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Theorem fwddifval 35129
Description: Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)
Hypotheses
Ref Expression
fwddifval.1 (πœ‘ β†’ 𝐴 βŠ† β„‚)
fwddifval.2 (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)
fwddifval.3 (πœ‘ β†’ 𝑋 ∈ 𝐴)
fwddifval.4 (πœ‘ β†’ (𝑋 + 1) ∈ 𝐴)
Assertion
Ref Expression
fwddifval (πœ‘ β†’ (( β–³ β€˜πΉ)β€˜π‘‹) = ((πΉβ€˜(𝑋 + 1)) βˆ’ (πΉβ€˜π‘‹)))

Proof of Theorem fwddifval
Dummy variables π‘₯ 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fwddif 35126 . . . 4 β–³ = (𝑓 ∈ (β„‚ ↑pm β„‚) ↦ (π‘₯ ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((π‘“β€˜(π‘₯ + 1)) βˆ’ (π‘“β€˜π‘₯))))
2 dmeq 5903 . . . . . 6 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
32eleq2d 2819 . . . . . 6 (𝑓 = 𝐹 β†’ ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
42, 3rabeqbidv 3449 . . . . 5 (𝑓 = 𝐹 β†’ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5 fveq1 6890 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯ + 1)) = (πΉβ€˜(π‘₯ + 1)))
6 fveq1 6890 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
75, 6oveq12d 7426 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘“β€˜(π‘₯ + 1)) βˆ’ (π‘“β€˜π‘₯)) = ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯)))
84, 7mpteq12dv 5239 . . . 4 (𝑓 = 𝐹 β†’ (π‘₯ ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((π‘“β€˜(π‘₯ + 1)) βˆ’ (π‘“β€˜π‘₯))) = (π‘₯ ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯))))
9 fwddifval.2 . . . . 5 (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)
10 fwddifval.1 . . . . 5 (πœ‘ β†’ 𝐴 βŠ† β„‚)
11 cnex 11190 . . . . . 6 β„‚ ∈ V
12 elpm2r 8838 . . . . . 6 (((β„‚ ∈ V ∧ β„‚ ∈ V) ∧ (𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚)) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
1311, 11, 12mpanl12 700 . . . . 5 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
149, 10, 13syl2anc 584 . . . 4 (πœ‘ β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
159fdmd 6728 . . . . . 6 (πœ‘ β†’ dom 𝐹 = 𝐴)
1611a1i 11 . . . . . . 7 (πœ‘ β†’ β„‚ ∈ V)
1716, 10ssexd 5324 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ V)
1815, 17eqeltrd 2833 . . . . 5 (πœ‘ β†’ dom 𝐹 ∈ V)
19 rabexg 5331 . . . . 5 (dom 𝐹 ∈ V β†’ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20 mptexg 7222 . . . . 5 ({𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V β†’ (π‘₯ ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯))) ∈ V)
2118, 19, 203syl 18 . . . 4 (πœ‘ β†’ (π‘₯ ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯))) ∈ V)
221, 8, 14, 21fvmptd3 7021 . . 3 (πœ‘ β†’ ( β–³ β€˜πΉ) = (π‘₯ ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯))))
2315eleq2d 2819 . . . . 5 (πœ‘ β†’ ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
2415, 23rabeqbidv 3449 . . . 4 (πœ‘ β†’ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
2524mpteq1d 5243 . . 3 (πœ‘ β†’ (π‘₯ ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯))) = (π‘₯ ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯))))
2622, 25eqtrd 2772 . 2 (πœ‘ β†’ ( β–³ β€˜πΉ) = (π‘₯ ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯))))
27 fvoveq1 7431 . . . 4 (π‘₯ = 𝑋 β†’ (πΉβ€˜(π‘₯ + 1)) = (πΉβ€˜(𝑋 + 1)))
28 fveq2 6891 . . . 4 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
2927, 28oveq12d 7426 . . 3 (π‘₯ = 𝑋 β†’ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯)) = ((πΉβ€˜(𝑋 + 1)) βˆ’ (πΉβ€˜π‘‹)))
3029adantl 482 . 2 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯)) = ((πΉβ€˜(𝑋 + 1)) βˆ’ (πΉβ€˜π‘‹)))
31 fwddifval.3 . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐴)
32 fwddifval.4 . . 3 (πœ‘ β†’ (𝑋 + 1) ∈ 𝐴)
33 oveq1 7415 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 + 1) = (𝑋 + 1))
3433eleq1d 2818 . . . 4 (𝑦 = 𝑋 β†’ ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
3534elrab 3683 . . 3 (𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
3631, 32, 35sylanbrc 583 . 2 (πœ‘ β†’ 𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37 ovexd 7443 . 2 (πœ‘ β†’ ((πΉβ€˜(𝑋 + 1)) βˆ’ (πΉβ€˜π‘‹)) ∈ V)
3826, 30, 36, 37fvmptd 7005 1 (πœ‘ β†’ (( β–³ β€˜πΉ)β€˜π‘‹) = ((πΉβ€˜(𝑋 + 1)) βˆ’ (πΉβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   βŠ† wss 3948   ↦ cmpt 5231  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑pm cpm 8820  β„‚cc 11107  1c1 11110   + caddc 11112   βˆ’ cmin 11443   β–³ cfwddif 35125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-pm 8822  df-fwddif 35126
This theorem is referenced by: (None)
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