Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eluzrabdioph | Structured version Visualization version GIF version |
Description: Diophantine set builder for membership in a fixed upper set of integers. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
Ref | Expression |
---|---|
eluzrabdioph | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ (ℤ≥‘𝑀)} ∈ (Dioph‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabdiophlem1 40603 | . . . . 5 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ) | |
2 | eluz 12578 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐴)) | |
3 | 2 | ex 412 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝐴 ∈ ℤ → (𝐴 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐴))) |
4 | 3 | ralimdv 3105 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐴))) |
5 | 4 | imp 406 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐴)) |
6 | 1, 5 | sylan2 592 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐴)) |
7 | rabbi 3314 | . . . 4 ⊢ (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐴) ↔ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ (ℤ≥‘𝑀)} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝑀 ≤ 𝐴}) | |
8 | 6, 7 | sylib 217 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ (ℤ≥‘𝑀)} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝑀 ≤ 𝐴}) |
9 | 8 | 3adant1 1128 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ (ℤ≥‘𝑀)} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝑀 ≤ 𝐴}) |
10 | ovex 7301 | . . . 4 ⊢ (1...𝑁) ∈ V | |
11 | mzpconstmpt 40542 | . . . 4 ⊢ (((1...𝑁) ∈ V ∧ 𝑀 ∈ ℤ) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑀) ∈ (mzPoly‘(1...𝑁))) | |
12 | 10, 11 | mpan 686 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑀) ∈ (mzPoly‘(1...𝑁))) |
13 | lerabdioph 40607 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑀) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝑀 ≤ 𝐴} ∈ (Dioph‘𝑁)) | |
14 | 12, 13 | syl3an2 1162 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝑀 ≤ 𝐴} ∈ (Dioph‘𝑁)) |
15 | 9, 14 | eqeltrd 2840 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ (ℤ≥‘𝑀)} ∈ (Dioph‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∀wral 3065 {crab 3069 Vcvv 3430 class class class wbr 5078 ↦ cmpt 5161 ‘cfv 6430 (class class class)co 7268 ↑m cmap 8589 1c1 10856 ≤ cle 10994 ℕ0cn0 12216 ℤcz 12302 ℤ≥cuz 12564 ...cfz 13221 mzPolycmzp 40524 Diophcdioph 40557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-oadd 8285 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-hash 14026 df-mzpcl 40525 df-mzp 40526 df-dioph 40558 |
This theorem is referenced by: elnnrabdioph 40609 rmydioph 40816 expdiophlem2 40824 |
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