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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 11597 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 − 1)) → (𝑁 = (𝑀 − 1) ∨ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)))) | |
2 | zcn 11259 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
3 | ax-1cn 9873 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
4 | npcan 10169 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀) | |
5 | 2, 3, 4 | sylancl 693 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) + 1) = 𝑀) |
6 | 5 | oveq2d 6565 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = (𝑀...𝑀)) |
7 | uncom 3719 | . . . . . . . 8 ⊢ (∅ ∪ {𝑀}) = ({𝑀} ∪ ∅) | |
8 | un0 3919 | . . . . . . . 8 ⊢ ({𝑀} ∪ ∅) = {𝑀} | |
9 | 7, 8 | eqtri 2632 | . . . . . . 7 ⊢ (∅ ∪ {𝑀}) = {𝑀} |
10 | zre 11258 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
11 | 10 | ltm1d 10835 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀) |
12 | peano2zm 11297 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
13 | fzn 12228 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) | |
14 | 12, 13 | mpdan 699 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
15 | 11, 14 | mpbid 221 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅) |
16 | 5 | sneqd 4137 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → {((𝑀 − 1) + 1)} = {𝑀}) |
17 | 15, 16 | uneq12d 3730 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (∅ ∪ {𝑀})) |
18 | fzsn 12254 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
19 | 9, 17, 18 | 3eqtr4a 2670 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (𝑀...𝑀)) |
20 | 6, 19 | eqtr4d 2647 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) |
21 | oveq1 6556 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑁 + 1) = ((𝑀 − 1) + 1)) | |
22 | 21 | oveq2d 6565 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = (𝑀...((𝑀 − 1) + 1))) |
23 | oveq2 6557 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...𝑁) = (𝑀...(𝑀 − 1))) | |
24 | 21 | sneqd 4137 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → {(𝑁 + 1)} = {((𝑀 − 1) + 1)}) |
25 | 23, 24 | uneq12d 3730 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → ((𝑀...𝑁) ∪ {(𝑁 + 1)}) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) |
26 | 22, 25 | eqeq12d 2625 | . . . . 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 236 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))) |
28 | 27 | imp 444 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = (𝑀 − 1)) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
29 | 5 | fveq2d 6107 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
30 | 29 | eleq2d 2673 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝑀))) |
31 | 30 | biimpa 500 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
32 | fzsuc 12258 | . . . 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 822 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 = (𝑀 − 1) ∨ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
35 | 1, 34 | sylan2 490 | 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 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∅c0 3874 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: fseq1p1m1 12283 fzennn 12629 fsumm1 14324 fprodm1 14536 prmreclem4 15461 ppiprm 24677 ppinprm 24678 chtprm 24679 chtnprm 24680 eupap1 26503 eupath2lem3 26506 poimirlem3 32582 poimirlem4 32583 mapfzcons 36297 |
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