Theorem shftfvalg 10760

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 109	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉)
2		simpl 108	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → 𝐴 ∈ ℂ)
3		simplr 520	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑥 ∈ ℂ)
4		simpll 519	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝐴 ∈ ℂ)
5	3, 4	subcld 8209	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ ℂ)
6		vex 2729	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7		breldmg 4810	. . . . . . . . . . 11 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ 𝑦 ∈ V ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ dom 𝐹)
8	6, 7	mp3an2 1315	. . . . . . . . . 10 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ dom 𝐹)
9	5, 8	sylancom 417	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ dom 𝐹)
10		npcan 8107	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
11	10	eqcomd 2171	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
12	11	ancoms 266	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
13	12	adantr 274	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
14		oveq1 5849	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 − 𝐴) → (𝑤 + 𝐴) = ((𝑥 − 𝐴) + 𝐴))
15	14	eqeq2d 2177	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 − 𝐴) → (𝑥 = (𝑤 + 𝐴) ↔ 𝑥 = ((𝑥 − 𝐴) + 𝐴)))
16	15	rspcev 2830	. . . . . . . . 9 ⊢ (((𝑥 − 𝐴) ∈ dom 𝐹 ∧ 𝑥 = ((𝑥 − 𝐴) + 𝐴)) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
17	9, 13, 16	syl2anc 409	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
18		vex 2729	. . . . . . . . 9 ⊢ 𝑥 ∈ V
19		eqeq1 2172	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
20	19	rexbidv 2467	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
21	18, 20	elab 2870	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
22	17, 21	sylibr 133	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)})
23		brelrng 4835	. . . . . . . . 9 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ 𝑦 ∈ V ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
24	6, 23	mp3an2 1315	. . . . . . . 8 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
25	5, 24	sylancom 417	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
26	22, 25	jca 304	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹))
27	26	expl 376	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)))
28	27	ssopab2dv 4256	. . . 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 4610	. . . 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 3191	. . 3 ⊢ (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹))
31		dmexg 4868	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
32		abrexexg 6086	. . . . 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 4869	. . . 4 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
35		xpexg 4718	. . . 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 409	. . 3 ⊢ (𝐹 ∈ 𝑉 → ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V)
37		ssexg 4121	. . 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 287	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V)
39		elex 2737	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
40		breq 3984	. . . . . 6 ⊢ (𝑧 = 𝐹 → ((𝑥 − 𝑤)𝑧𝑦 ↔ (𝑥 − 𝑤)𝐹𝑦))
41	40	anbi2d 460	. . . . 5 ⊢ (𝑧 = 𝐹 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)))
42	41	opabbidv 4048	. . . 4 ⊢ (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)})
43		oveq2 5850	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 − 𝑤) = (𝑥 − 𝐴))
44	43	breq1d 3992	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑥 − 𝑤)𝐹𝑦 ↔ (𝑥 − 𝐴)𝐹𝑦))
45	44	anbi2d 460	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)))
46	45	opabbidv 4048	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
47		df-shft 10757	. . . 4 ⊢ shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)})
48	42, 46, 47	ovmpog 5976	. . 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 1261	. 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 1228	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {cab 2151  ∃wrex 2445  Vcvv 2726   ⊆ wss 3116   class class class wbr 3982  {copab 4042   × cxp 4602  dom cdm 4604  ran crn 4605  (class class class)co 5842  ℂcc 7751   + caddc 7756   − cmin 8069   shift cshi 10756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-shft 10757

This theorem is referenced by:  ovshftex  10761  shftfibg  10762  2shfti  10773