Theorem extdgval 34219

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 34210	. . 3 ⊢ Rel /FldExt
2	1	brrelex1i 5703	. 2 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ V)
3		elrelimasn 6076	. . . 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 6888	. 2 ⊢ (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V)
7		simpl 488	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
8	7	fveq2d 6877	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸))
9		simpr 490	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
10	9	fveq2d 6877	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
11	8, 10	fveq12d 6880	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
12	11	fveq2d 6877	. . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
13		sneq 4593	. . . 4 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
14	13	imaeq2d 6050	. . 3 ⊢ (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸}))
15		df-extdg 34208	. . 3 ⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
16	12, 14, 15	ovmpox 7561	. 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 1398	1 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  {csn 4583   class class class wbr 5102   “ cima 5650  Rel wrel 5652  ‘cfv 6527  (class class class)co 7408  Basecbs 17349  subringAlg csra 21408  dimcldim 34165  /_FldExtcfldext 34204  [:]cextdg 34206

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-fldext 34207  df-extdg 34208

This theorem is used by:  extdgcl  34222  extdggt0  34223  extdgid  34226  extdgmul  34229  extdg1id  34232  ccfldextdgrr  34238  fldextrspunlem1  34241  fldextrspundgle  34244  finextalg  34264  algextdeglem4  34286