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Theorem extdgval 33705
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 33697 . . 3 Rel /FldExt
21brrelex1i 5741 . 2 (𝐸/FldExt𝐹𝐸 ∈ V)
3 elrelimasn 6104 . . . 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 6921 . 2 (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V)
7 simpl 482 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑒 = 𝐸)
87fveq2d 6910 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸))
9 simpr 484 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑓 = 𝐹)
109fveq2d 6910 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
118, 10fveq12d 6913 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
1211fveq2d 6910 . . 3 ((𝑒 = 𝐸𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
13 sneq 4636 . . . 4 (𝑒 = 𝐸 → {𝑒} = {𝐸})
1413imaeq2d 6078 . . 3 (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸}))
15 df-extdg 33694 . . 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 1373 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 1540  wcel 2108  Vcvv 3480  {csn 4626   class class class wbr 5143  cima 5688  Rel wrel 5690  cfv 6561  (class class class)co 7431  Basecbs 17247  subringAlg csra 21170  dimcldim 33649  /FldExtcfldext 33689  [:]cextdg 33692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fldext 33693  df-extdg 33694
This theorem is referenced by:  extdgcl  33707  extdggt0  33708  extdgid  33711  extdgmul  33714  extdg1id  33716  ccfldextdgrr  33722  fldextrspunlem1  33725  fldextrspundgle  33728  algextdeglem4  33761
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