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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.27dlem2 | Structured version Visualization version GIF version |
Description: Lemma for rmydioph 39489. This theorem is used along with the next three to efficiently infer steps like 7 ∈ (1...;10). (Contributed by Stefan O'Rear, 11-Oct-2014.) |
Ref | Expression |
---|---|
jm2.27dlem2.1 | ⊢ 𝐴 ∈ (1...𝐵) |
jm2.27dlem2.2 | ⊢ 𝐶 = (𝐵 + 1) |
jm2.27dlem2.3 | ⊢ 𝐵 ∈ ℕ |
Ref | Expression |
---|---|
jm2.27dlem2 | ⊢ 𝐴 ∈ (1...𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jm2.27dlem2.1 | . . 3 ⊢ 𝐴 ∈ (1...𝐵) | |
2 | elfzelz 12896 | . . 3 ⊢ (𝐴 ∈ (1...𝐵) → 𝐴 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝐴 ∈ ℤ |
4 | elfzle1 12898 | . . 3 ⊢ (𝐴 ∈ (1...𝐵) → 1 ≤ 𝐴) | |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ 1 ≤ 𝐴 |
6 | 3 | zrei 11975 | . . . 4 ⊢ 𝐴 ∈ ℝ |
7 | jm2.27dlem2.3 | . . . . 5 ⊢ 𝐵 ∈ ℕ | |
8 | 7 | nnrei 11635 | . . . 4 ⊢ 𝐵 ∈ ℝ |
9 | elfzle2 12899 | . . . . 5 ⊢ (𝐴 ∈ (1...𝐵) → 𝐴 ≤ 𝐵) | |
10 | 1, 9 | ax-mp 5 | . . . 4 ⊢ 𝐴 ≤ 𝐵 |
11 | letrp1 11472 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ (𝐵 + 1)) | |
12 | 6, 8, 10, 11 | mp3an 1452 | . . 3 ⊢ 𝐴 ≤ (𝐵 + 1) |
13 | jm2.27dlem2.2 | . . 3 ⊢ 𝐶 = (𝐵 + 1) | |
14 | 12, 13 | breqtrri 5084 | . 2 ⊢ 𝐴 ≤ 𝐶 |
15 | 1z 12000 | . . 3 ⊢ 1 ∈ ℤ | |
16 | nnz 11992 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
17 | peano2z 12011 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐵 + 1) ∈ ℤ) | |
18 | 7, 16, 17 | mp2b 10 | . . . 4 ⊢ (𝐵 + 1) ∈ ℤ |
19 | 13, 18 | eqeltri 2906 | . . 3 ⊢ 𝐶 ∈ ℤ |
20 | elfz1 12885 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (1...𝐶) ↔ (𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶))) | |
21 | 15, 19, 20 | mp2an 688 | . 2 ⊢ (𝐴 ∈ (1...𝐶) ↔ (𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶)) |
22 | 3, 5, 14, 21 | mpbir3an 1333 | 1 ⊢ 𝐴 ∈ (1...𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 ≤ cle 10664 ℕcn 11626 ℤcz 11969 ...cfz 12880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 |
This theorem is referenced by: rmydioph 39489 expdiophlem2 39497 |
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