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| Mirrors > Home > MPE Home > Th. List > fzoend | Structured version Visualization version GIF version | ||
| Description: The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| fzoend | ⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . . . 5 ⊢ (𝐴 ∈ (𝐴..^𝐵) → 𝐴 ∈ (𝐴..^𝐵)) | |
| 2 | elfzoel2 13699 | . . . . . 6 ⊢ (𝐴 ∈ (𝐴..^𝐵) → 𝐵 ∈ ℤ) | |
| 3 | fzoval 13701 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) | |
| 4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) | 
| 5 | 1, 4 | eleqtrd 2842 | . . . 4 ⊢ (𝐴 ∈ (𝐴..^𝐵) → 𝐴 ∈ (𝐴...(𝐵 − 1))) | 
| 6 | elfzuz3 13562 | . . . 4 ⊢ (𝐴 ∈ (𝐴...(𝐵 − 1)) → (𝐵 − 1) ∈ (ℤ≥‘𝐴)) | |
| 7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (ℤ≥‘𝐴)) | 
| 8 | eluzfz2 13573 | . . 3 ⊢ ((𝐵 − 1) ∈ (ℤ≥‘𝐴) → (𝐵 − 1) ∈ (𝐴...(𝐵 − 1))) | |
| 9 | 7, 8 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴...(𝐵 − 1))) | 
| 10 | 9, 4 | eleqtrrd 2843 | 1 ⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 1c1 11157 − cmin 11493 ℤcz 12615 ℤ≥cuz 12879 ...cfz 13548 ..^cfzo 13695 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-neg 11496 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 | 
| This theorem is referenced by: fzo0end 13798 ssfzo12 13799 fzoopth 13802 lswccatn0lsw 14630 efgsdmi 19751 efgs1b 19755 clwlkclwwlklem2 30020 | 
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