Theorem cntzval 19260

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 19259	. . . 4 ⊢ (𝑀 ∈ V → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
5	4	fveq1d 6863	. . 3 ⊢ (𝑀 ∈ V → (𝑍‘𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆))
6	1	fvexi 6875	. . . . 5 ⊢ 𝐵 ∈ V
7	6	elpw2 5292	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
8		raleq 3298	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
9	8	rabbidv 3416	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
10		eqid 2730	. . . . 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 6971	. . . 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 235	. . 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 2785	. 2 ⊢ ((𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
15		0fv 6905	. . . 4 ⊢ (∅‘𝑆) = ∅
16		fvprc 6853	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
17	3, 16	eqtrid 2777	. . . . 5 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
18	17	fveq1d 6863	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆))
19		ssrab2 4046	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ 𝐵
20		fvprc 6853	. . . . . . 7 ⊢ (¬ 𝑀 ∈ V → (Base‘𝑀) = ∅)
21	1, 20	eqtrid 2777	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → 𝐵 = ∅)
22	19, 21	sseqtrid 3992	. . . . 5 ⊢ (¬ 𝑀 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅)
23		ss0 4368	. . . . 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 2791	. . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
26	25	adantr 480	. 2 ⊢ ((¬ 𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
27	14, 26	pm2.61ian 811	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  Cntzccntz 19254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-cntz 19256

This theorem is referenced by:  elcntz  19261  cntzsnval  19263  sscntz  19265  cntzssv  19267  cntziinsn  19276  cmnbascntr  19742