Theorem cntrval 19350

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 6907	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2		cntrval.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	eqtr4di 2793	. . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4		fveq2 6907	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		cntrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	eqtr4di 2793	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	fveq12d 6914	. . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵))
8		df-cntr 19349	. . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9		fvex 6920	. . . 4 ⊢ (𝑍‘𝐵) ∈ V
10	7, 8, 9	fvmpt 7016	. . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵))
11	10	eqcomd 2741	. 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
12		0fv 6951	. . 3 ⊢ (∅‘𝐵) = ∅
13		fvprc 6899	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
14	2, 13	eqtrid 2787	. . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
15	14	fveq1d 6909	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵))
16		fvprc 6899	. . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
17	12, 15, 16	3eqtr4a 2801	. 2 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
18	11, 17	pm2.61i 182	1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  ‘cfv 6563  Basecbs 17245  Cntzccntz 19346  Cntrccntr 19347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-cntr 19349

This theorem is referenced by:  elcntr  19361  cntrss  19362  cntri  19363  cntrsubgnsg  19374  cntrnsg  19375  oppgcntr  19399  cmnbascntr  19838  cntrcmnd  19875  cntrabl  19876  primefld  20823  rng2idl1cntr  21333  cntrcrng  33056