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Mirrors > Home > ILE Home > Th. List > Mathboxes > sumdc2 | GIF version |
Description: Alternate proof of sumdc 11159, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 11159). (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 2203 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
4 | 3 | dcbid 824 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑦 ∈ 𝐴)) |
5 | 4 | rspccv 2790 | . . . . . 6 ⊢ (∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴 → (𝑦 ∈ (ℤ≥‘𝑀) → DECID 𝑦 ∈ 𝐴)) |
6 | exmiddc 822 | . . . . . 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 13174 | . . 3 ⊢ (𝜑 → 𝐴 DECIDin (ℤ≥‘𝑀)) |
10 | sumdc2.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
11 | uzdcinzz 13176 | . . . 4 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) DECIDin ℤ) | |
12 | 10, 11 | syl 14 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝑀) DECIDin ℤ) |
13 | 1, 9, 12 | decidin 13175 | . 2 ⊢ (𝜑 → 𝐴 DECIDin ℤ) |
14 | sumdc2.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
15 | df-dcin 13172 | . . 3 ⊢ (𝐴 DECIDin ℤ ↔ ∀𝑧 ∈ ℤ DECID 𝑧 ∈ 𝐴) | |
16 | nfv 1509 | . . . . . 6 ⊢ Ⅎ𝑧DECID 𝑁 ∈ 𝐴 | |
17 | 16 | rspct 2786 | . . . . 5 ⊢ (∀𝑧(𝑧 = 𝑁 → (DECID 𝑧 ∈ 𝐴 ↔ DECID 𝑁 ∈ 𝐴)) → (𝑁 ∈ ℤ → (∀𝑧 ∈ ℤ DECID 𝑧 ∈ 𝐴 → DECID 𝑁 ∈ 𝐴))) |
18 | eleq1 2203 | . . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑧 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
19 | 18 | dcbid 824 | . . . . 5 ⊢ (𝑧 = 𝑁 → (DECID 𝑧 ∈ 𝐴 ↔ DECID 𝑁 ∈ 𝐴)) |
20 | 17, 19 | mpg 1428 | . . . 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 698 DECID wdc 820 = wceq 1332 ∈ wcel 1481 ∀wral 2417 ⊆ wss 3076 ‘cfv 5131 ℤcz 9078 ℤ≥cuz 9350 DECIDin wdcin 13171 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-dcin 13172 |
This theorem is referenced by: (None) |
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