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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.27dlem2 | Structured version Visualization version GIF version |
Description: Lemma for rmydioph 41381. 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 13447 | . . 3 ⊢ (𝐴 ∈ (1...𝐵) → 𝐴 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝐴 ∈ ℤ |
4 | elfzle1 13450 | . . 3 ⊢ (𝐴 ∈ (1...𝐵) → 1 ≤ 𝐴) | |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ 1 ≤ 𝐴 |
6 | 3 | zrei 12510 | . . . 4 ⊢ 𝐴 ∈ ℝ |
7 | jm2.27dlem2.3 | . . . . 5 ⊢ 𝐵 ∈ ℕ | |
8 | 7 | nnrei 12167 | . . . 4 ⊢ 𝐵 ∈ ℝ |
9 | elfzle2 13451 | . . . . 5 ⊢ (𝐴 ∈ (1...𝐵) → 𝐴 ≤ 𝐵) | |
10 | 1, 9 | ax-mp 5 | . . . 4 ⊢ 𝐴 ≤ 𝐵 |
11 | letrp1 12004 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ (𝐵 + 1)) | |
12 | 6, 8, 10, 11 | mp3an 1462 | . . 3 ⊢ 𝐴 ≤ (𝐵 + 1) |
13 | jm2.27dlem2.2 | . . 3 ⊢ 𝐶 = (𝐵 + 1) | |
14 | 12, 13 | breqtrri 5133 | . 2 ⊢ 𝐴 ≤ 𝐶 |
15 | 1z 12538 | . . 3 ⊢ 1 ∈ ℤ | |
16 | nnz 12525 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
17 | peano2z 12549 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐵 + 1) ∈ ℤ) | |
18 | 7, 16, 17 | mp2b 10 | . . . 4 ⊢ (𝐵 + 1) ∈ ℤ |
19 | 13, 18 | eqeltri 2830 | . . 3 ⊢ 𝐶 ∈ ℤ |
20 | elfz1 13435 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (1...𝐶) ↔ (𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶))) | |
21 | 15, 19, 20 | mp2an 691 | . 2 ⊢ (𝐴 ∈ (1...𝐶) ↔ (𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶)) |
22 | 3, 5, 14, 21 | mpbir3an 1342 | 1 ⊢ 𝐴 ∈ (1...𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝcr 11055 1c1 11057 + caddc 11059 ≤ cle 11195 ℕcn 12158 ℤcz 12504 ...cfz 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 |
This theorem is referenced by: rmydioph 41381 expdiophlem2 41389 |
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