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Mirrors > Home > MPE Home > Th. List > fztp | Structured version Visualization version GIF version |
Description: A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.) |
Ref | Expression |
---|---|
fztp | ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzid 12835 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
2 | peano2uz 12883 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) | |
3 | fzsuc 13546 | . . 3 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (𝑀...((𝑀 + 1) + 1)) = ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)})) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 + 1) + 1)) = ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)})) |
5 | zcn 12561 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
6 | ax-1cn 11165 | . . . . . 6 ⊢ 1 ∈ ℂ | |
7 | addass 11194 | . . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) | |
8 | 6, 6, 7 | mp3an23 1449 | . . . . 5 ⊢ (𝑀 ∈ ℂ → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) |
10 | df-2 12273 | . . . . 5 ⊢ 2 = (1 + 1) | |
11 | 10 | oveq2i 7413 | . . . 4 ⊢ (𝑀 + 2) = (𝑀 + (1 + 1)) |
12 | 9, 11 | eqtr4di 2782 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1) + 1) = (𝑀 + 2)) |
13 | 12 | oveq2d 7418 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 + 1) + 1)) = (𝑀...(𝑀 + 2))) |
14 | fzpr 13554 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) | |
15 | 12 | sneqd 4633 | . . . 4 ⊢ (𝑀 ∈ ℤ → {((𝑀 + 1) + 1)} = {(𝑀 + 2)}) |
16 | 14, 15 | uneq12d 4157 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)})) |
17 | df-tp 4626 | . . 3 ⊢ {𝑀, (𝑀 + 1), (𝑀 + 2)} = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)}) | |
18 | 16, 17 | eqtr4di 2782 | . 2 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
19 | 4, 13, 18 | 3eqtr3d 2772 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3939 {csn 4621 {cpr 4623 {ctp 4625 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 1c1 11108 + caddc 11110 2c2 12265 ℤcz 12556 ℤ≥cuz 12820 ...cfz 13482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 |
This theorem is referenced by: fztpval 13561 fz0tp 13600 fz0to4untppr 13602 fzo0to3tp 13716 fzo1to4tp 13718 1cubr 26693 rabren3dioph 42067 nnsum4primesodd 46974 nnsum4primesoddALTV 46975 |
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