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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.27dlem2 | Structured version Visualization version GIF version |
Description: Lemma for rmydioph 39609. 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 12907 | . . 3 ⊢ (𝐴 ∈ (1...𝐵) → 𝐴 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝐴 ∈ ℤ |
4 | elfzle1 12909 | . . 3 ⊢ (𝐴 ∈ (1...𝐵) → 1 ≤ 𝐴) | |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ 1 ≤ 𝐴 |
6 | 3 | zrei 11986 | . . . 4 ⊢ 𝐴 ∈ ℝ |
7 | jm2.27dlem2.3 | . . . . 5 ⊢ 𝐵 ∈ ℕ | |
8 | 7 | nnrei 11646 | . . . 4 ⊢ 𝐵 ∈ ℝ |
9 | elfzle2 12910 | . . . . 5 ⊢ (𝐴 ∈ (1...𝐵) → 𝐴 ≤ 𝐵) | |
10 | 1, 9 | ax-mp 5 | . . . 4 ⊢ 𝐴 ≤ 𝐵 |
11 | letrp1 11483 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ (𝐵 + 1)) | |
12 | 6, 8, 10, 11 | mp3an 1457 | . . 3 ⊢ 𝐴 ≤ (𝐵 + 1) |
13 | jm2.27dlem2.2 | . . 3 ⊢ 𝐶 = (𝐵 + 1) | |
14 | 12, 13 | breqtrri 5092 | . 2 ⊢ 𝐴 ≤ 𝐶 |
15 | 1z 12011 | . . 3 ⊢ 1 ∈ ℤ | |
16 | nnz 12003 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
17 | peano2z 12022 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐵 + 1) ∈ ℤ) | |
18 | 7, 16, 17 | mp2b 10 | . . . 4 ⊢ (𝐵 + 1) ∈ ℤ |
19 | 13, 18 | eqeltri 2909 | . . 3 ⊢ 𝐶 ∈ ℤ |
20 | elfz1 12896 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (1...𝐶) ↔ (𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶))) | |
21 | 15, 19, 20 | mp2an 690 | . 2 ⊢ (𝐴 ∈ (1...𝐶) ↔ (𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶)) |
22 | 3, 5, 14, 21 | mpbir3an 1337 | 1 ⊢ 𝐴 ∈ (1...𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 (class class class)co 7155 ℝcr 10535 1c1 10537 + caddc 10539 ≤ cle 10675 ℕcn 11637 ℤcz 11980 ...cfz 12891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 |
This theorem is referenced by: rmydioph 39609 expdiophlem2 39617 |
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