	⊢ (𝜑 → (chr‘𝑅) = 0)

**Assertion**
Ref	Expression
qqhucn	⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cn_u𝑉))

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 ⊢ (𝜑 → 𝑅 ∈ DivRing)
2		qqhucn.4	. . . 4 ⊢ (𝜑 → (chr‘𝑅) = 0)
3		qqhucn.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2736	. . . . 5 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
5		eqid 2736	. . . . 5 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6	3, 4, 5	qqhf 34022	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
7	1, 2, 6	syl2anc 584	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅):ℚ⟶𝐵)
8		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
9		qqhucn.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ NrmRing)
10		nrgngp 24606	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
11	9, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ NrmGrp)
12	11	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmGrp)
13	7	ffvelcdmda 7079	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵)
14	13	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵)
15	7	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (ℚHom‘𝑅):ℚ⟶𝐵)
16	15	ffvelcdmda 7079	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ 𝐵)
17		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
18		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
19		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
20	17, 3, 18, 19	ngpdsr 24549	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ NrmGrp ∧ ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵 ∧ ((ℚHom‘𝑅)‘𝑞) ∈ 𝐵) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝))))
21	12, 14, 16, 20	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝))))
22		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℚ)
23		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℚ)
24		qsubdrg 21392	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
25	24	simpli 483	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
26		subrgsubg 20542	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
27	25, 26	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
28		cnfldsub 21365	. . . . . . . . . . . . . . . . . 18 ⊢ − = (-_g‘ℂ_fld)
29		qqhucn.q	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
30		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (-_g‘𝑄) = (-_g‘𝑄)
31	28, 29, 30	subgsub 19126	. . . . . . . . . . . . . . . . 17 ⊢ ((ℚ ∈ (SubGrp‘ℂ_fld) ∧ 𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
32	27, 31	mp3an1 1450	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
33	22, 23, 32	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
34	33	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞 − 𝑝)) = ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)))
35	3, 4, 5, 29	qqhghm 34024	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
36	1, 2, 35	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
37	36	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
38	29	qrngbas 27587	. . . . . . . . . . . . . . . 16 ⊢ ℚ = (Base‘𝑄)
39	38, 30, 18	ghmsub 19212	. . . . . . . . . . . . . . 15 ⊢ (((ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅) ∧ 𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)))
40	37, 22, 23, 39	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)))
41	34, 40	eqtr2d 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)) = ((ℚHom‘𝑅)‘(𝑞 − 𝑝)))
42	41	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝))) = ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))))
43	9, 1	elind 4180	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (NrmRing ∩ DivRing))
44	43	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ (NrmRing ∩ DivRing))
45		qqhucn.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ NrmMod)
46	45	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑍 ∈ NrmMod)
47	2	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
48		qsubcl 12989	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
49	22, 23, 48	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
50		qqhucn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
51	17, 50	qqhnm 34026	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑞 − 𝑝) ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))) = (abs‘(𝑞 − 𝑝)))
52	44, 46, 47, 49, 51	syl31anc 1375	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))) = (abs‘(𝑞 − 𝑝)))
53	21, 42, 52	3eqtrd 2775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = (abs‘(𝑞 − 𝑝)))
54	14, 16	ovresd 7579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) = (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)))
55		qsscn 12981	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
56	55, 23	sselid 3961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℂ)
57	55, 22	sselid 3961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
58		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
59	58	cnmetdval 24714	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
60	56, 57, 59	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
61	23, 22	ovresd 7579	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (𝑝(abs ∘ − )𝑞))
62	57, 56	abssubd 15477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (abs‘(𝑞 − 𝑝)) = (abs‘(𝑝 − 𝑞)))
63	60, 61, 62	3eqtr4d 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (abs‘(𝑞 − 𝑝)))
64	53, 54, 63	3eqtr4rd 2782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)))
65	64	breq1d 5134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 ↔ (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
66	65	biimpd 229	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
67	66	ralrimiva 3133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
68	67	ralrimiva 3133	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
69	68	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
70		breq2 5128	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
71	70	imbi1d 341	. . . . . . 7 ⊢ (𝑑 = 𝑒 → (((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒) ↔ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒)))
