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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 12891 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
2 | peano2uz 12941 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) | |
3 | fzsuc 13608 | . . 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 12616 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
6 | ax-1cn 11211 | . . . . . 6 ⊢ 1 ∈ ℂ | |
7 | addass 11240 | . . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) + 1) = (𝑀 + (1 + 1))) | |
8 | 6, 6, 7 | mp3an23 1452 | . . . . 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 12327 | . . . . 5 ⊢ 2 = (1 + 1) | |
11 | 10 | oveq2i 7442 | . . . 4 ⊢ (𝑀 + 2) = (𝑀 + (1 + 1)) |
12 | 9, 11 | eqtr4di 2793 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1) + 1) = (𝑀 + 2)) |
13 | 12 | oveq2d 7447 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 + 1) + 1)) = (𝑀...(𝑀 + 2))) |
14 | fzpr 13616 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) | |
15 | 12 | sneqd 4643 | . . . 4 ⊢ (𝑀 ∈ ℤ → {((𝑀 + 1) + 1)} = {(𝑀 + 2)}) |
16 | 14, 15 | uneq12d 4179 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)})) |
17 | df-tp 4636 | . . 3 ⊢ {𝑀, (𝑀 + 1), (𝑀 + 2)} = ({𝑀, (𝑀 + 1)} ∪ {(𝑀 + 2)}) | |
18 | 16, 17 | eqtr4di 2793 | . 2 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 + 1)) ∪ {((𝑀 + 1) + 1)}) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
19 | 4, 13, 18 | 3eqtr3d 2783 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {csn 4631 {cpr 4633 {ctp 4635 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 2c2 12319 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 |
This theorem is referenced by: fztpval 13623 fz0tp 13665 fz0to5un2tp 13668 fzo0to3tp 13788 fzo1to4tp 13790 1cubr 26900 rabren3dioph 42803 nnsum4primesodd 47721 nnsum4primesoddALTV 47722 |
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