![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fargshiftfv | Structured version Visualization version GIF version |
Description: If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.) |
Ref | Expression |
---|---|
fargshift.g | ⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1))) |
Ref | Expression |
---|---|
fargshiftfv | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺‘𝑋) = (𝐹‘(𝑋 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6293 | . . . . 5 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁)) | |
2 | fseq1hash 13484 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁) | |
3 | oveq2 6932 | . . . . . . . . 9 ⊢ (𝑁 = (♯‘𝐹) → (0..^𝑁) = (0..^(♯‘𝐹))) | |
4 | 3 | eqcoms 2786 | . . . . . . . 8 ⊢ ((♯‘𝐹) = 𝑁 → (0..^𝑁) = (0..^(♯‘𝐹))) |
5 | 4 | eleq2d 2845 | . . . . . . 7 ⊢ ((♯‘𝐹) = 𝑁 → (𝑋 ∈ (0..^𝑁) ↔ 𝑋 ∈ (0..^(♯‘𝐹)))) |
6 | 5 | biimpd 221 | . . . . . 6 ⊢ ((♯‘𝐹) = 𝑁 → (𝑋 ∈ (0..^𝑁) → 𝑋 ∈ (0..^(♯‘𝐹)))) |
7 | 2, 6 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (𝑋 ∈ (0..^𝑁) → 𝑋 ∈ (0..^(♯‘𝐹)))) |
8 | 1, 7 | sylan2 586 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → 𝑋 ∈ (0..^(♯‘𝐹)))) |
9 | 8 | imp 397 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑋 ∈ (0..^𝑁)) → 𝑋 ∈ (0..^(♯‘𝐹))) |
10 | fvex 6461 | . . 3 ⊢ (𝐹‘(𝑋 + 1)) ∈ V | |
11 | fvoveq1 6947 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1))) | |
12 | fargshift.g | . . . 4 ⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1))) | |
13 | 11, 12 | fvmptg 6542 | . . 3 ⊢ ((𝑋 ∈ (0..^(♯‘𝐹)) ∧ (𝐹‘(𝑋 + 1)) ∈ V) → (𝐺‘𝑋) = (𝐹‘(𝑋 + 1))) |
14 | 9, 10, 13 | sylancl 580 | . 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑋 ∈ (0..^𝑁)) → (𝐺‘𝑋) = (𝐹‘(𝑋 + 1))) |
15 | 14 | ex 403 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺‘𝑋) = (𝐹‘(𝑋 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ↦ cmpt 4967 dom cdm 5357 Fn wfn 6132 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 0cc0 10274 1c1 10275 + caddc 10277 ℕ0cn0 11646 ...cfz 12647 ..^cfzo 12788 ♯chash 13439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-hash 13440 |
This theorem is referenced by: fargshiftf1 42419 fargshiftfva 42421 |
Copyright terms: Public domain | W3C validator |