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Theorem fzval2 12542
Description: An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
fzval2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

Proof of Theorem fzval2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fzval 12541 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
2 zssre 11596 . . . . . . 7 ℤ ⊆ ℝ
3 ressxr 10295 . . . . . . 7 ℝ ⊆ ℝ*
42, 3sstri 3753 . . . . . 6 ℤ ⊆ ℝ*
54sseli 3740 . . . . 5 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
64sseli 3740 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ*)
7 iccval 12427 . . . . 5 ((𝑀 ∈ ℝ*𝑁 ∈ ℝ*) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀𝑘𝑘𝑁)})
85, 6, 7syl2an 495 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀[,]𝑁) = {𝑘 ∈ ℝ* ∣ (𝑀𝑘𝑘𝑁)})
98ineq1d 3956 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀[,]𝑁) ∩ ℤ) = ({𝑘 ∈ ℝ* ∣ (𝑀𝑘𝑘𝑁)} ∩ ℤ))
10 inrab2 4043 . . . 4 ({𝑘 ∈ ℝ* ∣ (𝑀𝑘𝑘𝑁)} ∩ ℤ) = {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀𝑘𝑘𝑁)}
11 sseqin2 3960 . . . . . 6 (ℤ ⊆ ℝ* ↔ (ℝ* ∩ ℤ) = ℤ)
124, 11mpbi 220 . . . . 5 (ℝ* ∩ ℤ) = ℤ
13 rabeq 3332 . . . . 5 ((ℝ* ∩ ℤ) = ℤ → {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀𝑘𝑘𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
1412, 13ax-mp 5 . . . 4 {𝑘 ∈ (ℝ* ∩ ℤ) ∣ (𝑀𝑘𝑘𝑁)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)}
1510, 14eqtri 2782 . . 3 ({𝑘 ∈ ℝ* ∣ (𝑀𝑘𝑘𝑁)} ∩ ℤ) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)}
169, 15syl6req 2811 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)} = ((𝑀[,]𝑁) ∩ ℤ))
171, 16eqtrd 2794 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  cin 3714  wss 3715   class class class wbr 4804  (class class class)co 6814  cr 10147  *cxr 10285  cle 10287  cz 11589  [,]cicc 12391  ...cfz 12539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-xr 10290  df-neg 10481  df-z 11590  df-icc 12395  df-fz 12540
This theorem is referenced by:  dvfsumle  24003  dvfsumabs  24005  taylplem1  24336  taylplem2  24337  taylpfval  24338  dvtaylp  24343  ppisval  25050
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