Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fzosplitsnm1 | Structured version Visualization version GIF version |
Description: Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
Ref | Expression |
---|---|
fzosplitsnm1 | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐴..^𝐵) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 12254 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘(𝐴 + 1)) → 𝐵 ∈ ℤ) | |
2 | 1 | zcnd 12089 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘(𝐴 + 1)) → 𝐵 ∈ ℂ) |
3 | 2 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → 𝐵 ∈ ℂ) |
4 | ax-1cn 10595 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | npcan 10895 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐵 − 1) + 1) = 𝐵) | |
6 | 5 | eqcomd 2827 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → 𝐵 = ((𝐵 − 1) + 1)) |
7 | 3, 4, 6 | sylancl 588 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → 𝐵 = ((𝐵 − 1) + 1)) |
8 | 7 | oveq2d 7172 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐴..^𝐵) = (𝐴..^((𝐵 − 1) + 1))) |
9 | eluzp1m1 12269 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐵 − 1) ∈ (ℤ≥‘𝐴)) | |
10 | 1 | adantl 484 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → 𝐵 ∈ ℤ) |
11 | peano2zm 12026 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐵 − 1) ∈ ℤ) | |
12 | uzid 12259 | . . . . 5 ⊢ ((𝐵 − 1) ∈ ℤ → (𝐵 − 1) ∈ (ℤ≥‘(𝐵 − 1))) | |
13 | peano2uz 12302 | . . . . 5 ⊢ ((𝐵 − 1) ∈ (ℤ≥‘(𝐵 − 1)) → ((𝐵 − 1) + 1) ∈ (ℤ≥‘(𝐵 − 1))) | |
14 | 10, 11, 12, 13 | 4syl 19 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → ((𝐵 − 1) + 1) ∈ (ℤ≥‘(𝐵 − 1))) |
15 | elfzuzb 12903 | . . . 4 ⊢ ((𝐵 − 1) ∈ (𝐴...((𝐵 − 1) + 1)) ↔ ((𝐵 − 1) ∈ (ℤ≥‘𝐴) ∧ ((𝐵 − 1) + 1) ∈ (ℤ≥‘(𝐵 − 1)))) | |
16 | 9, 14, 15 | sylanbrc 585 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐵 − 1) ∈ (𝐴...((𝐵 − 1) + 1))) |
17 | fzosplit 13071 | . . 3 ⊢ ((𝐵 − 1) ∈ (𝐴...((𝐵 − 1) + 1)) → (𝐴..^((𝐵 − 1) + 1)) = ((𝐴..^(𝐵 − 1)) ∪ ((𝐵 − 1)..^((𝐵 − 1) + 1)))) | |
18 | 16, 17 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐴..^((𝐵 − 1) + 1)) = ((𝐴..^(𝐵 − 1)) ∪ ((𝐵 − 1)..^((𝐵 − 1) + 1)))) |
19 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘(𝐴 + 1)) → (𝐵 − 1) ∈ ℤ) |
20 | 19 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐵 − 1) ∈ ℤ) |
21 | fzosn 13109 | . . . 4 ⊢ ((𝐵 − 1) ∈ ℤ → ((𝐵 − 1)..^((𝐵 − 1) + 1)) = {(𝐵 − 1)}) | |
22 | 20, 21 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → ((𝐵 − 1)..^((𝐵 − 1) + 1)) = {(𝐵 − 1)}) |
23 | 22 | uneq2d 4139 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → ((𝐴..^(𝐵 − 1)) ∪ ((𝐵 − 1)..^((𝐵 − 1) + 1))) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) |
24 | 8, 18, 23 | 3eqtrd 2860 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐴..^𝐵) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 {csn 4567 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 − cmin 10870 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: elfzonlteqm1 13114 pthdlem1 27547 clwwlkccatlem 27767 cycpmco2lem7 30774 cycpmrn 30785 |
Copyright terms: Public domain | W3C validator |