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Mirrors > Home > MPE Home > Th. List > vdwap0 | Structured version Visualization version GIF version |
Description: Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
Ref | Expression |
---|---|
vdwap0 | ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4328 | . . . . . 6 ⊢ ¬ 𝑚 ∈ ∅ | |
2 | 1 | pm2.21i 119 | . . . . 5 ⊢ (𝑚 ∈ ∅ → ¬ 𝑥 = (𝐴 + (𝑚 · 𝐷))) |
3 | risefall0lem 15957 | . . . . 5 ⊢ (0...(0 − 1)) = ∅ | |
4 | 2, 3 | eleq2s 2852 | . . . 4 ⊢ (𝑚 ∈ (0...(0 − 1)) → ¬ 𝑥 = (𝐴 + (𝑚 · 𝐷))) |
5 | 4 | nrex 3075 | . . 3 ⊢ ¬ ∃𝑚 ∈ (0...(0 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷)) |
6 | 0nn0 12474 | . . . 4 ⊢ 0 ∈ ℕ0 | |
7 | vdwapval 16893 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘0)𝐷) ↔ ∃𝑚 ∈ (0...(0 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷)))) | |
8 | 6, 7 | mp3an1 1449 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘0)𝐷) ↔ ∃𝑚 ∈ (0...(0 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷)))) |
9 | 5, 8 | mtbiri 327 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ¬ 𝑥 ∈ (𝐴(AP‘0)𝐷)) |
10 | 9 | eq0rdv 4402 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∅c0 4320 ‘cfv 6535 (class class class)co 7396 0cc0 11097 1c1 11098 + caddc 11100 · cmul 11102 − cmin 11431 ℕcn 12199 ℕ0cn0 12459 ...cfz 13471 APcvdwa 16885 |
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-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-vdwap 16888 |
This theorem is referenced by: vdwap1 16897 vdwmc2 16899 vdwlem13 16913 |
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