Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  extdgval Structured version   Visualization version   GIF version

Theorem extdgval 33682
Description: Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Assertion
Ref Expression
extdgval (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))

Proof of Theorem extdgval
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfldext 33674 . . 3 Rel /FldExt
21brrelex1i 5745 . 2 (𝐸/FldExt𝐹𝐸 ∈ V)
3 elrelimasn 6106 . . . 4 (Rel /FldExt → (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹))
41, 3ax-mp 5 . . 3 (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹)
54biimpri 228 . 2 (𝐸/FldExt𝐹𝐹 ∈ (/FldExt “ {𝐸}))
6 fvexd 6922 . 2 (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V)
7 simpl 482 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑒 = 𝐸)
87fveq2d 6911 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸))
9 simpr 484 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑓 = 𝐹)
109fveq2d 6911 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
118, 10fveq12d 6914 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
1211fveq2d 6911 . . 3 ((𝑒 = 𝐸𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
13 sneq 4641 . . . 4 (𝑒 = 𝐸 → {𝑒} = {𝐸})
1413imaeq2d 6080 . . 3 (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸}))
15 df-extdg 33671 . . 3 [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
1612, 14, 15ovmpox 7586 . 2 ((𝐸 ∈ V ∧ 𝐹 ∈ (/FldExt “ {𝐸}) ∧ (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
172, 5, 6, 16syl3anc 1370 1 (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   class class class wbr 5148  cima 5692  Rel wrel 5694  cfv 6563  (class class class)co 7431  Basecbs 17245  subringAlg csra 21188  dimcldim 33626  /FldExtcfldext 33666  [:]cextdg 33669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fldext 33670  df-extdg 33671
This theorem is referenced by:  extdgcl  33684  extdggt0  33685  extdgid  33688  extdgmul  33689  extdg1id  33691  ccfldextdgrr  33697  algextdeglem4  33726
  Copyright terms: Public domain W3C validator