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Theorem exfzdc 10233
Description: Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
Hypotheses
Ref Expression
exfzdc.1 (𝜑𝑀 ∈ ℤ)
exfzdc.2 (𝜑𝑁 ∈ ℤ)
exfzdc.3 ((𝜑𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)
Assertion
Ref Expression
exfzdc (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓)
Distinct variable groups:   𝑛,𝑀   𝑛,𝑁   𝜑,𝑛
Allowed substitution hint:   𝜓(𝑛)

Proof of Theorem exfzdc
Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exfzdc.1 . . . . 5 (𝜑𝑀 ∈ ℤ)
2 exfzdc.2 . . . . 5 (𝜑𝑁 ∈ ℤ)
3 eluz 9535 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
41, 2, 3syl2anc 411 . . . 4 (𝜑 → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
54biimpar 297 . . 3 ((𝜑𝑀𝑁) → 𝑁 ∈ (ℤ𝑀))
6 simpl 109 . . 3 ((𝜑𝑀𝑁) → 𝜑)
7 eluzfz2 10025 . . . 4 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
8 oveq2 5878 . . . . . . . 8 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
98rexeqdv 2679 . . . . . . 7 (𝑤 = 𝑀 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑀)𝜓))
109dcbid 838 . . . . . 6 (𝑤 = 𝑀 → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...𝑀)𝜓))
1110imbi2d 230 . . . . 5 (𝑤 = 𝑀 → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...𝑀)𝜓)))
12 oveq2 5878 . . . . . . . 8 (𝑤 = 𝑦 → (𝑀...𝑤) = (𝑀...𝑦))
1312rexeqdv 2679 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑦)𝜓))
1413dcbid 838 . . . . . 6 (𝑤 = 𝑦 → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...𝑦)𝜓))
1514imbi2d 230 . . . . 5 (𝑤 = 𝑦 → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...𝑦)𝜓)))
16 oveq2 5878 . . . . . . . 8 (𝑤 = (𝑦 + 1) → (𝑀...𝑤) = (𝑀...(𝑦 + 1)))
1716rexeqdv 2679 . . . . . . 7 (𝑤 = (𝑦 + 1) → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
1817dcbid 838 . . . . . 6 (𝑤 = (𝑦 + 1) → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
1918imbi2d 230 . . . . 5 (𝑤 = (𝑦 + 1) → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
20 oveq2 5878 . . . . . . . 8 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
2120rexeqdv 2679 . . . . . . 7 (𝑤 = 𝑁 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
2221dcbid 838 . . . . . 6 (𝑤 = 𝑁 → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...𝑁)𝜓))
2322imbi2d 230 . . . . 5 (𝑤 = 𝑁 → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓)))
24 eluzfz1 10024 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
2524adantl 277 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → 𝑀 ∈ (𝑀...𝑁))
26 exfzdc.3 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)
2726ralrimiva 2550 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
2827adantr 276 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
29 nfsbc1v 2981 . . . . . . . . . 10 𝑛[𝑀 / 𝑛]𝜓
3029nfdc 1659 . . . . . . . . 9 𝑛DECID [𝑀 / 𝑛]𝜓
31 sbceq1a 2972 . . . . . . . . . 10 (𝑛 = 𝑀 → (𝜓[𝑀 / 𝑛]𝜓))
3231dcbid 838 . . . . . . . . 9 (𝑛 = 𝑀 → (DECID 𝜓DECID [𝑀 / 𝑛]𝜓))
3330, 32rspc 2835 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓DECID [𝑀 / 𝑛]𝜓))
3425, 28, 33sylc 62 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ𝑀)) → DECID [𝑀 / 𝑛]𝜓)
351adantr 276 . . . . . . . . . . 11 ((𝜑𝑁 ∈ (ℤ𝑀)) → 𝑀 ∈ ℤ)
36 fzsn 10059 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3735, 36syl 14 . . . . . . . . . 10 ((𝜑𝑁 ∈ (ℤ𝑀)) → (𝑀...𝑀) = {𝑀})
3837rexeqdv 2679 . . . . . . . . 9 ((𝜑𝑁 ∈ (ℤ𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ ∃𝑛 ∈ {𝑀}𝜓))
39 rexsns 3631 . . . . . . . . 9 (∃𝑛 ∈ {𝑀}𝜓[𝑀 / 𝑛]𝜓)
4038, 39bitrdi 196 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓[𝑀 / 𝑛]𝜓))
4140dcbid 838 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ𝑀)) → (DECID𝑛 ∈ (𝑀...𝑀)𝜓DECID [𝑀 / 𝑛]𝜓))
4234, 41mpbird 167 . . . . . 6 ((𝜑𝑁 ∈ (ℤ𝑀)) → DECID𝑛 ∈ (𝑀...𝑀)𝜓)
4342expcom 116 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (𝜑DECID𝑛 ∈ (𝑀...𝑀)𝜓))
44 simpr 110 . . . . . . . . . 10 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ (𝑀...𝑦)𝜓)
45 fzofzp1 10220 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑀..^𝑁) → (𝑦 + 1) ∈ (𝑀...𝑁))
4645ad2antrr 488 . . . . . . . . . . . 12 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑦 + 1) ∈ (𝑀...𝑁))
4727ad2antlr 489 . . . . . . . . . . . 12 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
48 nfsbc1v 2981 . . . . . . . . . . . . . 14 𝑛[(𝑦 + 1) / 𝑛]𝜓
4948nfdc 1659 . . . . . . . . . . . . 13 𝑛DECID [(𝑦 + 1) / 𝑛]𝜓
50 sbceq1a 2972 . . . . . . . . . . . . . 14 (𝑛 = (𝑦 + 1) → (𝜓[(𝑦 + 1) / 𝑛]𝜓))
