qqhucn - Mathbox for Thierry Arnoux

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Theorem qqhucn 31842

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 2738	. . . . 5 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
5		eqid 2738	. . . . 5 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6	3, 4, 5	qqhf 31836	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
7	1, 2, 6	syl2anc 583	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅):ℚ⟶𝐵)
8		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
9		qqhucn.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ NrmRing)
10		nrgngp 23732	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
11	9, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ NrmGrp)
12	11	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmGrp)
13	7	ffvelrnda 6943	. . . . . . . . . . . . . 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	ffvelrnda 6943	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ 𝐵)
17		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
18		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
19		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
20	17, 3, 18, 19	ngpdsr 23667	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℚ)
24		qsubdrg 20562	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
25	24	simpli 483	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
26		subrgsubg 19945	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
27	25, 26	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
28		cnfldsub 20538	. . . . . . . . . . . . . . . . . 18 ⊢ − = (-_g‘ℂ_fld)
29		qqhucn.q	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
30		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (-_g‘𝑄) = (-_g‘𝑄)
31	28, 29, 30	subgsub 18682	. . . . . . . . . . . . . . . . 17 ⊢ ((ℚ ∈ (SubGrp‘ℂ_fld) ∧ 𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
32	27, 31	mp3an1 1446	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
33	22, 23, 32	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
34	33	fveq2d 6760	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞 − 𝑝)) = ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)))
35	3, 4, 5, 29	qqhghm 31838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
36	1, 2, 35	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
37	36	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
38	29	qrngbas 26672	. . . . . . . . . . . . . . . 16 ⊢ ℚ = (Base‘𝑄)
39	38, 30, 18	ghmsub 18757	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)))
41	34, 40	eqtr2d 2779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)) = ((ℚHom‘𝑅)‘(𝑞 − 𝑝)))
42	41	fveq2d 6760	. . . . . . . . . . . 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 4124	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (NrmRing ∩ DivRing))
44	43	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ (NrmRing ∩ DivRing))
45		qqhucn.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ NrmMod)
46	45	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑍 ∈ NrmMod)
47	2	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
48		qsubcl 12637	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
49	22, 23, 48	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
50		qqhucn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
51	17, 50	qqhnm 31840	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))) = (abs‘(𝑞 − 𝑝)))
53	21, 42, 52	3eqtrd 2782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = (abs‘(𝑞 − 𝑝)))
54	14, 16	ovresd 7417	. . . . . . . . . . 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 12629	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
56	55, 23	sselid 3915	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℂ)
57	55, 22	sselid 3915	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
58		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
59	58	cnmetdval 23840	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
60	56, 57, 59	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
61	23, 22	ovresd 7417	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (𝑝(abs ∘ − )𝑞))
62	57, 56	abssubd 15093	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (abs‘(𝑞 − 𝑝)) = (abs‘(𝑝 − 𝑞)))
63	60, 61, 62	3eqtr4d 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (abs‘(𝑞 − 𝑝)))
64	53, 54, 63	3eqtr4rd 2789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)))
65	64	breq1d 5080	. . . . . . . . 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 228	. . . . . . . 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 3107	. . . . . . 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 3107	. . . . . 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 5074	. . . . . . . 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 3122	. . . . . 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 3552	. . . . 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 583	. . . 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 3107	. . 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 2738	. . . 4 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
