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Mirrors > Home > ILE Home > Th. List > btwnapz | GIF version |
Description: A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.) |
Ref | Expression |
---|---|
btwnapz.a | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
btwnapz.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
btwnapz.c | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
btwnapz.ab | ⊢ (𝜑 → 𝐴 < 𝐵) |
btwnapz.ba | ⊢ (𝜑 → 𝐵 < (𝐴 + 1)) |
Ref | Expression |
---|---|
btwnapz | ⊢ (𝜑 → 𝐵 # 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnapz.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | 1 | zred 9166 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
3 | 2 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐴) → 𝐶 ∈ ℝ) |
4 | btwnapz.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐴) → 𝐵 ∈ ℝ) |
6 | btwnapz.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
7 | 6 | zred 9166 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | 7 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐴) → 𝐴 ∈ ℝ) |
9 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐴) → 𝐶 ≤ 𝐴) | |
10 | btwnapz.ab | . . . . 5 ⊢ (𝜑 → 𝐴 < 𝐵) | |
11 | 10 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐴) → 𝐴 < 𝐵) |
12 | 3, 8, 5, 9, 11 | lelttrd 7880 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐴) → 𝐶 < 𝐵) |
13 | 3, 5, 12 | gtapd 8392 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐴) → 𝐵 # 𝐶) |
14 | 4 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐶) → 𝐵 ∈ ℝ) |
15 | 2 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐶) → 𝐶 ∈ ℝ) |
16 | peano2re 7891 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
17 | 7, 16 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ) |
18 | 17 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐶) → (𝐴 + 1) ∈ ℝ) |
19 | btwnapz.ba | . . . . 5 ⊢ (𝜑 → 𝐵 < (𝐴 + 1)) | |
20 | 19 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐶) → 𝐵 < (𝐴 + 1)) |
21 | zltp1le 9101 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
22 | 6, 1, 21 | syl2anc 408 | . . . . 5 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) |
23 | 22 | biimpa 294 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐶) → (𝐴 + 1) ≤ 𝐶) |
24 | 14, 18, 15, 20, 23 | ltletrd 8178 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐶) → 𝐵 < 𝐶) |
25 | 14, 15, 24 | ltapd 8393 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐶) → 𝐵 # 𝐶) |
26 | zlelttric 9092 | . . 3 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐶 ≤ 𝐴 ∨ 𝐴 < 𝐶)) | |
27 | 1, 6, 26 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ∨ 𝐴 < 𝐶)) |
28 | 13, 25, 27 | mpjaodan 787 | 1 ⊢ (𝜑 → 𝐵 # 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℝcr 7612 1c1 7614 + caddc 7616 < clt 7793 ≤ cle 7794 # cap 8336 ℤcz 9047 |
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-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
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-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-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: eirraplem 11472 |
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