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Mirrors > Home > MPE Home > Th. List > fzostep1 | Structured version Visualization version 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 13684 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) | |
2 | uzid 12889 | . . . 4 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ (ℤ≥‘𝐵)) | |
3 | peano2uz 12937 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐵) → (𝐵 + 1) ∈ (ℤ≥‘𝐵)) | |
4 | fzoss1 13713 | . . . 4 ⊢ ((𝐵 + 1) ∈ (ℤ≥‘𝐵) → ((𝐵 + 1)..^(𝐶 + 1)) ⊆ (𝐵..^(𝐶 + 1))) | |
5 | 1, 2, 3, 4 | 4syl 19 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐵 + 1)..^(𝐶 + 1)) ⊆ (𝐵..^(𝐶 + 1))) |
6 | 1z 12644 | . . . 4 ⊢ 1 ∈ ℤ | |
7 | fzoaddel 13739 | . . . 4 ⊢ ((𝐴 ∈ (𝐵..^𝐶) ∧ 1 ∈ ℤ) → (𝐴 + 1) ∈ ((𝐵 + 1)..^(𝐶 + 1))) | |
8 | 6, 7 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐴 + 1) ∈ ((𝐵 + 1)..^(𝐶 + 1))) |
9 | 5, 8 | sseldd 3980 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐴 + 1) ∈ (𝐵..^(𝐶 + 1))) |
10 | elfzoel2 13685 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) | |
11 | elfzolt3 13696 | . . . . 5 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 < 𝐶) | |
12 | zre 12614 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
13 | zre 12614 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℝ) | |
14 | ltle 11352 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 → 𝐵 ≤ 𝐶)) | |
15 | 12, 13, 14 | syl2an 594 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 < 𝐶 → 𝐵 ≤ 𝐶)) |
16 | 1, 10, 15 | syl2anc 582 | . . . . 5 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵 < 𝐶 → 𝐵 ≤ 𝐶)) |
17 | 11, 16 | mpd 15 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ≤ 𝐶) |
18 | eluz2 12880 | . . . 4 ⊢ (𝐶 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ≤ 𝐶)) | |
19 | 1, 10, 17, 18 | syl3anbrc 1340 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ (ℤ≥‘𝐵)) |
20 | fzosplitsni 13798 | . . 3 ⊢ (𝐶 ∈ (ℤ≥‘𝐵) → ((𝐴 + 1) ∈ (𝐵..^(𝐶 + 1)) ↔ ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶))) | |
21 | 19, 20 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^(𝐶 + 1)) ↔ ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶))) |
22 | 9, 21 | mpbid 231 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 1c1 11159 + caddc 11161 < clt 11298 ≤ cle 11299 ℤcz 12610 ℤ≥cuz 12874 ..^cfzo 13681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 |
This theorem is referenced by: psgnunilem5 19492 psdmul 22160 |
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