Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > sumdc2 | GIF version |
Description: Alternate proof of sumdc 11308, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 11308). (Contributed by BJ, 19-Feb-2022.) |
Ref | Expression |
---|---|
sumdc2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
sumdc2.ss | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) |
sumdc2.dc | ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴) |
sumdc2.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
sumdc2 | ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumdc2.ss | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
2 | sumdc2.dc | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴) | |
3 | eleq1 2233 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
4 | 3 | dcbid 833 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑦 ∈ 𝐴)) |
5 | 4 | rspccv 2831 | . . . . . 6 ⊢ (∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴 → (𝑦 ∈ (ℤ≥‘𝑀) → DECID 𝑦 ∈ 𝐴)) |
6 | exmiddc 831 | . . . . . 6 ⊢ (DECID 𝑦 ∈ 𝐴 → (𝑦 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐴)) | |
7 | 5, 6 | syl6 33 | . . . . 5 ⊢ (∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴 → (𝑦 ∈ (ℤ≥‘𝑀) → (𝑦 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐴))) |
8 | 2, 7 | syl 14 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ (ℤ≥‘𝑀) → (𝑦 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐴))) |
9 | 8 | decidr 13752 | . . 3 ⊢ (𝜑 → 𝐴 DECIDin (ℤ≥‘𝑀)) |
10 | sumdc2.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
11 | uzdcinzz 13754 | . . . 4 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) DECIDin ℤ) | |
12 | 10, 11 | syl 14 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝑀) DECIDin ℤ) |
13 | 1, 9, 12 | decidin 13753 | . 2 ⊢ (𝜑 → 𝐴 DECIDin ℤ) |
14 | sumdc2.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
15 | df-dcin 13750 | . . 3 ⊢ (𝐴 DECIDin ℤ ↔ ∀𝑧 ∈ ℤ DECID 𝑧 ∈ 𝐴) | |
16 | nfv 1521 | . . . . . 6 ⊢ Ⅎ𝑧DECID 𝑁 ∈ 𝐴 | |
17 | 16 | rspct 2827 | . . . . 5 ⊢ (∀𝑧(𝑧 = 𝑁 → (DECID 𝑧 ∈ 𝐴 ↔ DECID 𝑁 ∈ 𝐴)) → (𝑁 ∈ ℤ → (∀𝑧 ∈ ℤ DECID 𝑧 ∈ 𝐴 → DECID 𝑁 ∈ 𝐴))) |
18 | eleq1 2233 | . . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑧 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
19 | 18 | dcbid 833 | . . . . 5 ⊢ (𝑧 = 𝑁 → (DECID 𝑧 ∈ 𝐴 ↔ DECID 𝑁 ∈ 𝐴)) |
20 | 17, 19 | mpg 1444 | . . . 4 ⊢ (𝑁 ∈ ℤ → (∀𝑧 ∈ ℤ DECID 𝑧 ∈ 𝐴 → DECID 𝑁 ∈ 𝐴)) |
21 | 20 | com12 30 | . . 3 ⊢ (∀𝑧 ∈ ℤ DECID 𝑧 ∈ 𝐴 → (𝑁 ∈ ℤ → DECID 𝑁 ∈ 𝐴)) |
22 | 15, 21 | sylbi 120 | . 2 ⊢ (𝐴 DECIDin ℤ → (𝑁 ∈ ℤ → DECID 𝑁 ∈ 𝐴)) |
23 | 13, 14, 22 | sylc 62 | 1 ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 703 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 ‘cfv 5196 ℤcz 9199 ℤ≥cuz 9474 DECIDin wdcin 13749 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 df-uz 9475 df-dcin 13750 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |