Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1fzopredsuc | Structured version Visualization version GIF version |
Description: Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020.) |
Ref | Expression |
---|---|
1fzopredsuc | ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = (({0} ∪ (1..^𝑁)) ∪ {𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0uz 12673 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
2 | fzopredsuc 45059 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = (({0} ∪ ((0 + 1)..^𝑁)) ∪ {𝑁})) | |
3 | 0p1e1 12145 | . . . . . 6 ⊢ (0 + 1) = 1 | |
4 | 3 | oveq1i 7317 | . . . . 5 ⊢ ((0 + 1)..^𝑁) = (1..^𝑁) |
5 | 4 | uneq2i 4100 | . . . 4 ⊢ ({0} ∪ ((0 + 1)..^𝑁)) = ({0} ∪ (1..^𝑁)) |
6 | 5 | uneq1i 4099 | . . 3 ⊢ (({0} ∪ ((0 + 1)..^𝑁)) ∪ {𝑁}) = (({0} ∪ (1..^𝑁)) ∪ {𝑁}) |
7 | 2, 6 | eqtrdi 2792 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = (({0} ∪ (1..^𝑁)) ∪ {𝑁})) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = (({0} ∪ (1..^𝑁)) ∪ {𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 ∪ cun 3890 {csn 4565 ‘cfv 6458 (class class class)co 7307 0cc0 10921 1c1 10922 + caddc 10924 ℕ0cn0 12283 ℤ≥cuz 12632 ...cfz 13289 ..^cfzo 13432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-nn 12024 df-n0 12284 df-z 12370 df-uz 12633 df-fz 13290 df-fzo 13433 |
This theorem is referenced by: el1fzopredsuc 45061 |
Copyright terms: Public domain | W3C validator |