Theorem cntrval 14145

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 5729	. . . 4 ⊢ (𝑘 ∈ (𝑍‘𝐵) → ∃𝑗 𝑗 ∈ 𝑍)
2		df-cntz 14142	. . . . . . 7 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
3	2	mptrcl 5788	. . . . . 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 14143	. . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9	8	mptrcl 5788	. . 3 ⊢ (𝑘 ∈ (Cntr‘𝑀) → 𝑀 ∈ V)
10		fveq2 5695	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
11	10, 4	eqtr4di 2289	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
12		fveq2 5695	. . . . . . . 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 5702	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵))
16		id 19	. . . . . 6 ⊢ (𝑀 ∈ V → 𝑀 ∈ V)
17	4	cntzex 14144	. . . . . . 7 ⊢ (𝑀 ∈ V → 𝑍 ∈ V)
18		basfn 13463	. . . . . . . . 9 ⊢ Base Fn V
19		funfvex 5712	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑀 ∈ dom Base) → (Base‘𝑀) ∈ V)
20	19	funfni 5483	. . . . . . . . 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 5714	. . . . . . 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 5799	. . . . 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 4192   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  Basecbs 13404  +_gcplusg 13484  Cntzccntz 14140  Cntrccntr 14141

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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-inn 9308  df-ndx 13407  df-slot 13408  df-base 13410  df-cntz 14142  df-cntr 14143

This theorem is used by:  elcntr  14157  cntrss  14158  cntri  14159  cntrsubgnsg  14169  cntrnsg  14170