Theorem shftfvalg 11378

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 529	. . . . . . . . . . 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 8489	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ ℂ)
6		vex 2805	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7		breldmg 4937	. . . . . . . . . . 11 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ 𝑦 ∈ V ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 − 𝐴) ∈ dom 𝐹)
8	6, 7	mp3an2 1361	. . . . . . . . . 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 8387	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
11	10	eqcomd 2237	. . . . . . . . . . 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 6024	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 − 𝐴) → (𝑤 + 𝐴) = ((𝑥 − 𝐴) + 𝐴))
15	14	eqeq2d 2243	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 − 𝐴) → (𝑥 = (𝑤 + 𝐴) ↔ 𝑥 = ((𝑥 − 𝐴) + 𝐴)))
16	15	rspcev 2910	. . . . . . . . 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 2805	. . . . . . . . 9 ⊢ 𝑥 ∈ V
19		eqeq1 2238	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
20	19	rexbidv 2533	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
21	18, 20	elab 2950	. . . . . . . 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 4963	. . . . . . . . 9 ⊢ (((𝑥 − 𝐴) ∈ ℂ ∧ 𝑦 ∈ V ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
24	6, 23	mp3an2 1361	. . . . . . . 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 4373	. . . 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 4731	. . . 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 3276	. . 3 ⊢ (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹))
31		dmexg 4996	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
32		abrexexg 6279	. . . . 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 4997	. . . 4 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
35		xpexg 4840	. . . 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 4228	. . 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 2814	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
40		breq 4090	. . . . . 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 4155	. . . 4 ⊢ (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)})
43		oveq2 6025	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 − 𝑤) = (𝑥 − 𝐴))
44	43	breq1d 4098	. . . . . 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 4155	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
47		df-shft 11375	. . . 4 ⊢ shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)})
48	42, 46, 47	ovmpog 6155	. . 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 1306	. 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 1273	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {cab 2217  ∃wrex 2511  Vcvv 2802   ⊆ wss 3200   class class class wbr 4088  {copab 4149   × cxp 4723  dom cdm 4725  ran crn 4726  (class class class)co 6017  ℂcc 8029   + caddc 8034   − cmin 8349   shift cshi 11374

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351  df-shft 11375

This theorem is referenced by:  ovshftex  11379  shftfibg  11380  2shfti  11391