Theorem cntrval 14151

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 variables 𝑘 𝑚 𝑗 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvm 5732	. . . 4 ⊢ (𝑘 ∈ (𝑍‘𝐵) → ∃𝑗 𝑗 ∈ 𝑍)
2		df-cntz 14148	. . . . . . 7 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
3	2	mptrcl 5791	. . . . . 6 ⊢ (𝑗 ∈ (Cntz‘𝑀) → 𝑀 ∈ V)
4		cntrval.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
5	3, 4	eleq2s 2333	. . . . 5 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ∈ V)
6	5	exlimiv 1651	. . . 4 ⊢ (∃𝑗 𝑗 ∈ 𝑍 → 𝑀 ∈ V)
7	1, 6	syl 14	. . 3 ⊢ (𝑘 ∈ (𝑍‘𝐵) → 𝑀 ∈ V)
8		df-cntr 14149	. . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9	8	mptrcl 5791	. . 3 ⊢ (𝑘 ∈ (Cntr‘𝑀) → 𝑀 ∈ V)
10		fveq2 5698	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
11	10, 4	eqtr4di 2289	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
12		fveq2 5698	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
13		cntrval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
14	12, 13	eqtr4di 2289	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
15	11, 14	fveq12d 5705	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵))
16		id 19	. . . . . 6 ⊢ (𝑀 ∈ V → 𝑀 ∈ V)
17	4	cntzex 14150	. . . . . . 7 ⊢ (𝑀 ∈ V → 𝑍 ∈ V)
18		basfn 13469	. . . . . . . . 9 ⊢ Base Fn V
19		funfvex 5715	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑀 ∈ dom Base) → (Base‘𝑀) ∈ V)
20	19	funfni 5486	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑀 ∈ V) → (Base‘𝑀) ∈ V)
21	18, 20	mpan 428	. . . . . . . 8 ⊢ (𝑀 ∈ V → (Base‘𝑀) ∈ V)
22	13, 21	eqeltrid 2325	. . . . . . 7 ⊢ (𝑀 ∈ V → 𝐵 ∈ V)
23		fvexg 5717	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ 𝐵 ∈ V) → (𝑍‘𝐵) ∈ V)
24	17, 22, 23	syl2anc 415	. . . . . 6 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) ∈ V)
25	8, 15, 16, 24	fvmptd3 5802	. . . . 5 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵))
26	25	eqcomd 2244	. . . 4 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀))
27	26	eleq2d 2308	. . 3 ⊢ (𝑀 ∈ V → (𝑘 ∈ (𝑍‘𝐵) ↔ 𝑘 ∈ (Cntr‘𝑀)))
28	7, 9, 27	pm5.21nii 716	. 2 ⊢ (𝑘 ∈ (𝑍‘𝐵) ↔ 𝑘 ∈ (Cntr‘𝑀))
29	28	eqriv 2235	1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀)

This proof depends on syntax axioms:   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821  𝒫 cpw 3688   ↦ cmpt 4193   Fn wfn 5375  ‘cfv 5380  (class class class)co 6088  Basecbs 13410  +_gcplusg 13490  Cntzccntz 14146  Cntrccntr 14147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-inn 9314  df-ndx 13413  df-slot 13414  df-base 13416  df-cntz 14148  df-cntr 14149

This theorem is used by:  elcntr  14163  cntrss  14164  cntri  14165  cntrsubgnsg  14175  cntrnsg  14176