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Mirrors > Home > MPE Home > Th. List > fzsuc | Structured version Visualization version GIF version |
Description: Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fzsuc | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2uz 11855 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
2 | eluzfz2 12463 | . . . . 5 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
4 | peano2fzr 12468 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → 𝑁 ∈ (𝑀...(𝑁 + 1))) | |
5 | 3, 4 | mpdan 705 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...(𝑁 + 1))) |
6 | fzsplit 12481 | . . 3 ⊢ (𝑁 ∈ (𝑀...(𝑁 + 1)) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ ((𝑁 + 1)...(𝑁 + 1)))) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ ((𝑁 + 1)...(𝑁 + 1)))) |
8 | eluzelz 11810 | . . . 4 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ ℤ) | |
9 | fzsn 12497 | . . . 4 ⊢ ((𝑁 + 1) ∈ ℤ → ((𝑁 + 1)...(𝑁 + 1)) = {(𝑁 + 1)}) | |
10 | 1, 8, 9 | 3syl 18 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑁 + 1)...(𝑁 + 1)) = {(𝑁 + 1)}) |
11 | 10 | uneq2d 3875 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∪ ((𝑁 + 1)...(𝑁 + 1))) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
12 | 7, 11 | eqtrd 2758 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1596 ∈ wcel 2103 ∪ cun 3678 {csn 4285 ‘cfv 6001 (class class class)co 6765 1c1 10050 + caddc 10052 ℤcz 11490 ℤ≥cuz 11800 ...cfz 12440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-iun 4630 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-om 7183 df-1st 7285 df-2nd 7286 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-er 7862 df-en 8073 df-dom 8074 df-sdom 8075 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-nn 11134 df-n0 11406 df-z 11491 df-uz 11801 df-fz 12441 |
This theorem is referenced by: elfzp1 12505 fztp 12511 fzsuc2 12512 fzdifsuc 12514 bpoly3 14909 prmind2 15521 vdwlem6 15813 gsummptfzsplit 18453 telgsumfzslem 18506 imasdsf1olem 22300 voliunlem1 23439 chtub 25057 2sqlem10 25273 dchrisum0flb 25319 pntpbnd1 25395 wlkp1 26709 iuninc 29607 esumfzf 30361 cvmliftlem10 31504 poimirlem2 33643 iunp1 39651 sge0p1 41051 |
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