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Mirrors > Home > MPE Home > Th. List > fzsuc2 | Structured version Visualization version GIF version |
Description: Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.) |
Ref | Expression |
---|---|
fzsuc2 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzp1 12859 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 − 1)) → (𝑁 = (𝑀 − 1) ∨ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)))) | |
2 | zcn 12559 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
3 | ax-1cn 11164 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
4 | npcan 11465 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀) | |
5 | 2, 3, 4 | sylancl 586 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) + 1) = 𝑀) |
6 | 5 | oveq2d 7421 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = (𝑀...𝑀)) |
7 | uncom 4152 | . . . . . . . 8 ⊢ (∅ ∪ {𝑀}) = ({𝑀} ∪ ∅) | |
8 | un0 4389 | . . . . . . . 8 ⊢ ({𝑀} ∪ ∅) = {𝑀} | |
9 | 7, 8 | eqtri 2760 | . . . . . . 7 ⊢ (∅ ∪ {𝑀}) = {𝑀} |
10 | zre 12558 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
11 | 10 | ltm1d 12142 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀) |
12 | peano2zm 12601 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
13 | fzn 13513 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) | |
14 | 12, 13 | mpdan 685 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
15 | 11, 14 | mpbid 231 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅) |
16 | 5 | sneqd 4639 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → {((𝑀 − 1) + 1)} = {𝑀}) |
17 | 15, 16 | uneq12d 4163 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (∅ ∪ {𝑀})) |
18 | fzsn 13539 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
19 | 9, 17, 18 | 3eqtr4a 2798 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (𝑀...𝑀)) |
20 | 6, 19 | eqtr4d 2775 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) |
21 | oveq1 7412 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑁 + 1) = ((𝑀 − 1) + 1)) | |
22 | 21 | oveq2d 7421 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = (𝑀...((𝑀 − 1) + 1))) |
23 | oveq2 7413 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...𝑁) = (𝑀...(𝑀 − 1))) | |
24 | 21 | sneqd 4639 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → {(𝑁 + 1)} = {((𝑀 − 1) + 1)}) |
25 | 23, 24 | uneq12d 4163 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → ((𝑀...𝑁) ∪ {(𝑁 + 1)}) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) |
26 | 22, 25 | eqeq12d 2748 | . . . . 5 ⊢ (𝑁 = (𝑀 − 1) → ((𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ↔ (𝑀...((𝑀 − 1) + 1)) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}))) |
27 | 20, 26 | syl5ibrcom 246 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))) |
28 | 27 | imp 407 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = (𝑀 − 1)) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
29 | 5 | fveq2d 6892 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
30 | 29 | eleq2d 2819 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝑀))) |
31 | 30 | biimpa 477 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
32 | fzsuc 13544 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | |
33 | 31, 32 | syl 17 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
34 | 28, 33 | jaodan 956 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 = (𝑀 − 1) ∨ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
35 | 1, 34 | sylan2 593 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∪ cun 3945 ∅c0 4321 {csn 4627 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 1c1 11107 + caddc 11109 < clt 11244 − cmin 11440 ℤcz 12554 ℤ≥cuz 12818 ...cfz 13480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 |
This theorem is referenced by: fseq1p1m1 13571 fzennn 13929 fsumm1 15693 fprodm1 15907 prmreclem4 16848 ppiprm 26644 ppinprm 26645 chtprm 26646 chtnprm 26647 poimirlem3 36479 poimirlem4 36480 lcmfunnnd 40865 mapfzcons 41439 |
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