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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 9364 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
2 | peano2uz 9405 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) | |
3 | fzsuc 9880 | . . 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 9083 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
6 | ax-1cn 7737 | . . . . . 6 ⊢ 1 ∈ ℂ | |
7 | addass 7774 | . . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) | |
8 | 6, 6, 7 | mp3an23 1308 | . . . . 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 8803 | . . . . 5 ⊢ 2 = (1 + 1) | |
11 | 10 | oveq2i 5793 | . . . 4 ⊢ (𝑀 + 2) = (𝑀 + (1 + 1)) |
12 | 9, 11 | eqtr4di 2191 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1) + 1) = (𝑀 + 2)) |
13 | 12 | oveq2d 5798 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 + 1) + 1)) = (𝑀...(𝑀 + 2))) |
14 | fzpr 9888 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) | |
15 | 12 | sneqd 3545 | . . . 4 ⊢ (𝑀 ∈ ℤ → {((𝑀 + 1) + 1)} = {(𝑀 + 2)}) |
16 | 14, 15 | uneq12d 3236 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)})) |
17 | df-tp 3540 | . . 3 ⊢ {𝑀, (𝑀 + 1), (𝑀 + 2)} = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)}) | |
18 | 16, 17 | eqtr4di 2191 | . 2 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
19 | 4, 13, 18 | 3eqtr3d 2181 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ∪ cun 3074 {csn 3532 {cpr 3533 {ctp 3534 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 1c1 7645 + caddc 7647 2c2 8795 ℤcz 9078 ℤ≥cuz 9350 ...cfz 9821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-tp 3540 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 |
This theorem is referenced by: fztpval 9894 fz0tp 9932 fzo0to3tp 10027 |
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