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Mirrors > Home > ILE Home > Th. List > fzostep1 | GIF version |
Description: Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
fzostep1 | ⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoel1 9915 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) | |
2 | uzid 9333 | . . . 4 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ (ℤ≥‘𝐵)) | |
3 | peano2uz 9371 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐵) → (𝐵 + 1) ∈ (ℤ≥‘𝐵)) | |
4 | fzoss1 9941 | . . . 4 ⊢ ((𝐵 + 1) ∈ (ℤ≥‘𝐵) → ((𝐵 + 1)..^(𝐶 + 1)) ⊆ (𝐵..^(𝐶 + 1))) | |
5 | 1, 2, 3, 4 | 4syl 18 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐵 + 1)..^(𝐶 + 1)) ⊆ (𝐵..^(𝐶 + 1))) |
6 | 1z 9073 | . . . 4 ⊢ 1 ∈ ℤ | |
7 | fzoaddel 9962 | . . . 4 ⊢ ((𝐴 ∈ (𝐵..^𝐶) ∧ 1 ∈ ℤ) → (𝐴 + 1) ∈ ((𝐵 + 1)..^(𝐶 + 1))) | |
8 | 6, 7 | mpan2 421 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐴 + 1) ∈ ((𝐵 + 1)..^(𝐶 + 1))) |
9 | 5, 8 | sseldd 3093 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐴 + 1) ∈ (𝐵..^(𝐶 + 1))) |
10 | elfzoel2 9916 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) | |
11 | elfzolt3 9927 | . . . . 5 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 < 𝐶) | |
12 | zre 9051 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
13 | zre 9051 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℝ) | |
14 | ltle 7844 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 → 𝐵 ≤ 𝐶)) | |
15 | 12, 13, 14 | syl2an 287 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 < 𝐶 → 𝐵 ≤ 𝐶)) |
16 | 1, 10, 15 | syl2anc 408 | . . . . 5 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵 < 𝐶 → 𝐵 ≤ 𝐶)) |
17 | 11, 16 | mpd 13 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ≤ 𝐶) |
18 | eluz2 9325 | . . . 4 ⊢ (𝐶 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ≤ 𝐶)) | |
19 | 1, 10, 17, 18 | syl3anbrc 1165 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ (ℤ≥‘𝐵)) |
20 | fzosplitsni 10005 | . . 3 ⊢ (𝐶 ∈ (ℤ≥‘𝐵) → ((𝐴 + 1) ∈ (𝐵..^(𝐶 + 1)) ↔ ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶))) | |
21 | 19, 20 | syl 14 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^(𝐶 + 1)) ↔ ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶))) |
22 | 9, 21 | mpbid 146 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ⊆ wss 3066 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 ℝcr 7612 1c1 7614 + caddc 7616 < clt 7793 ≤ cle 7794 ℤcz 9047 ℤ≥cuz 9319 ..^cfzo 9912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-fzo 9913 |
This theorem is referenced by: (None) |
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