Theorem isrrext 34399

Description: Express the property "𝑅 is an extension of ℝ". (Contributed by Thierry Arnoux, 2-May-2018.)

Hypotheses

Ref	Expression
isrrext.b	⊢ 𝐵 = (Base‘𝑅)
isrrext.v	⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))
isrrext.z	⊢ 𝑍 = (ℤMod‘𝑅)

Assertion

Ref	Expression
isrrext	⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))

Proof of Theorem isrrext

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3921	. . 3 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) ↔ (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
2	1	anbi1i 635	. 2 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))) ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
3		fveq2 6881	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ℤMod‘𝑟) = (ℤMod‘𝑅))
4	3	eleq1d 2848	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod))
5		isrrext.z	. . . . . . 7 ⊢ 𝑍 = (ℤMod‘𝑅)
6	5	eleq1i 2854	. . . . . 6 ⊢ (𝑍 ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod)
7	4, 6	bitr4di 292	. . . . 5 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ 𝑍 ∈ NrmMod))
8		fveqeq2 6890	. . . . 5 ⊢ (𝑟 = 𝑅 → ((chr‘𝑟) = 0 ↔ (chr‘𝑅) = 0))
9	7, 8	anbi12d 643	. . . 4 ⊢ (𝑟 = 𝑅 → (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ↔ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0)))
10		eleq1 2851	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ CUnifSp ↔ 𝑅 ∈ CUnifSp))
11		fveq2 6881	. . . . . 6 ⊢ (𝑟 = 𝑅 → (UnifSt‘𝑟) = (UnifSt‘𝑅))
12		fveq2 6881	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (dist‘𝑟) = (dist‘𝑅))
13		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14		isrrext.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
15	13, 14	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
16	15	sqxpeqd 5693	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) × (Base‘𝑟)) = (𝐵 × 𝐵))
17	12, 16	reseq12d 5979	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = ((dist‘𝑅) ↾ (𝐵 × 𝐵)))
18		isrrext.v	. . . . . . . 8 ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))
19	17, 18	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = 𝐷)
20	19	fveq2d 6885	. . . . . 6 ⊢ (𝑟 = 𝑅 → (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) = (metUnif‘𝐷))
21	11, 20	eqeq12d 2779	. . . . 5 ⊢ (𝑟 = 𝑅 → ((UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) ↔ (UnifSt‘𝑅) = (metUnif‘𝐷)))
22	10, 21	anbi12d 643	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))) ↔ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
23	9, 22	anbi12d 643	. . 3 ⊢ (𝑟 = 𝑅 → ((((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))))) ↔ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
24		df-rrext 34398	. . 3 ⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
25	23, 24	elrab2 3654	. 2 ⊢ (𝑅 ∈ ℝExt ↔ (𝑅 ∈ (NrmRing ∩ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
26		3anass 1111	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))) ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
27	2, 25, 26	3bitr4i 306	1 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∩ cin 3904   × cxp 5659   ↾ cres 5663  ‘cfv 6536  0cc0 11104  Basecbs 17273  distcds 17323  DivRingcdr 20836  metUnifcmetu 21522  ℤModczlm 21659  chrcchr 21660  UnifStcuss 24419  CUnifSpccusp 24462  NrmRingcnrg 24745  NrmModcnlm 24746   ℝExt crrext 34393

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-res 5673  df-iota 6492  df-fv 6544  df-rrext 34398

This theorem is used by:  rrextnrg  34400  rrextdrg  34401  rrextnlm  34402  rrextchr  34403  rrextcusp  34404  rrextust  34407  rerrext  34408  cnrrext  34409