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Theorem extdgval 31065
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 31057 . . 3 Rel /FldExt
21brrelex1i 5589 . 2 (𝐸/FldExt𝐹𝐸 ∈ V)
3 elrelimasn 5934 . . . 4 (Rel /FldExt → (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹))
41, 3ax-mp 5 . . 3 (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹)
54biimpri 231 . 2 (𝐸/FldExt𝐹𝐹 ∈ (/FldExt “ {𝐸}))
6 fvexd 6666 . 2 (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V)
7 simpl 486 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑒 = 𝐸)
87fveq2d 6655 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸))
9 simpr 488 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → 𝑓 = 𝐹)
109fveq2d 6655 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
118, 10fveq12d 6658 . . . 4 ((𝑒 = 𝐸𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
1211fveq2d 6655 . . 3 ((𝑒 = 𝐸𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
13 sneq 4558 . . . 4 (𝑒 = 𝐸 → {𝑒} = {𝐸})
1413imaeq2d 5910 . . 3 (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸}))
15 df-extdg 31054 . . 3 [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
1612, 14, 15ovmpox 7285 . 2 ((𝐸 ∈ V ∧ 𝐹 ∈ (/FldExt “ {𝐸}) ∧ (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
172, 5, 6, 16syl3anc 1368 1 (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3479  {csn 4548   class class class wbr 5047  cima 5539  Rel wrel 5541  cfv 6336  (class class class)co 7138  Basecbs 16472  subringAlg csra 19926  dimcldim 31020  /FldExtcfldext 31049  [:]cextdg 31052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-fldext 31053  df-extdg 31054
This theorem is referenced by:  extdgcl  31067  extdggt0  31068  extdgid  31071  extdgmul  31072  extdg1id  31074  ccfldextdgrr  31078
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