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Mirrors > Home > MPE Home > Th. List > fzdifsuc | Structured version Visualization version GIF version |
Description: Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.) |
Ref | Expression |
---|---|
fzdifsuc | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzsuc 12705 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | |
2 | 1 | difeq1d 3950 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}) = (((𝑀...𝑁) ∪ {(𝑁 + 1)}) ∖ {(𝑁 + 1)})) |
3 | uncom 3980 | . . 3 ⊢ ({(𝑁 + 1)} ∪ (𝑀...𝑁)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}) | |
4 | ssun2 4000 | . . . 4 ⊢ {(𝑁 + 1)} ⊆ ((𝑀...𝑁) ∪ {(𝑁 + 1)}) | |
5 | incom 4028 | . . . . . 6 ⊢ ({(𝑁 + 1)} ∩ (𝑀...𝑁)) = ((𝑀...𝑁) ∩ {(𝑁 + 1)}) | |
6 | fzp1disj 12717 | . . . . . 6 ⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ | |
7 | 5, 6 | eqtri 2802 | . . . . 5 ⊢ ({(𝑁 + 1)} ∩ (𝑀...𝑁)) = ∅ |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ({(𝑁 + 1)} ∩ (𝑀...𝑁)) = ∅) |
9 | uneqdifeq 4281 | . . . 4 ⊢ (({(𝑁 + 1)} ⊆ ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ∧ ({(𝑁 + 1)} ∩ (𝑀...𝑁)) = ∅) → (({(𝑁 + 1)} ∪ (𝑀...𝑁)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ↔ (((𝑀...𝑁) ∪ {(𝑁 + 1)}) ∖ {(𝑁 + 1)}) = (𝑀...𝑁))) | |
10 | 4, 8, 9 | sylancr 581 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (({(𝑁 + 1)} ∪ (𝑀...𝑁)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ↔ (((𝑀...𝑁) ∪ {(𝑁 + 1)}) ∖ {(𝑁 + 1)}) = (𝑀...𝑁))) |
11 | 3, 10 | mpbii 225 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝑀...𝑁) ∪ {(𝑁 + 1)}) ∖ {(𝑁 + 1)}) = (𝑀...𝑁)) |
12 | 2, 11 | eqtr2d 2815 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∖ cdif 3789 ∪ cun 3790 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 {csn 4398 ‘cfv 6135 (class class class)co 6922 1c1 10273 + caddc 10275 ℤ≥cuz 11992 ...cfz 12643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 |
This theorem is referenced by: fzdifsuc2 40433 dvnmul 41086 |
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