fvffz0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fvffz0		Structured version Visualization version GIF version

Theorem fvffz0 13619

Description: The function value of a function from a finite interval of nonnegative integers. (Contributed by AV, 13-Feb-2021.)

**Assertion**
Ref	Expression
fvffz0	⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < 𝑁) ∧ 𝑃:(0...𝑁)⟶𝑉) → (𝑃‘𝐼) ∈ 𝑉)

**Proof of Theorem fvffz0**
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 12481	. . . . . 6 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℝ)
5		nn0re 12481	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
6		ltle 11302	. . . . . 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 13592	. . . 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 7088	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 5149 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 < clt 11248 ≤ cle 11249 ℕ₀cn0 12472 ...cfz 13484

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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485

This theorem is referenced by: upgr3v3e3cycl 29433 upgr4cycl4dv4e 29438

W3C validator