Theorem cntrval 19338

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 6905	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2		cntrval.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	eqtr4di 2794	. . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4		fveq2 6905	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		cntrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	eqtr4di 2794	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	fveq12d 6912	. . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵))
8		df-cntr 19337	. . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9		fvex 6918	. . . 4 ⊢ (𝑍‘𝐵) ∈ V
10	7, 8, 9	fvmpt 7015	. . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵))
11	10	eqcomd 2742	. 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
12		0fv 6949	. . 3 ⊢ (∅‘𝐵) = ∅
13		fvprc 6897	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
14	2, 13	eqtrid 2788	. . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
15	14	fveq1d 6907	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵))
16		fvprc 6897	. . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
17	12, 15, 16	3eqtr4a 2802	. 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 2107  Vcvv 3479  ∅c0 4332  ‘cfv 6560  Basecbs 17248  Cntzccntz 19334  Cntrccntr 19335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-cntr 19337

This theorem is referenced by:  elcntr  19349  cntrss  19350  cntri  19351  cntrsubgnsg  19362  cntrnsg  19363  oppgcntr  19385  cmnbascntr  19824  cntrcmnd  19861  cntrabl  19862  primefld  20807  rng2idl1cntr  21316  cntrcrng  33074