Theorem isrrext 33990

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 3930	. . 3 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) ↔ (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
2	1	anbi1i 624	. 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 6858	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ℤMod‘𝑟) = (ℤMod‘𝑅))
4	3	eleq1d 2813	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod))
5		isrrext.z	. . . . . . 7 ⊢ 𝑍 = (ℤMod‘𝑅)
6	5	eleq1i 2819	. . . . . 6 ⊢ (𝑍 ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod)
7	4, 6	bitr4di 289	. . . . 5 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ 𝑍 ∈ NrmMod))
8		fveqeq2 6867	. . . . 5 ⊢ (𝑟 = 𝑅 → ((chr‘𝑟) = 0 ↔ (chr‘𝑅) = 0))
9	7, 8	anbi12d 632	. . . 4 ⊢ (𝑟 = 𝑅 → (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ↔ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0)))
10		eleq1 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ CUnifSp ↔ 𝑅 ∈ CUnifSp))
11		fveq2 6858	. . . . . 6 ⊢ (𝑟 = 𝑅 → (UnifSt‘𝑟) = (UnifSt‘𝑅))
12		fveq2 6858	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (dist‘𝑟) = (dist‘𝑅))
13		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14		isrrext.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
15	13, 14	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
16	15	sqxpeqd 5670	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) × (Base‘𝑟)) = (𝐵 × 𝐵))
17	12, 16	reseq12d 5951	. . . . . . . 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 2782	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = 𝐷)
20	19	fveq2d 6862	. . . . . 6 ⊢ (𝑟 = 𝑅 → (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) = (metUnif‘𝐷))
21	11, 20	eqeq12d 2745	. . . . 5 ⊢ (𝑟 = 𝑅 → ((UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) ↔ (UnifSt‘𝑅) = (metUnif‘𝐷)))
22	10, 21	anbi12d 632	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))) ↔ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
23	9, 22	anbi12d 632	. . 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 33989	. . 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 3662	. 2 ⊢ (𝑅 ∈ ℝExt ↔ (𝑅 ∈ (NrmRing ∩ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
26		3anass 1094	. 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 303	1 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∩ cin 3913   × cxp 5636   ↾ cres 5640  ‘cfv 6511  0cc0 11068  Basecbs 17179  distcds 17229  DivRingcdr 20638  metUnifcmetu 21255  ℤModczlm 21410  chrcchr 21411  UnifStcuss 24141  CUnifSpccusp 24184  NrmRingcnrg 24467  NrmModcnlm 24468   ℝExt crrext 33984

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-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-res 5650  df-iota 6464  df-fv 6519  df-rrext 33989

This theorem is referenced by:  rrextnrg  33991  rrextdrg  33992  rrextnlm  33993  rrextchr  33994  rrextcusp  33995  rrextust  33998  rerrext  33999  cnrrext  34000