Theorem cntrval 18925

Description: Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cntrval.b	⊢ 𝐵 = (Base‘𝑀)
cntrval.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntrval	⊢ (𝑍‘𝐵) = (Cntr‘𝑀)

Proof of Theorem cntrval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2		cntrval.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	eqtr4di 2796	. . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4		fveq2 6774	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		cntrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	eqtr4di 2796	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	fveq12d 6781	. . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵))
8		df-cntr 18924	. . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9		fvex 6787	. . . 4 ⊢ (𝑍‘𝐵) ∈ V
10	7, 8, 9	fvmpt 6875	. . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵))
11	10	eqcomd 2744	. 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
12		0fv 6813	. . 3 ⊢ (∅‘𝐵) = ∅
13		fvprc 6766	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
14	2, 13	eqtrid 2790	. . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
15	14	fveq1d 6776	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵))
16		fvprc 6766	. . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
17	12, 15, 16	3eqtr4a 2804	. 2 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
18	11, 17	pm2.61i 182	1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  ‘cfv 6433  Basecbs 16912  Cntzccntz 18921  Cntrccntr 18922

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-cntr 18924

This theorem is referenced by:  cntrss  18936  cntri  18937  cntrsubgnsg  18947  cntrnsg  18948  oppgcntr  18972  cntrcmnd  19443  cntrabl  19444  primefld  20073  cntrcrng  31322