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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 12916 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 − 1)) → (𝑁 = (𝑀 − 1) ∨ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)))) | |
2 | zcn 12615 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
3 | ax-1cn 11210 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
4 | npcan 11514 | . . . . . . . 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 7446 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = (𝑀...𝑀)) |
7 | uncom 4167 | . . . . . . . 8 ⊢ (∅ ∪ {𝑀}) = ({𝑀} ∪ ∅) | |
8 | un0 4399 | . . . . . . . 8 ⊢ ({𝑀} ∪ ∅) = {𝑀} | |
9 | 7, 8 | eqtri 2762 | . . . . . . 7 ⊢ (∅ ∪ {𝑀}) = {𝑀} |
10 | zre 12614 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
11 | 10 | ltm1d 12197 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀) |
12 | peano2zm 12657 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
13 | fzn 13576 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) | |
14 | 12, 13 | mpdan 687 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
15 | 11, 14 | mpbid 232 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅) |
16 | 5 | sneqd 4642 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → {((𝑀 − 1) + 1)} = {𝑀}) |
17 | 15, 16 | uneq12d 4178 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (∅ ∪ {𝑀})) |
18 | fzsn 13602 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
19 | 9, 17, 18 | 3eqtr4a 2800 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (𝑀...𝑀)) |
20 | 6, 19 | eqtr4d 2777 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) |
21 | oveq1 7437 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑁 + 1) = ((𝑀 − 1) + 1)) | |
22 | 21 | oveq2d 7446 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = (𝑀...((𝑀 − 1) + 1))) |
23 | oveq2 7438 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...𝑁) = (𝑀...(𝑀 − 1))) | |
24 | 21 | sneqd 4642 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → {(𝑁 + 1)} = {((𝑀 − 1) + 1)}) |
25 | 23, 24 | uneq12d 4178 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → ((𝑀...𝑁) ∪ {(𝑁 + 1)}) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) |
26 | 22, 25 | eqeq12d 2750 | . . . . 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 247 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))) |
28 | 27 | imp 406 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = (𝑀 − 1)) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
29 | 5 | fveq2d 6910 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
30 | 29 | eleq2d 2824 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝑀))) |
31 | 30 | biimpa 476 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
32 | fzsuc 13607 | . . . 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 959 | . 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 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ∪ cun 3960 ∅c0 4338 {csn 4630 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 < clt 11292 − cmin 11489 ℤcz 12610 ℤ≥cuz 12875 ...cfz 13543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 |
This theorem is referenced by: fseq1p1m1 13634 fzennn 14005 fsumm1 15783 fprodm1 15999 prmreclem4 16952 ppiprm 27208 ppinprm 27209 chtprm 27210 chtnprm 27211 poimirlem3 37609 poimirlem4 37610 lcmfunnnd 41993 mapfzcons 42703 |
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