Theorem cntrval 19231

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 6822	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2		cntrval.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	eqtr4di 2784	. . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4		fveq2 6822	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		cntrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	eqtr4di 2784	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	fveq12d 6829	. . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵))
8		df-cntr 19230	. . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9		fvex 6835	. . . 4 ⊢ (𝑍‘𝐵) ∈ V
10	7, 8, 9	fvmpt 6929	. . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵))
11	10	eqcomd 2737	. 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
12		0fv 6863	. . 3 ⊢ (∅‘𝐵) = ∅
13		fvprc 6814	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
14	2, 13	eqtrid 2778	. . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
15	14	fveq1d 6824	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵))
16		fvprc 6814	. . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
17	12, 15, 16	3eqtr4a 2792	. 2 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
18	11, 17	pm2.61i 182	1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4280  ‘cfv 6481  Basecbs 17120  Cntzccntz 19227  Cntrccntr 19228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-cntr 19230

This theorem is referenced by:  elcntr  19242  cntrss  19243  cntri  19244  cntrsubgnsg  19255  cntrnsg  19256  oppgcntr  19277  cmnbascntr  19717  cntrcmnd  19754  cntrabl  19755  primefld  20720  rng2idl1cntr  21242  cntrcrng  33050