Theorem shftfvalg 10965

Description: The value of the sequence shifter operation is a function on ℂ. 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
shftfvalg	⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem shftfvalg

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉)
2		simpl 109	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → 𝐴 ∈ ℂ)
3		simplr 528	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑥 ∈ ℂ)
4		simpll 527	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝐴 ∈ ℂ)
5	3, 4	subcld 8332	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ ℂ)
6		vex 2763	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7		breldmg 4869	. . . . . . . . . . 11 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ 𝑦 ∈ V ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ dom 𝐹)
8	6, 7	mp3an2 1336	. . . . . . . . . 10 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ dom 𝐹)
9	5, 8	sylancom 420	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ dom 𝐹)
10		npcan 8230	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
11	10	eqcomd 2199	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
12	11	ancoms 268	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
13	12	adantr 276	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
14		oveq1 5926	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 − 𝐴) → (𝑤 + 𝐴) = ((𝑥 − 𝐴) + 𝐴))
15	14	eqeq2d 2205	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 − 𝐴) → (𝑥 = (𝑤 + 𝐴) ↔ 𝑥 = ((𝑥 − 𝐴) + 𝐴)))
16	15	rspcev 2865	. . . . . . . . 9 ⊢ (((𝑥 − 𝐴) ∈ dom 𝐹 ∧ 𝑥 = ((𝑥 − 𝐴) + 𝐴)) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
17	9, 13, 16	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
18		vex 2763	. . . . . . . . 9 ⊢ 𝑥 ∈ V
19		eqeq1 2200	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
20	19	rexbidv 2495	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
21	18, 20	elab 2905	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
22	17, 21	sylibr 134	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)})
23		brelrng 4894	. . . . . . . . 9 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ 𝑦 ∈ V ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
24	6, 23	mp3an2 1336	. . . . . . . 8 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
25	5, 24	sylancom 420	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
26	22, 25	jca 306	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹))
27	26	expl 378	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)))
28	27	ssopab2dv 4310	. . . 4 ⊢ (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)})
29		df-xp 4666	. . . 4 ⊢ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)}
30	28, 29	sseqtrrdi 3229	. . 3 ⊢ (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹))
31		dmexg 4927	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
32		abrexexg 6172	. . . . 5 ⊢ (dom 𝐹 ∈ V → {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V)
33	31, 32	syl 14	. . . 4 ⊢ (𝐹 ∈ 𝑉 → {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V)
34		rnexg 4928	. . . 4 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
35		xpexg 4774	. . . 4 ⊢ (({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V ∧ ran 𝐹 ∈ V) → ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V)
36	33, 34, 35	syl2anc 411	. . 3 ⊢ (𝐹 ∈ 𝑉 → ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V)
37		ssexg 4169	. . 3 ⊢ (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∧ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V)
38	30, 36, 37	syl2an 289	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V)
39		elex 2771	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
40		breq 4032	. . . . . 6 ⊢ (𝑧 = 𝐹 → ((𝑥 − 𝑤)𝑧𝑦 ↔ (𝑥 − 𝑤)𝐹𝑦))
41	40	anbi2d 464	. . . . 5 ⊢ (𝑧 = 𝐹 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)))
42	41	opabbidv 4096	. . . 4 ⊢ (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)})
43		oveq2 5927	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 − 𝑤) = (𝑥 − 𝐴))
44	43	breq1d 4040	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑥 − 𝑤)𝐹𝑦 ↔ (𝑥 − 𝐴)𝐹𝑦))
45	44	anbi2d 464	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)))
46	45	opabbidv 4096	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
47		df-shft 10962	. . . 4 ⊢ shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)})
48	42, 46, 47	ovmpog 6054	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
49	39, 48	syl3an1 1282	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
50	1, 2, 38, 49	syl3anc 1249	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473  Vcvv 2760   ⊆ wss 3154   class class class wbr 4030  {copab 4090   × cxp 4658  dom cdm 4660  ran crn 4661  (class class class)co 5919  ℂcc 7872   + caddc 7877   − cmin 8192   shift cshi 10961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-sub 8194  df-shft 10962

This theorem is referenced by:  ovshftex  10966  shftfibg  10967  2shfti  10978