Theorem shftfval 14993

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.)

Hypothesis

Ref	Expression
shftfval.1	⊢ 𝐹 ∈ V

Assertion

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

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

Proof of Theorem shftfval

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

Step	Hyp	Ref	Expression
1		ovex 7391	. . . . . . . . . 10 ⊢ (𝑥 − 𝐴) ∈ V
2		vex 3444	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	1, 2	breldm 5857	. . . . . . . . 9 ⊢ ((𝑥 − 𝐴)𝐹𝑦 → (𝑥 − 𝐴) ∈ dom 𝐹)
4		npcan 11389	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
5	4	eqcomd 2742	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
6	5	ancoms 458	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 = ((𝑥 − 𝐴) + 𝐴))
7		oveq1 7365	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 − 𝐴) → (𝑤 + 𝐴) = ((𝑥 − 𝐴) + 𝐴))
8	7	rspceeqv 3599	. . . . . . . . 9 ⊢ (((𝑥 − 𝐴) ∈ dom 𝐹 ∧ 𝑥 = ((𝑥 − 𝐴) + 𝐴)) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
9	3, 6, 8	syl2anr 597	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
10		vex 3444	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
12	11	rexbidv 3160	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
13	10, 12	elab 3634	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
14	9, 13	sylibr 234	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)})
15	1, 2	brelrn 5891	. . . . . . . 8 ⊢ ((𝑥 − 𝐴)𝐹𝑦 → 𝑦 ∈ ran 𝐹)
16	15	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
17	14, 16	jca 511	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹))
18	17	expl 457	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)))
19	18	ssopab2dv 5499	. . . 4 ⊢ (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)})
20		df-xp 5630	. . . 4 ⊢ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)}
21	19, 20	sseqtrrdi 3975	. . 3 ⊢ (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹))
22		shftfval.1	. . . . . 6 ⊢ 𝐹 ∈ V
23	22	dmex 7851	. . . . 5 ⊢ dom 𝐹 ∈ V
24	23	abrexex 7906	. . . 4 ⊢ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V
25	22	rnex 7852	. . . 4 ⊢ ran 𝐹 ∈ V
26	24, 25	xpex 7698	. . 3 ⊢ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V
27		ssexg 5268	. . 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)
28	21, 26, 27	sylancl 586	. 2 ⊢ (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V)
29		breq 5100	. . . . . 6 ⊢ (𝑧 = 𝐹 → ((𝑥 − 𝑤)𝑧𝑦 ↔ (𝑥 − 𝑤)𝐹𝑦))
30	29	anbi2d 630	. . . . 5 ⊢ (𝑧 = 𝐹 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)))
31	30	opabbidv 5164	. . . 4 ⊢ (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)})
32		oveq2 7366	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 − 𝑤) = (𝑥 − 𝐴))
33	32	breq1d 5108	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑥 − 𝑤)𝐹𝑦 ↔ (𝑥 − 𝐴)𝐹𝑦))
34	33	anbi2d 630	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)))
35	34	opabbidv 5164	. . . 4 ⊢ (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
36		df-shft 14990	. . . 4 ⊢ shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝑤)𝑧𝑦)})
37	31, 35, 36	ovmpog 7517	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
38	22, 37	mp3an1 1450	. 2 ⊢ ((𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
39	28, 38	mpdan 687	1 ⊢ (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901   class class class wbr 5098  {copab 5160   × cxp 5622  dom cdm 5624  ran crn 5625  (class class class)co 7358  ℂcc 11024   + caddc 11029   − cmin 11364   shift cshi 14989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-sub 11366  df-shft 14990

This theorem is referenced by:  shftdm  14994  shftfib  14995  shftfn  14996  2shfti  15003  shftidt2  15004