5150dcbid 838 . . . . . . . . . . . . 13 (𝑛 = (𝑦 + 1) → (DECID 𝜓DECID [(𝑦 + 1) / 𝑛]𝜓))
5249, 51rspc 2835 . . . . . . . . . . . 12 ((𝑦 + 1) ∈ (𝑀...𝑁) → (∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓DECID [(𝑦 + 1) / 𝑛]𝜓))
5346, 47, 52sylc 62 . . . . . . . . . . 11 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID [(𝑦 + 1) / 𝑛]𝜓)
54 rexsns 3631 . . . . . . . . . . . 12 (∃𝑛 ∈ {(𝑦 + 1)}𝜓[(𝑦 + 1) / 𝑛]𝜓)
5554dcbii 840 . . . . . . . . . . 11 (DECID𝑛 ∈ {(𝑦 + 1)}𝜓DECID [(𝑦 + 1) / 𝑛]𝜓)
5653, 55sylibr 134 . . . . . . . . . 10 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ {(𝑦 + 1)}𝜓)
57 dcor 935 . . . . . . . . . 10 (DECID𝑛 ∈ (𝑀...𝑦)𝜓 → (DECID𝑛 ∈ {(𝑦 + 1)}𝜓DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓)))
5844, 56, 57sylc 62 . . . . . . . . 9 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
59 rexun 3315 . . . . . . . . . 10 (∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
6059dcbii 840 . . . . . . . . 9 (DECID𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
6158, 60sylibr 134 . . . . . . . 8 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓)
62 elfzouz 10144 . . . . . . . . . . . 12 (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ𝑀))
6362ad2antrr 488 . . . . . . . . . . 11 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → 𝑦 ∈ (ℤ𝑀))
64 fzsuc 10062 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
6563, 64syl 14 . . . . . . . . . 10 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
6665rexeqdv 2679 . . . . . . . . 9 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
6766dcbid 838 . . . . . . . 8 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓DECID𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
6861, 67mpbird 167 . . . . . . 7 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)
6968exp31 364 . . . . . 6 (𝑦 ∈ (𝑀..^𝑁) → (𝜑 → (DECID𝑛 ∈ (𝑀...𝑦)𝜓DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
7069a2d 26 . . . . 5 (𝑦 ∈ (𝑀..^𝑁) → ((𝜑DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (𝜑DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
7111, 15, 19, 23, 43, 70fzind2 10232 . . . 4 (𝑁 ∈ (𝑀...𝑁) → (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓))
727, 71syl 14 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓))
735, 6, 72sylc 62 . 2 ((𝜑𝑀𝑁) → DECID𝑛 ∈ (𝑀...𝑁)𝜓)
74 rex0 3440 . . . . 5 ¬ ∃𝑛 ∈ ∅ 𝜓
75 zltnle 9293 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀𝑁))
762, 1, 75syl2anc 411 . . . . . . . 8 (𝜑 → (𝑁 < 𝑀 ↔ ¬ 𝑀𝑁))
7776biimpar 297 . . . . . . 7 ((𝜑 ∧ ¬ 𝑀𝑁) → 𝑁 < 𝑀)
78 fzn 10035 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
791, 2, 78syl2anc 411 . . . . . . . 8 (𝜑 → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
8079adantr 276 . . . . . . 7 ((𝜑 ∧ ¬ 𝑀𝑁) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
8177, 80mpbid 147 . . . . . 6 ((𝜑 ∧ ¬ 𝑀𝑁) → (𝑀...𝑁) = ∅)
8281rexeqdv 2679 . . . . 5 ((𝜑 ∧ ¬ 𝑀𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ ∃𝑛 ∈ ∅ 𝜓))
8374, 82mtbiri 675 . . . 4 ((𝜑 ∧ ¬ 𝑀𝑁) → ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓)
8483olcd 734 . . 3 ((𝜑 ∧ ¬ 𝑀𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
85 df-dc 835 . . 3 (DECID𝑛 ∈ (𝑀...𝑁)𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
8684, 85sylibr 134 . 2 ((𝜑 ∧ ¬ 𝑀𝑁) → DECID𝑛 ∈ (𝑀...𝑁)𝜓)
87 zdcle 9323 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀𝑁)
88 exmiddc 836 . . . 4 (DECID 𝑀𝑁 → (𝑀𝑁 ∨ ¬ 𝑀𝑁))
8987, 88syl 14 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ∨ ¬ 𝑀𝑁))
901, 2, 89syl2anc 411 . 2 (𝜑 → (𝑀𝑁 ∨ ¬ 𝑀𝑁))
9173, 86, 90mpjaodan 798 1 (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  wrex 2456  [wsbc 2962  cun 3127  c0 3422  {csn 3592   class class class wbr 4001  cfv 5213  (class class class)co 5870  1c1 7807   + caddc 7809   < clt 7986  cle 7987  cz 9247  cuz 9522  ...cfz 10002  ..^cfzo 10135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-addcom 7906  ax-addass 7908  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-0id 7914  ax-rnegex 7915  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-apti 7921  ax-pre-ltadd 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-fv 5221  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-inn 8914  df-n0 9171  df-z 9248  df-uz 9523  df-fz 10003  df-fzo 10136
This theorem is referenced by:  prmind2  12110
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