Theorem cntzval 14153

Description: Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cntzfval.b	⊢ 𝐵 = (Base‘𝑀)
cntzfval.p	⊢ + = (+g‘𝑀)
cntzfval.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzval	⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})

Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐵   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑦)   𝑍(𝑥, 𝑦)

Proof of Theorem cntzval

Dummy variables 𝑚 𝑠 𝑤 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvm 5732	. . . . 5 ⊢ (𝑤 ∈ (𝑍‘𝑆) → ∃𝑗 𝑗 ∈ 𝑍)
2		df-cntz 14148	. . . . . . . 8 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
3	2	mptrcl 5791	. . . . . . 7 ⊢ (𝑗 ∈ (Cntz‘𝑀) → 𝑀 ∈ V)
4		cntzfval.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝑀)
5	3, 4	eleq2s 2333	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ∈ V)
6	5	exlimiv 1651	. . . . 5 ⊢ (∃𝑗 𝑗 ∈ 𝑍 → 𝑀 ∈ V)
7	1, 6	syl 14	. . . 4 ⊢ (𝑤 ∈ (𝑍‘𝑆) → 𝑀 ∈ V)
8	7	a1i 9	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝑤 ∈ (𝑍‘𝑆) → 𝑀 ∈ V))
9		elrabi 2979	. . . . 5 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} → 𝑤 ∈ 𝐵)
10		cntzfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
11	10	basmex 13470	. . . . 5 ⊢ (𝑤 ∈ 𝐵 → 𝑀 ∈ V)
12	9, 11	syl 14	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} → 𝑀 ∈ V)
13	12	a1i 9	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} → 𝑀 ∈ V))
14		raleq 2749	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
15	14	rabbidv 2810	. . . . . 6 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
16		cntzfval.p	. . . . . . . 8 ⊢ + = (+g‘𝑀)
17	10, 16, 4	cntzfval 14152	. . . . . . 7 ⊢ (𝑀 ∈ V → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
18	17	adantr 276	. . . . . 6 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
19		simpr 110	. . . . . . 7 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
20		basfn 13469	. . . . . . . . . 10 ⊢ Base Fn V
21		simpl 109	. . . . . . . . . 10 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → 𝑀 ∈ V)
22		funfvex 5715	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝑀 ∈ dom Base) → (Base‘𝑀) ∈ V)
23	22	funfni 5486	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝑀 ∈ V) → (Base‘𝑀) ∈ V)
24	20, 21, 23	sylancr 418	. . . . . . . . 9 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (Base‘𝑀) ∈ V)
25	10, 24	eqeltrid 2325	. . . . . . . 8 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → 𝐵 ∈ V)
26		elpw2g 4294	. . . . . . . 8 ⊢ (𝐵 ∈ V → (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵))
27	25, 26	syl 14	. . . . . . 7 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵))
28	19, 27	mpbird 167	. . . . . 6 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵)
29		eqid 2238	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}
30	29, 25	rabexd 4283	. . . . . 6 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ∈ V)
31	15, 18, 28, 30	fvmptd4 5803	. . . . 5 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
32	31	eleq2d 2308	. . . 4 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (𝑤 ∈ (𝑍‘𝑆) ↔ 𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
33	32	expcom 116	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝑀 ∈ V → (𝑤 ∈ (𝑍‘𝑆) ↔ 𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})))
34	8, 13, 33	pm5.21ndd 717	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝑤 ∈ (𝑍‘𝑆) ↔ 𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
35	34	eqrdv 2236	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688   ↦ cmpt 4193   Fn wfn 5375  ‘cfv 5380  (class class class)co 6088  Basecbs 13410  +_gcplusg 13490  Cntzccntz 14146

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

This theorem is used by:  elcntz  14154  cntzsnval  14156  sscntz  14158  cntzssv  14160