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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzdif2 | Structured version Visualization version GIF version |
Description: Split the last element of a finite set of sequential integers. More generic than fzsuc 13596. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
Ref | Expression |
---|---|
fzdif2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzspl 32695 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) | |
2 | 1 | difeq1d 4117 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (((𝑀...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁})) |
3 | difun2 4475 | . . 3 ⊢ (((𝑀...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((𝑀...(𝑁 − 1)) ∖ {𝑁}) | |
4 | 2, 3 | eqtrdi 2782 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = ((𝑀...(𝑁 − 1)) ∖ {𝑁})) |
5 | eluzelz 12878 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
6 | uzid 12883 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
7 | uznfz 13632 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) | |
8 | 5, 6, 7 | 3syl 18 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) |
9 | disjsn 4710 | . . . 4 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) | |
10 | 8, 9 | sylibr 233 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
11 | disjdif2 4474 | . . 3 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ → ((𝑀...(𝑁 − 1)) ∖ {𝑁}) = (𝑀...(𝑁 − 1))) | |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...(𝑁 − 1)) ∖ {𝑁}) = (𝑀...(𝑁 − 1))) |
13 | 4, 12 | eqtrd 2766 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ∖ cdif 3943 ∪ cun 3944 ∩ cin 3945 ∅c0 4322 {csn 4623 ‘cfv 6546 (class class class)co 7416 1c1 11150 − cmin 11485 ℤcz 12604 ℤ≥cuz 12868 ...cfz 13532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 |
This theorem is referenced by: submat1n 33633 submatres 33634 madjusmdetlem1 33655 madjusmdetlem2 33656 madjusmdetlem3 33657 |
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