Theorem isrrext 33946

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 3992	. . 3 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) ↔ (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
2	1	anbi1i 623	. 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 6920	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ℤMod‘𝑟) = (ℤMod‘𝑅))
4	3	eleq1d 2829	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod))
5		isrrext.z	. . . . . . 7 ⊢ 𝑍 = (ℤMod‘𝑅)
6	5	eleq1i 2835	. . . . . 6 ⊢ (𝑍 ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod)
7	4, 6	bitr4di 289	. . . . 5 ⊢ (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ 𝑍 ∈ NrmMod))
8		fveqeq2 6929	. . . . 5 ⊢ (𝑟 = 𝑅 → ((chr‘𝑟) = 0 ↔ (chr‘𝑅) = 0))
9	7, 8	anbi12d 631	. . . 4 ⊢ (𝑟 = 𝑅 → (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ↔ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0)))
10		eleq1 2832	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ CUnifSp ↔ 𝑅 ∈ CUnifSp))
11		fveq2 6920	. . . . . 6 ⊢ (𝑟 = 𝑅 → (UnifSt‘𝑟) = (UnifSt‘𝑅))
12		fveq2 6920	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (dist‘𝑟) = (dist‘𝑅))
13		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14		isrrext.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
15	13, 14	eqtr4di 2798	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
16	15	sqxpeqd 5732	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) × (Base‘𝑟)) = (𝐵 × 𝐵))
17	12, 16	reseq12d 6010	. . . . . . . 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 2798	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = 𝐷)
20	19	fveq2d 6924	. . . . . 6 ⊢ (𝑟 = 𝑅 → (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) = (metUnif‘𝐷))
21	11, 20	eqeq12d 2756	. . . . 5 ⊢ (𝑟 = 𝑅 → ((UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) ↔ (UnifSt‘𝑅) = (metUnif‘𝐷)))
22	10, 21	anbi12d 631	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))) ↔ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
23	9, 22	anbi12d 631	. . 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 33945	. . 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 3711	. 2 ⊢ (𝑅 ∈ ℝExt ↔ (𝑅 ∈ (NrmRing ∩ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
26		3anass 1095	. 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 1087   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   × cxp 5698   ↾ cres 5702  ‘cfv 6573  0cc0 11184  Basecbs 17258  distcds 17320  DivRingcdr 20751  metUnifcmetu 21378  ℤModczlm 21534  chrcchr 21535  UnifStcuss 24283  CUnifSpccusp 24327  NrmRingcnrg 24613  NrmModcnlm 24614   ℝExt crrext 33940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-res 5712  df-iota 6525  df-fv 6581  df-rrext 33945

This theorem is referenced by:  rrextnrg  33947  rrextdrg  33948  rrextnlm  33949  rrextchr  33950  rrextcusp  33951  rrextust  33954  rerrext  33955  cnrrext  33956