Theorem isrrext 34193

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 3899	. . 3 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) ↔ (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
2	1	anbi1i 630	. 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 6828	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ℤMod‘𝑟) = (ℤMod‘𝑅))
4	3	eleq1d 2824	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod))
5		isrrext.z	. . . . . . 7 ⊢ 𝑍 = (ℤMod‘𝑅)
6	5	eleq1i 2830	. . . . . 6 ⊢ (𝑍 ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod)
7	4, 6	bitr4di 290	. . . . 5 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ 𝑍 ∈ NrmMod))
8		fveqeq2 6837	. . . . 5 ⊢ (𝑟 = 𝑅 → ((chr‘𝑟) = 0 ↔ (chr‘𝑅) = 0))
9	7, 8	anbi12d 638	. . . 4 ⊢ (𝑟 = 𝑅 → (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ↔ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0)))
10		eleq1 2827	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ CUnifSp ↔ 𝑅 ∈ CUnifSp))
11		fveq2 6828	. . . . . 6 ⊢ (𝑟 = 𝑅 → (UnifSt‘𝑟) = (UnifSt‘𝑅))
12		fveq2 6828	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (dist‘𝑟) = (dist‘𝑅))
13		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14		isrrext.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
15	13, 14	eqtr4di 2792	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
16	15	sqxpeqd 5651	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) × (Base‘𝑟)) = (𝐵 × 𝐵))
17	12, 16	reseq12d 5933	. . . . . . . 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 2792	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = 𝐷)
20	19	fveq2d 6832	. . . . . 6 ⊢ (𝑟 = 𝑅 → (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) = (metUnif‘𝐷))
21	11, 20	eqeq12d 2755	. . . . 5 ⊢ (𝑟 = 𝑅 → ((UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) ↔ (UnifSt‘𝑅) = (metUnif‘𝐷)))
22	10, 21	anbi12d 638	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))) ↔ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
23	9, 22	anbi12d 638	. . 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 34192	. . 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 3632	. 2 ⊢ (𝑅 ∈ ℝExt ↔ (𝑅 ∈ (NrmRing ∩ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
26		3anass 1100	. 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 304	1 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   × cxp 5617   ↾ cres 5621  ‘cfv 6486  0cc0 11030  Basecbs 17171  distcds 17221  DivRingcdr 20702  metUnifcmetu 21339  ℤModczlm 21476  chrcchr 21477  UnifStcuss 24237  CUnifSpccusp 24280  NrmRingcnrg 24563  NrmModcnlm 24564   ℝExt crrext 34187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-xp 5625  df-res 5631  df-iota 6442  df-fv 6494  df-rrext 34192

This theorem is referenced by:  rrextnrg  34194  rrextdrg  34195  rrextnlm  34196  rrextchr  34197  rrextcusp  34198  rrextust  34201  rerrext  34202  cnrrext  34203