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Mirrors > Home > ILE Home > Th. List > sumdc | GIF version |
Description: Decidability of a subset of upper integers. (Contributed by Jim Kingdon, 1-Jan-2022.) |
Ref | Expression |
---|---|
sumdc.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
sumdc.ss | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) |
sumdc.dc | ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴) |
sumdc.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
sumdc | ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumdc.dc | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴) | |
2 | eleq1 2202 | . . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
3 | 2 | dcbid 823 | . . . 4 ⊢ (𝑥 = 𝑁 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑁 ∈ 𝐴)) |
4 | 3 | rspcv 2785 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴 → DECID 𝑁 ∈ 𝐴)) |
5 | 1, 4 | mpan9 279 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → DECID 𝑁 ∈ 𝐴) |
6 | sumdc.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
7 | 6 | ssneld 3099 | . . . . 5 ⊢ (𝜑 → (¬ 𝑁 ∈ (ℤ≥‘𝑀) → ¬ 𝑁 ∈ 𝐴)) |
8 | 7 | imp 123 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ (ℤ≥‘𝑀)) → ¬ 𝑁 ∈ 𝐴) |
9 | 8 | olcd 723 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑁 ∈ 𝐴 ∨ ¬ 𝑁 ∈ 𝐴)) |
10 | df-dc 820 | . . 3 ⊢ (DECID 𝑁 ∈ 𝐴 ↔ (𝑁 ∈ 𝐴 ∨ ¬ 𝑁 ∈ 𝐴)) | |
11 | 9, 10 | sylibr 133 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ (ℤ≥‘𝑀)) → DECID 𝑁 ∈ 𝐴) |
12 | sumdc.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | sumdc.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
14 | eluzdc 9404 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ≥‘𝑀)) | |
15 | 12, 13, 14 | syl2anc 408 | . . 3 ⊢ (𝜑 → DECID 𝑁 ∈ (ℤ≥‘𝑀)) |
16 | exmiddc 821 | . . 3 ⊢ (DECID 𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑁 ∈ (ℤ≥‘𝑀))) | |
17 | 15, 16 | syl 14 | . 2 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑁 ∈ (ℤ≥‘𝑀))) |
18 | 5, 11, 17 | mpjaodan 787 | 1 ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 ‘cfv 5123 ℤcz 9054 ℤ≥cuz 9326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 |
This theorem is referenced by: sumeq2 11128 prodeq2 11326 |
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