77		qqhucn.v	. . . 4 ⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))
78		0z 12260	. . . . . 6 ⊢ 0 ∈ ℤ
79		zq 12623	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
80		ne0i 4265	. . . . . 6 ⊢ (0 ∈ ℚ → ℚ ≠ ∅)
81	78, 79, 80	mp2b 10	. . . . 5 ⊢ ℚ ≠ ∅
82	81	a1i 11	. . . 4 ⊢ (𝜑 → ℚ ≠ ∅)
83		drngring 19913	. . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
84		eqid 2738	. . . . . 6 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
85	3, 84	ringidcl 19722	. . . . 5 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
86		ne0i 4265	. . . . 5 ⊢ ((1_r‘𝑅) ∈ 𝐵 → 𝐵 ≠ ∅)
87	1, 83, 85, 86	4syl 19	. . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅)
88		cnfldxms 23846	. . . . . . . 8 ⊢ ℂ_fld ∈ ∞MetSp
89		qex 12630	. . . . . . . 8 ⊢ ℚ ∈ V
90		ressxms 23587	. . . . . . . 8 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
91	88, 89, 90	mp2an 688	. . . . . . 7 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
92	29, 91	eqeltri 2835	. . . . . 6 ⊢ 𝑄 ∈ ∞MetSp
93		cnfldds 20520	. . . . . . . . 9 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
94	29, 93	ressds 17039	. . . . . . . 8 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
95	89, 94	ax-mp 5	. . . . . . 7 ⊢ (abs ∘ − ) = (dist‘𝑄)
96	38, 95	xmsxmet2 23520	. . . . . 6 ⊢ (𝑄 ∈ ∞MetSp → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
97	92, 96	mp1i 13	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
98		xmetpsmet 23409	. . . . 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 23663	. . . . . 6 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
101	3, 19	xmsxmet2 23520	. . . . . 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 23409	. . . . 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 23633	. . 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 709	. 2 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
107		qqhucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑄)
108	29	fveq2i 6759	. . . . . 6 ⊢ (UnifSt‘𝑄) = (UnifSt‘(ℂ_fld ↾_s ℚ))
109		ressuss 23322	. . . . . . 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 2770	. . . . 5 ⊢ 𝑈 = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ))
112		eqid 2738	. . . . . . 7 ⊢ (UnifSt‘ℂ_fld) = (UnifSt‘ℂ_fld)
113	112	cnflduss 24425	. . . . . 6 ⊢ (UnifSt‘ℂ_fld) = (metUnif‘(abs ∘ − ))
114	113	oveq1i 7265	. . . . 5 ⊢ ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ)) = ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ))
115		cnxmet 23842	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
116		xmetpsmet 23409	. . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ))
117	115, 116	ax-mp 5	. . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ)
118		restmetu 23632	. . . . . 6 ⊢ ((ℚ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℚ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
119	81, 117, 55, 118	mp3an 1459	. . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
120	111, 114, 119	3eqtri 2770	. . . 4 ⊢ 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
121	120	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
122	121	oveq1d 7270	. 2 ⊢ (𝜑 → (𝑈 Cn_u𝑉) = ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
123	106, 122	eleqtrrd 2842	1 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cn_u𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 class class class wbr 5070 × cxp 5578 ↾ cres 5582 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 0cc0 10802 < clt 10940 − cmin 11135 ℤcz 12249 ℚcq 12617 ℝ⁺crp 12659 abscabs 14873 Basecbs 16840 ↾_s cress 16867 distcds 16897 ↾_t crest 17048 -_gcsg 18494 SubGrpcsubg 18664 GrpHom cghm 18746 1_rcur 19652 Ringcrg 19698 /_rcdvr 19839 DivRingcdr 19906 SubRingcsubrg 19935 PsMetcpsmet 20494 ∞Metcxmet 20495 metUnifcmetu 20501 ℂ_fldccnfld 20510 ℤRHomczrh 20613 ℤModczlm 20614 chrcchr 20615 UnifStcuss 23313 Cn_ucucn 23335 ∞MetSpcxms 23378 normcnm 23638 NrmGrpcngp 23639 NrmRingcnrg 23641 NrmModcnlm 23642 ℚHomcqqh 31822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ico 13014 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-numer 16367 df-denom 16368 df-gz 16559 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-0g 17069 df-topgen 17071 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-od 19051 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-abv 19992 df-lmod 20040 df-nzr 20442 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-metu 20509 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zlm 20618 df-chr 20619 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-fil 22905 df-ust 23260 df-uss 23316 df-ucn 23336 df-xms 23381 df-ms 23382 df-nm 23644 df-ngp 23645 df-nrg 23647 df-nlm 23648 df-qqh 31823

This theorem is referenced by: rrhcn 31847

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