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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 2233 | . . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
3 | 2 | dcbid 833 | . . . 4 ⊢ (𝑥 = 𝑁 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑁 ∈ 𝐴)) |
4 | 3 | rspcv 2830 | . . 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 3149 | . . . . 5 ⊢ (𝜑 → (¬ 𝑁 ∈ (ℤ≥‘𝑀) → ¬ 𝑁 ∈ 𝐴)) |
8 | 7 | imp 123 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ (ℤ≥‘𝑀)) → ¬ 𝑁 ∈ 𝐴) |
9 | 8 | olcd 729 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑁 ∈ 𝐴 ∨ ¬ 𝑁 ∈ 𝐴)) |
10 | df-dc 830 | . . 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 9569 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ≥‘𝑀)) | |
15 | 12, 13, 14 | syl2anc 409 | . . 3 ⊢ (𝜑 → DECID 𝑁 ∈ (ℤ≥‘𝑀)) |
16 | exmiddc 831 | . . 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 793 | 1 ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 703 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 ‘cfv 5198 ℤcz 9212 ℤ≥cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: sumeq2 11322 prodeq2 11520 |
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