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Mirrors > Home > ILE Home > Th. List > fztp | 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 8633 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
2 | peano2uz 8671 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) | |
3 | fzsuc 9086 | . . 3 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (𝑀...((𝑀 + 1) + 1)) = ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)})) | |
4 | 1, 2, 3 | 3syl 17 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 + 1) + 1)) = ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)})) |
5 | zcn 8356 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
6 | ax-1cn 7069 | . . . . . 6 ⊢ 1 ∈ ℂ | |
7 | addass 7103 | . . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) | |
8 | 6, 6, 7 | mp3an23 1260 | . . . . 5 ⊢ (𝑀 ∈ ℂ → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) |
9 | 5, 8 | syl 14 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) |
10 | df-2 8098 | . . . . 5 ⊢ 2 = (1 + 1) | |
11 | 10 | oveq2i 5543 | . . . 4 ⊢ (𝑀 + 2) = (𝑀 + (1 + 1)) |
12 | 9, 11 | syl6eqr 2131 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1) + 1) = (𝑀 + 2)) |
13 | 12 | oveq2d 5548 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 + 1) + 1)) = (𝑀...(𝑀 + 2))) |
14 | fzpr 9094 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) | |
15 | 12 | sneqd 3411 | . . . 4 ⊢ (𝑀 ∈ ℤ → {((𝑀 + 1) + 1)} = {(𝑀 + 2)}) |
16 | 14, 15 | uneq12d 3127 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)})) |
17 | df-tp 3406 | . . 3 ⊢ {𝑀, (𝑀 + 1), (𝑀 + 2)} = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)}) | |
18 | 16, 17 | syl6eqr 2131 | . 2 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
19 | 4, 13, 18 | 3eqtr3d 2121 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 ∪ cun 2971 {csn 3398 {cpr 3399 {ctp 3400 ‘cfv 4922 (class class class)co 5532 ℂcc 6979 1c1 6982 + caddc 6984 2c2 8089 ℤcz 8351 ℤ≥cuz 8619 ...cfz 9029 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-tp 3406 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-2 8098 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 |
This theorem is referenced by: fztpval 9100 fz0tp 9135 fzo0to3tp 9228 |
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