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Theorem fwddifval 34800
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 34797 . . . 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 5863 . . . . . 6 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
32eleq2d 2820 . . . . . 6 (𝑓 = 𝐹 β†’ ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
42, 3rabeqbidv 3423 . . . . 5 (𝑓 = 𝐹 β†’ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5 fveq1 6845 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯ + 1)) = (πΉβ€˜(π‘₯ + 1)))
6 fveq1 6845 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
75, 6oveq12d 7379 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘“β€˜(π‘₯ + 1)) βˆ’ (π‘“β€˜π‘₯)) = ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯)))
84, 7mpteq12dv 5200 . . . 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 11140 . . . . . 6 β„‚ ∈ V
12 elpm2r 8789 . . . . . 6 (((β„‚ ∈ V ∧ β„‚ ∈ V) ∧ (𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚)) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
1311, 11, 12mpanl12 701 . . . . 5 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
149, 10, 13syl2anc 585 . . . 4 (πœ‘ β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
159fdmd 6683 . . . . . 6 (πœ‘ β†’ dom 𝐹 = 𝐴)
1611a1i 11 . . . . . . 7 (πœ‘ β†’ β„‚ ∈ V)
1716, 10ssexd 5285 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ V)
1815, 17eqeltrd 2834 . . . . 5 (πœ‘ β†’ dom 𝐹 ∈ V)
19 rabexg 5292 . . . . 5 (dom 𝐹 ∈ V β†’ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20 mptexg 7175 . . . . 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 6975 . . 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 2820 . . . . 5 (πœ‘ β†’ ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
2415, 23rabeqbidv 3423 . . . 4 (πœ‘ β†’ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
2524mpteq1d 5204 . . 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 2773 . 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 7384 . . . 4 (π‘₯ = 𝑋 β†’ (πΉβ€˜(π‘₯ + 1)) = (πΉβ€˜(𝑋 + 1)))
28 fveq2 6846 . . . 4 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
2927, 28oveq12d 7379 . . 3 (π‘₯ = 𝑋 β†’ ((πΉβ€˜(π‘₯ + 1)) βˆ’ (πΉβ€˜π‘₯)) = ((πΉβ€˜(𝑋 + 1)) βˆ’ (πΉβ€˜π‘‹)))
3029adantl 483 . 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 7368 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 + 1) = (𝑋 + 1))
3433eleq1d 2819 . . . 4 (𝑦 = 𝑋 β†’ ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
3534elrab 3649 . . 3 (𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
3631, 32, 35sylanbrc 584 . 2 (πœ‘ β†’ 𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37 ovexd 7396 . 2 (πœ‘ β†’ ((πΉβ€˜(𝑋 + 1)) βˆ’ (πΉβ€˜π‘‹)) ∈ V)
3826, 30, 36, 37fvmptd 6959 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 397   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3447   βŠ† wss 3914   ↦ cmpt 5192  dom cdm 5637  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑pm cpm 8772  β„‚cc 11057  1c1 11060   + caddc 11062   βˆ’ cmin 11393   β–³ cfwddif 34796
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-pm 8774  df-fwddif 34797
This theorem is referenced by: (None)
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