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Mirrors > Home > MPE Home > Th. List > fvffz0 | Structured version Visualization version GIF version |
Description: The function value of a function from a finite interval of nonnegative integers. (Contributed by AV, 13-Feb-2021.) |
Ref | Expression |
---|---|
fvffz0 | ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) ∧ 𝑃:(0...𝑁)⟶𝑉) → (𝑃‘𝐼) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) ∧ 𝑃:(0...𝑁)⟶𝑉) → 𝑃:(0...𝑁)⟶𝑉) | |
2 | simp2 1138 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) → 𝐼 ∈ ℕ0) | |
3 | simp1 1137 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) → 𝑁 ∈ ℕ0) | |
4 | nn0re 12468 | . . . . . 6 ⊢ (𝐼 ∈ ℕ0 → 𝐼 ∈ ℝ) | |
5 | nn0re 12468 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | ltle 11289 | . . . . . 6 ⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐼 < 𝑁 → 𝐼 ≤ 𝑁)) | |
7 | 4, 5, 6 | syl2anr 598 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → (𝐼 < 𝑁 → 𝐼 ≤ 𝑁)) |
8 | 7 | 3impia 1118 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) → 𝐼 ≤ 𝑁) |
9 | elfz2nn0 13579 | . . . 4 ⊢ (𝐼 ∈ (0...𝑁) ↔ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ≤ 𝑁)) | |
10 | 2, 3, 8, 9 | syl3anbrc 1344 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) → 𝐼 ∈ (0...𝑁)) |
11 | 10 | adantr 482 | . 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) ∧ 𝑃:(0...𝑁)⟶𝑉) → 𝐼 ∈ (0...𝑁)) |
12 | 1, 11 | ffvelcdmd 7075 | 1 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < 𝑁) ∧ 𝑃:(0...𝑁)⟶𝑉) → (𝑃‘𝐼) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5144 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 ℝcr 11096 0cc0 11097 < clt 11235 ≤ cle 11236 ℕ0cn0 12459 ...cfz 13471 |
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-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 |
This theorem is referenced by: upgr3v3e3cycl 29400 upgr4cycl4dv4e 29405 |
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