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Theorem shftfval 13854
 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 ∧ (𝑥𝐴)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.
StepHypRef Expression
1 ovex 6718 . . . . . . . . . 10 (𝑥𝐴) ∈ V
2 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
31, 2breldm 5361 . . . . . . . . 9 ((𝑥𝐴)𝐹𝑦 → (𝑥𝐴) ∈ dom 𝐹)
4 npcan 10328 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥𝐴) + 𝐴) = 𝑥)
54eqcomd 2657 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
65ancoms 468 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
7 oveq1 6697 . . . . . . . . . . 11 (𝑤 = (𝑥𝐴) → (𝑤 + 𝐴) = ((𝑥𝐴) + 𝐴))
87eqeq2d 2661 . . . . . . . . . 10 (𝑤 = (𝑥𝐴) → (𝑥 = (𝑤 + 𝐴) ↔ 𝑥 = ((𝑥𝐴) + 𝐴)))
98rspcev 3340 . . . . . . . . 9 (((𝑥𝐴) ∈ dom 𝐹𝑥 = ((𝑥𝐴) + 𝐴)) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
103, 6, 9syl2anr 494 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
11 vex 3234 . . . . . . . . 9 𝑥 ∈ V
12 eqeq1 2655 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
1312rexbidv 3081 . . . . . . . . 9 (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
1411, 13elab 3382 . . . . . . . 8 (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
1510, 14sylibr 224 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → 𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)})
161, 2brelrn 5388 . . . . . . . 8 ((𝑥𝐴)𝐹𝑦𝑦 ∈ ran 𝐹)
1716adantl 481 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
1815, 17jca 553 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹))
1918expl 647 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)))
2019ssopab2dv 5033 . . . 4 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)})
21 df-xp 5149 . . . 4 ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)}
2220, 21syl6sseqr 3685 . . 3 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹))
23 shftfval.1 . . . . . 6 𝐹 ∈ V
2423dmex 7141 . . . . 5 dom 𝐹 ∈ V
2524abrexex 7183 . . . 4 {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V
2623rnex 7142 . . . 4 ran 𝐹 ∈ V
2725, 26xpex 7004 . . 3 ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V
28 ssexg 4837 . . 3 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∧ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V)
2922, 27, 28sylancl 695 . 2 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V)
30 breq 4687 . . . . . 6 (𝑧 = 𝐹 → ((𝑥𝑤)𝑧𝑦 ↔ (𝑥𝑤)𝐹𝑦))
3130anbi2d 740 . . . . 5 (𝑧 = 𝐹 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)))
3231opabbidv 4749 . . . 4 (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)})
33 oveq2 6698 . . . . . . 7 (𝑤 = 𝐴 → (𝑥𝑤) = (𝑥𝐴))
3433breq1d 4695 . . . . . 6 (𝑤 = 𝐴 → ((𝑥𝑤)𝐹𝑦 ↔ (𝑥𝐴)𝐹𝑦))
3534anbi2d 740 . . . . 5 (𝑤 = 𝐴 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)))
3635opabbidv 4749 . . . 4 (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
37 df-shft 13851 . . . 4 shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)})
3832, 36, 37ovmpt2g 6837 . . 3 ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3923, 38mp3an1 1451 . 2 ((𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
4029, 39mpdan 703 1 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607   class class class wbr 4685  {copab 4745   × cxp 5141  dom cdm 5143  ran crn 5144  (class class class)co 6690  ℂcc 9972   + caddc 9977   − cmin 10304   shift cshi 13850 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-ltxr 10117  df-sub 10306  df-shft 13851 This theorem is referenced by:  shftdm  13855  shftfib  13856  shftfn  13857  2shfti  13864  shftidt2  13865
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