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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 2220 | . . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
3 | 2 | dcbid 824 | . . . 4 ⊢ (𝑥 = 𝑁 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑁 ∈ 𝐴)) |
4 | 3 | rspcv 2812 | . . 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 3130 | . . . . 5 ⊢ (𝜑 → (¬ 𝑁 ∈ (ℤ≥‘𝑀) → ¬ 𝑁 ∈ 𝐴)) |
8 | 7 | imp 123 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ (ℤ≥‘𝑀)) → ¬ 𝑁 ∈ 𝐴) |
9 | 8 | olcd 724 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑁 ∈ 𝐴 ∨ ¬ 𝑁 ∈ 𝐴)) |
10 | df-dc 821 | . . 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 9522 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ≥‘𝑀)) | |
15 | 12, 13, 14 | syl2anc 409 | . . 3 ⊢ (𝜑 → DECID 𝑁 ∈ (ℤ≥‘𝑀)) |
16 | exmiddc 822 | . . 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 788 | 1 ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 ‘cfv 5171 ℤcz 9168 ℤ≥cuz 9440 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 |
This theorem is referenced by: sumeq2 11260 prodeq2 11458 |
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