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Theorem extdgval 32400
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 32392 . . 3 Rel /FldExt
21brrelex1i 5689 . 2 (𝐸/FldExt𝐹𝐸 ∈ V)
3 elrelimasn 6038 . . . 4 (Rel /FldExt → (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹))
41, 3ax-mp 5 . . 3 (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹)
54biimpri 227 . 2 (𝐸/FldExt𝐹𝐹 ∈ (/FldExt “ {𝐸}))
6 fvexd 6858 . 2 (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V)
7 simpl 484 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑒 = 𝐸)
87fveq2d 6847 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸))
9 simpr 486 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑓 = 𝐹)
109fveq2d 6847 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
118, 10fveq12d 6850 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
1211fveq2d 6847 . . 3 ((𝑒 = 𝐸𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
13 sneq 4597 . . . 4 (𝑒 = 𝐸 → {𝑒} = {𝐸})
1413imaeq2d 6014 . . 3 (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸}))
15 df-extdg 32389 . . 3 [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
1612, 14, 15ovmpox 7509 . 2 ((𝐸 ∈ V ∧ 𝐹 ∈ (/FldExt “ {𝐸}) ∧ (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
172, 5, 6, 16syl3anc 1372 1 (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  {csn 4587   class class class wbr 5106  cima 5637  Rel wrel 5639  cfv 6497  (class class class)co 7358  Basecbs 17088  subringAlg csra 20645  dimcldim 32353  /FldExtcfldext 32384  [:]cextdg 32387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-fldext 32388  df-extdg 32389
This theorem is referenced by:  extdgcl  32402  extdggt0  32403  extdgid  32406  extdgmul  32407  extdg1id  32409  ccfldextdgrr  32413
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