Theorem cntzval 19363

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 variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		cntzfval.p	. . . . 5 ⊢ + = (+g‘𝑀)
3		cntzfval.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzfval 19362	. . . 4 ⊢ (𝑀 ∈ V → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
5	4	fveq1d 6871	. . 3 ⊢ (𝑀 ∈ V → (𝑍‘𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆))
6	1	fvexi 6883	. . . . 5 ⊢ 𝐵 ∈ V
7	6	elpw2 5292	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
8		raleq 3319	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
9	8	rabbidv 3423	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
10		eqid 2764	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
11	6	rabex 5297	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ∈ V
12	9, 10, 11	fvmpt 6977	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
13	7, 12	sylbir 237	. . 3 ⊢ (𝑆 ⊆ 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
14	5, 13	sylan9eq 2819	. 2 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
15		0fv 6910	. . . 4 ⊢ (∅‘𝑆) = ∅
16		fvprc 6861	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
17	3, 16	eqtrid 2811	. . . . 5 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
18	17	fveq1d 6871	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆))
19		ssrab2 4035	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ 𝐵
20		fvprc 6861	. . . . . . 7 ⊢ (¬ 𝑀 ∈ V → (Base‘𝑀) = ∅)
21	1, 20	eqtrid 2811	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → 𝐵 = ∅)
22	19, 21	sseqtrid 3980	. . . . 5 ⊢ (¬ 𝑀 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅)
23		ss0 4358	. . . . 5 ⊢ ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅ → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = ∅)
24	22, 23	syl 17	. . . 4 ⊢ (¬ 𝑀 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = ∅)
25	15, 18, 24	3eqtr4a 2825	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
26	25	adantr 484	. 2 ⊢ ((¬ 𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
27	14, 26	pm2.61ian 821	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  Vcvv 3456   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557   ↦ cmpt 5183  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  Cntzccntz 19357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-cntz 19359

This theorem is referenced by:  elcntz  19364  cntzsnval  19366  sscntz  19368  cntzssv  19370  cntziinsn  19379  cmnbascntr  19847