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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 3878 | . . . . . 6 ⊢ ¬ 𝑚 ∈ ∅ | |
2 | 1 | pm2.21i 115 | . . . . 5 ⊢ (𝑚 ∈ ∅ → ¬ 𝑥 = (𝐴 + (𝑚 · 𝐷))) |
3 | 0re 9919 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
4 | ltm1 10742 | . . . . . . 7 ⊢ (0 ∈ ℝ → (0 − 1) < 0) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (0 − 1) < 0 |
6 | 0z 11265 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | peano2zm 11297 | . . . . . . . 8 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ (0 − 1) ∈ ℤ |
9 | fzn 12228 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ (0 − 1) ∈ ℤ) → ((0 − 1) < 0 ↔ (0...(0 − 1)) = ∅)) | |
10 | 6, 8, 9 | mp2an 704 | . . . . . 6 ⊢ ((0 − 1) < 0 ↔ (0...(0 − 1)) = ∅) |
11 | 5, 10 | mpbi 219 | . . . . 5 ⊢ (0...(0 − 1)) = ∅ |
12 | 2, 11 | eleq2s 2706 | . . . 4 ⊢ (𝑚 ∈ (0...(0 − 1)) → ¬ 𝑥 = (𝐴 + (𝑚 · 𝐷))) |
13 | 12 | nrex 2983 | . . 3 ⊢ ¬ ∃𝑚 ∈ (0...(0 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷)) |
14 | 0nn0 11184 | . . . 4 ⊢ 0 ∈ ℕ0 | |
15 | vdwapval 15515 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘0)𝐷) ↔ ∃𝑚 ∈ (0...(0 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷)))) | |
16 | 14, 15 | mp3an1 1403 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘0)𝐷) ↔ ∃𝑚 ∈ (0...(0 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷)))) |
17 | 13, 16 | mtbiri 316 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ¬ 𝑥 ∈ (𝐴(AP‘0)𝐷)) |
18 | 17 | eq0rdv 3931 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 APcvdwa 15507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-vdwap 15510 |
This theorem is referenced by: vdwap1 15519 vdwmc2 15521 vdwlem13 15535 |
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