Theorem extdgval 34052

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.

Step	Hyp	Ref	Expression
1		relfldext 34043	. . 3 ⊢ Rel /FldExt
2	1	brrelex1i 5716	. 2 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ V)
3		elrelimasn 6087	. . . 4 ⊢ (Rel /FldExt → (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹))
4	1, 3	ax-mp 5	. . 3 ⊢ (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹)
5	4	biimpri 231	. 2 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ (/FldExt “ {𝐸}))
6		fvexd 6896	. 2 ⊢ (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V)
7		simpl 487	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
8	7	fveq2d 6885	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸))
9		simpr 489	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
10	9	fveq2d 6885	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
11	8, 10	fveq12d 6888	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
12	11	fveq2d 6885	. . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
13		sneq 4598	. . . 4 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
14	13	imaeq2d 6061	. . 3 ⊢ (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸}))
15		df-extdg 34041	. . 3 ⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
16	12, 14, 15	ovmpox 7565	. 2 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ (/FldExt “ {𝐸}) ∧ (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
17	2, 5, 6, 16	syl3anc 1397	1 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454  {csn 4588   class class class wbr 5108   “ cima 5663  Rel wrel 5665  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  subringAlg csra 21303  dimcldim 33998  /_FldExtcfldext 34037  [:]cextdg 34039

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fldext 34040  df-extdg 34041

This theorem is used by:  extdgcl  34055  extdggt0  34056  extdgid  34059  extdgmul  34062  extdg1id  34065  ccfldextdgrr  34071  fldextrspunlem1  34074  fldextrspundgle  34077  finextalg  34097  algextdeglem4  34119