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Theorem isrrext 30384
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 3947 . . 3 (𝑅 ∈ (NrmRing ∩ DivRing) ↔ (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
21anbi1i 610 . 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 6332 . . . . . . 7 (𝑟 = 𝑅 → (ℤMod‘𝑟) = (ℤMod‘𝑅))
43eleq1d 2835 . . . . . 6 (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod))
5 isrrext.z . . . . . . 7 𝑍 = (ℤMod‘𝑅)
65eleq1i 2841 . . . . . 6 (𝑍 ∈ NrmMod ↔ (ℤMod‘𝑅) ∈ NrmMod)
74, 6syl6bbr 278 . . . . 5 (𝑟 = 𝑅 → ((ℤMod‘𝑟) ∈ NrmMod ↔ 𝑍 ∈ NrmMod))
8 fveq2 6332 . . . . . 6 (𝑟 = 𝑅 → (chr‘𝑟) = (chr‘𝑅))
98eqeq1d 2773 . . . . 5 (𝑟 = 𝑅 → ((chr‘𝑟) = 0 ↔ (chr‘𝑅) = 0))
107, 9anbi12d 616 . . . 4 (𝑟 = 𝑅 → (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ↔ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0)))
11 eleq1 2838 . . . . 5 (𝑟 = 𝑅 → (𝑟 ∈ CUnifSp ↔ 𝑅 ∈ CUnifSp))
12 fveq2 6332 . . . . . 6 (𝑟 = 𝑅 → (UnifSt‘𝑟) = (UnifSt‘𝑅))
13 fveq2 6332 . . . . . . . . 9 (𝑟 = 𝑅 → (dist‘𝑟) = (dist‘𝑅))
14 fveq2 6332 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
15 isrrext.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
1614, 15syl6eqr 2823 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
1716sqxpeqd 5281 . . . . . . . . 9 (𝑟 = 𝑅 → ((Base‘𝑟) × (Base‘𝑟)) = (𝐵 × 𝐵))
1813, 17reseq12d 5535 . . . . . . . 8 (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = ((dist‘𝑅) ↾ (𝐵 × 𝐵)))
19 isrrext.v . . . . . . . 8 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))
2018, 19syl6eqr 2823 . . . . . . 7 (𝑟 = 𝑅 → ((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))) = 𝐷)
2120fveq2d 6336 . . . . . 6 (𝑟 = 𝑅 → (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) = (metUnif‘𝐷))
2212, 21eqeq12d 2786 . . . . 5 (𝑟 = 𝑅 → ((UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟)))) ↔ (UnifSt‘𝑅) = (metUnif‘𝐷)))
2311, 22anbi12d 616 . . . 4 (𝑟 = 𝑅 → ((𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))) ↔ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
2410, 23anbi12d 616 . . 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 30383 . . 3 ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
2624, 25elrab2 3518 . 2 (𝑅 ∈ ℝExt ↔ (𝑅 ∈ (NrmRing ∩ DivRing) ∧ ((𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))))
27 3anass 1080 . 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 292 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  cin 3722   × cxp 5247  cres 5251  cfv 6031  0cc0 10138  Basecbs 16064  distcds 16158  DivRingcdr 18957  metUnifcmetu 19952  ℤModczlm 20064  chrcchr 20065  UnifStcuss 22277  CUnifSpccusp 22321  NrmRingcnrg 22604  NrmModcnlm 22605   ℝExt crrext 30378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-rrext 30383
This theorem is referenced by:  rrextnrg  30385  rrextdrg  30386  rrextnlm  30387  rrextchr  30388  rrextcusp  30389  rrextust  30392  rerrext  30393  cnrrext  30394
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