72	71	2ralbidv 3209	. . . . . 6 ⊢ (𝑑 = 𝑒 → (∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒) ↔ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒)))
73	72	rspcev 3606	. . . . 5 ⊢ ((𝑒 ∈ ℝ⁺ ∧ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒)) → ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
74	8, 69, 73	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
75	74	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
76		eqid 2736	. . . 4 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
77		qqhucn.v	. . . 4 ⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))
78		0z 12604	. . . . . 6 ⊢ 0 ∈ ℤ
79		zq 12975	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
80		ne0i 4321	. . . . . 6 ⊢ (0 ∈ ℚ → ℚ ≠ ∅)
81	78, 79, 80	mp2b 10	. . . . 5 ⊢ ℚ ≠ ∅
82	81	a1i 11	. . . 4 ⊢ (𝜑 → ℚ ≠ ∅)
83		drngring 20701	. . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
84		eqid 2736	. . . . . 6 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
85	3, 84	ringidcl 20230	. . . . 5 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
86		ne0i 4321	. . . . 5 ⊢ ((1_r‘𝑅) ∈ 𝐵 → 𝐵 ≠ ∅)
87	1, 83, 85, 86	4syl 19	. . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅)
88		cnfldxms 24720	. . . . . . . 8 ⊢ ℂ_fld ∈ ∞MetSp
89		qex 12982	. . . . . . . 8 ⊢ ℚ ∈ V
90		ressxms 24469	. . . . . . . 8 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
91	88, 89, 90	mp2an 692	. . . . . . 7 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
92	29, 91	eqeltri 2831	. . . . . 6 ⊢ 𝑄 ∈ ∞MetSp
93		cnfldds 21332	. . . . . . . . 9 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
94	29, 93	ressds 17429	. . . . . . . 8 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
95	89, 94	ax-mp 5	. . . . . . 7 ⊢ (abs ∘ − ) = (dist‘𝑄)
96	38, 95	xmsxmet2 24403	. . . . . 6 ⊢ (𝑄 ∈ ∞MetSp → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
97	92, 96	mp1i 13	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
98		xmetpsmet 24292	. . . . 5 ⊢ (((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ) → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (PsMet‘ℚ))
99	97, 98	syl 17	. . . 4 ⊢ (𝜑 → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (PsMet‘ℚ))
100		ngpxms 24545	. . . . . 6 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
101	3, 19	xmsxmet2 24403	. . . . . 6 ⊢ (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
102	9, 10, 100, 101	4syl 19	. . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
103		xmetpsmet 24292	. . . . 5 ⊢ (((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (PsMet‘𝐵))
104	102, 103	syl 17	. . . 4 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (PsMet‘𝐵))
105	76, 77, 82, 87, 99, 104	metucn 24515	. . 3 ⊢ (𝜑 → ((ℚHom‘𝑅) ∈ ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉) ↔ ((ℚHom‘𝑅):ℚ⟶𝐵 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))))
106	7, 75, 105	mpbir2and 713	. 2 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
107		qqhucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑄)
108	29	fveq2i 6884	. . . . . 6 ⊢ (UnifSt‘𝑄) = (UnifSt‘(ℂ_fld ↾_s ℚ))
109		ressuss 24206	. . . . . . 7 ⊢ (ℚ ∈ V → (UnifSt‘(ℂ_fld ↾_s ℚ)) = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ)))
110	89, 109	ax-mp 5	. . . . . 6 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ))
111	107, 108, 110	3eqtri 2763	. . . . 5 ⊢ 𝑈 = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ))
112		eqid 2736	. . . . . . 7 ⊢ (UnifSt‘ℂ_fld) = (UnifSt‘ℂ_fld)
113	112	cnflduss 25313	. . . . . 6 ⊢ (UnifSt‘ℂ_fld) = (metUnif‘(abs ∘ − ))
114	113	oveq1i 7420	. . . . 5 ⊢ ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ)) = ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ))
115		cnxmet 24716	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
116		xmetpsmet 24292	. . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ))
117	115, 116	ax-mp 5	. . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ)
118		restmetu 24514	. . . . . 6 ⊢ ((ℚ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℚ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
119	81, 117, 55, 118	mp3an 1463	. . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
120	111, 114, 119	3eqtri 2763	. . . 4 ⊢ 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
121	120	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
122	121	oveq1d 7425	. 2 ⊢ (𝜑 → (𝑈 Cn_u𝑉) = ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
123	106, 122	eleqtrrd 2838	1 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cn_u𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 Vcvv 3464 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 class class class wbr 5124 × cxp 5657 ↾ cres 5661 ∘ ccom 5663 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 0cc0 11134 < clt 11274 − cmin 11471 ℤcz 12593 ℚcq 12969 ℝ⁺crp 13013 abscabs 15258 Basecbs 17233 ↾_s cress 17256 distcds 17285 ↾_t crest 17439 -_gcsg 18923 SubGrpcsubg 19108 GrpHom cghm 19200 1_rcur 20146 Ringcrg 20198 /_rcdvr 20365 SubRingcsubrg 20534 DivRingcdr 20694 PsMetcpsmet 21304 ∞Metcxmet 21305 metUnifcmetu 21311 ℂ_fldccnfld 21320 ℤRHomczrh 21465 ℤModczlm 21466 chrcchr 21467 UnifStcuss 24197 Cn_ucucn 24218 ∞MetSpcxms 24261 normcnm 24520 NrmGrpcngp 24521 NrmRingcnrg 24523 NrmModcnlm 24524 ℚHomcqqh 34006

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ico 13373 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-dvds 16278 df-gcd 16519 df-numer 16759 df-denom 16760 df-gz 16955 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-rest 17441 df-topn 17442 df-0g 17460 df-topgen 17462 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-od 19514 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-nzr 20478 df-subrng 20511 df-subrg 20535 df-drng 20696 df-abv 20774 df-lmod 20824 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-metu 21319 df-cnfld 21321 df-zring 21413 df-zrh 21469 df-zlm 21470 df-chr 21471 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-fil 23789 df-ust 24144 df-uss 24200 df-ucn 24219 df-xms 24264 df-ms 24265 df-nm 24526 df-ngp 24527 df-nrg 24529 df-nlm 24530 df-qqh 34007

This theorem is referenced by: rrhcn 34033

W3C validator