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Theorem fwddifval 34464
Description: Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)
Hypotheses
Ref Expression
fwddifval.1 (𝜑𝐴 ⊆ ℂ)
fwddifval.2 (𝜑𝐹:𝐴⟶ℂ)
fwddifval.3 (𝜑𝑋𝐴)
fwddifval.4 (𝜑 → (𝑋 + 1) ∈ 𝐴)
Assertion
Ref Expression
fwddifval (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))

Proof of Theorem fwddifval
Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fwddif 34461 . . . 4 △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))))
2 dmeq 5812 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
32eleq2d 2824 . . . . . 6 (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
42, 3rabeqbidv 3420 . . . . 5 (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5 fveq1 6773 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6 fveq1 6773 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
75, 6oveq12d 7293 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)))
84, 7mpteq12dv 5165 . . . 4 (𝑓 = 𝐹 → (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
9 fwddifval.2 . . . . 5 (𝜑𝐹:𝐴⟶ℂ)
10 fwddifval.1 . . . . 5 (𝜑𝐴 ⊆ ℂ)
11 cnex 10952 . . . . . 6 ℂ ∈ V
12 elpm2r 8633 . . . . . 6 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
1311, 11, 12mpanl12 699 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
149, 10, 13syl2anc 584 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
159fdmd 6611 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1611a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
1716, 10ssexd 5248 . . . . . 6 (𝜑𝐴 ∈ V)
1815, 17eqeltrd 2839 . . . . 5 (𝜑 → dom 𝐹 ∈ V)
19 rabexg 5255 . . . . 5 (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20 mptexg 7097 . . . . 5 ({𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) ∈ V)
2118, 19, 203syl 18 . . . 4 (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) ∈ V)
221, 8, 14, 21fvmptd3 6898 . . 3 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2315eleq2d 2824 . . . . 5 (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
2415, 23rabeqbidv 3420 . . . 4 (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
2524mpteq1d 5169 . . 3 (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2622, 25eqtrd 2778 . 2 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
27 fvoveq1 7298 . . . 4 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28 fveq2 6774 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2927, 28oveq12d 7293 . . 3 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
3029adantl 482 . 2 ((𝜑𝑥 = 𝑋) → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
31 fwddifval.3 . . 3 (𝜑𝑋𝐴)
32 fwddifval.4 . . 3 (𝜑 → (𝑋 + 1) ∈ 𝐴)
33 oveq1 7282 . . . . 5 (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
3433eleq1d 2823 . . . 4 (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
3534elrab 3624 . . 3 (𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
3631, 32, 35sylanbrc 583 . 2 (𝜑𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37 ovexd 7310 . 2 (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)) ∈ V)
3826, 30, 36, 37fvmptd 6882 1 (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {crab 3068  Vcvv 3432  wss 3887  cmpt 5157  dom cdm 5589  wf 6429  cfv 6433  (class class class)co 7275  pm cpm 8616  cc 10869  1c1 10872   + caddc 10874  cmin 11205  cfwddif 34460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-pm 8618  df-fwddif 34461
This theorem is referenced by: (None)
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