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Theorem isrrext 29372
Description: Express the property "𝑅 is an extension of ". (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.
StepHypRef Expression
1 elin 3758 . . 3 (𝑅 ∈ (NrmRing ∩ DivRing) ↔ (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
21anbi1i 727 . 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 6103 . . . . . . 7 (𝑟 = 𝑅 → (ℤMod‘𝑟) = (ℤMod‘𝑅))
43eleq1d 2672 . . . . . 6 (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod))
5 isrrext.z . . . . . . 7 𝑍 = (ℤMod‘𝑅)
65eleq1i 2679 . . . . . 6 (𝑍 ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod)
74, 6syl6bbr 277 . . . . 5 (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ 𝑍 ∈ NrmMod))
8 fveq2 6103 . . . . . 6 (𝑟 = 𝑅 → (chr‘𝑟) = (chr‘𝑅))
98eqeq1d 2612 . . . . 5 (𝑟 = 𝑅 → ((chr‘𝑟) = 0 ↔ (chr‘𝑅) = 0))
107, 9anbi12d 743 . . . 4 (𝑟 = 𝑅 → (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ↔ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0)))
11 eleq1 2676 . . . . 5 (𝑟 = 𝑅 → (𝑟 ∈ CUnifSp ↔ 𝑅 ∈ CUnifSp))
12 fveq2 6103 . . . . . 6 (𝑟 = 𝑅 → (UnifSt‘𝑟) = (UnifSt‘𝑅))
13 fveq2 6103 . . . . . . . . 9 (𝑟 = 𝑅 → (dist‘𝑟) = (dist‘𝑅))
14 fveq2 6103 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
15 isrrext.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
1614, 15syl6eqr 2662 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
1716sqxpeqd 5065 . . . . . . . . 9 (𝑟 = 𝑅 → ((Base‘𝑟) × (Base‘𝑟)) = (𝐵 × 𝐵))
1813, 17reseq12d 5318 . . . . . . . 8 (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = ((dist‘𝑅) ↾ (𝐵 × 𝐵)))
19 isrrext.v . . . . . . . 8 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))
2018, 19syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = 𝐷)
2120fveq2d 6107 . . . . . 6 (𝑟 = 𝑅 → (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) = (metUnif‘𝐷))
2212, 21eqeq12d 2625 . . . . 5 (𝑟 = 𝑅 → ((UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) ↔ (UnifSt‘𝑅) = (metUnif‘𝐷)))
2311, 22anbi12d 743 . . . 4 (𝑟 = 𝑅 → ((𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))) ↔ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
2410, 23anbi12d 743 . . 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)))))
25 df-rrext 29371 . . 3 ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
2624, 25elrab2 3333 . 2 (𝑅 ∈ ℝExt ↔ (𝑅 ∈ (NrmRing ∩ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
27 3anass 1035 . 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)))))
282, 26, 273bitr4i 291 1 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  cin 3539   × cxp 5036  cres 5040  cfv 5804  0cc0 9815  Basecbs 15695  distcds 15777  DivRingcdr 18570  metUnifcmetu 19558  ℤModczlm 19668  chrcchr 19669  UnifStcuss 21867  CUnifSpccusp 21911  NrmRingcnrg 22194  NrmModcnlm 22195   ℝExt crrext 29366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-rrext 29371
This theorem is referenced by:  rrextnrg  29373  rrextdrg  29374  rrextnlm  29375  rrextchr  29376  rrextcusp  29377  rrextust  29380  rerrext  29381  cnrrext  29382
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