Theorem cntrval 18451

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 6672	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2		cntrval.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	syl6eqr 2876	. . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4		fveq2 6672	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		cntrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	syl6eqr 2876	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	fveq12d 6679	. . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵))
8		df-cntr 18450	. . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9		fvex 6685	. . . 4 ⊢ (𝑍‘𝐵) ∈ V
10	7, 8, 9	fvmpt 6770	. . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵))
11	10	eqcomd 2829	. 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
12		0fv 6711	. . 3 ⊢ (∅‘𝐵) = ∅
13		fvprc 6665	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
14	2, 13	syl5eq 2870	. . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
15	14	fveq1d 6674	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵))
16		fvprc 6665	. . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
17	12, 15, 16	3eqtr4a 2884	. 2 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
18	11, 17	pm2.61i 184	1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ∅c0 4293  ‘cfv 6357  Basecbs 16485  Cntzccntz 18447  Cntrccntr 18448

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-cntr 18450

This theorem is referenced by:  cntrss  18462  cntri  18463  cntrsubgnsg  18473  cntrnsg  18474  oppgcntr  18495  cntrcmnd  18964  cntrabl  18965  primefld  19586  cntrcrng  30699