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Theorem exfzdc 9904
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 9235 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
41, 2, 3syl2anc 406 . . . 4 (𝜑 → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
54biimpar 293 . . 3 ((𝜑𝑀𝑁) → 𝑁 ∈ (ℤ𝑀))
6 simpl 108 . . 3 ((𝜑𝑀𝑁) → 𝜑)
7 eluzfz2 9699 . . . 4 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
8 oveq2 5734 . . . . . . . 8 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
98rexeqdv 2605 . . . . . . 7 (𝑤 = 𝑀 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑀)𝜓))
109dcbid 806 . . . . . 6 (𝑤 = 𝑀 → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...𝑀)𝜓))
1110imbi2d 229 . . . . 5 (𝑤 = 𝑀 → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...𝑀)𝜓)))
12 oveq2 5734 . . . . . . . 8 (𝑤 = 𝑦 → (𝑀...𝑤) = (𝑀...𝑦))
1312rexeqdv 2605 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑦)𝜓))
1413dcbid 806 . . . . . 6 (𝑤 = 𝑦 → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...𝑦)𝜓))
1514imbi2d 229 . . . . 5 (𝑤 = 𝑦 → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...𝑦)𝜓)))
16 oveq2 5734 . . . . . . . 8 (𝑤 = (𝑦 + 1) → (𝑀...𝑤) = (𝑀...(𝑦 + 1)))
1716rexeqdv 2605 . . . . . . 7 (𝑤 = (𝑦 + 1) → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
1817dcbid 806 . . . . . 6 (𝑤 = (𝑦 + 1) → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
1918imbi2d 229 . . . . 5 (𝑤 = (𝑦 + 1) → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
20 oveq2 5734 . . . . . . . 8 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
2120rexeqdv 2605 . . . . . . 7 (𝑤 = 𝑁 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
2221dcbid 806 . . . . . 6 (𝑤 = 𝑁 → (DECID𝑛 ∈ (𝑀...𝑤)𝜓DECID𝑛 ∈ (𝑀...𝑁)𝜓))
2322imbi2d 229 . . . . 5 (𝑤 = 𝑁 → ((𝜑DECID𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓)))
24 eluzfz1 9698 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
2524adantl 273 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → 𝑀 ∈ (𝑀...𝑁))
26 exfzdc.3 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)
2726ralrimiva 2477 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
2827adantr 272 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
29 nfsbc1v 2894 . . . . . . . . . 10 𝑛[𝑀 / 𝑛]𝜓
3029nfdc 1618 . . . . . . . . 9 𝑛DECID [𝑀 / 𝑛]𝜓
31 sbceq1a 2885 . . . . . . . . . 10 (𝑛 = 𝑀 → (𝜓[𝑀 / 𝑛]𝜓))
3231dcbid 806 . . . . . . . . 9 (𝑛 = 𝑀 → (DECID 𝜓DECID [𝑀 / 𝑛]𝜓))
3330, 32rspc 2752 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓DECID [𝑀 / 𝑛]𝜓))
3425, 28, 33sylc 62 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ𝑀)) → DECID [𝑀 / 𝑛]𝜓)
351adantr 272 . . . . . . . . . . 11 ((𝜑𝑁 ∈ (ℤ𝑀)) → 𝑀 ∈ ℤ)
36 fzsn 9733 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3735, 36syl 14 . . . . . . . . . 10 ((𝜑𝑁 ∈ (ℤ𝑀)) → (𝑀...𝑀) = {𝑀})
3837rexeqdv 2605 . . . . . . . . 9 ((𝜑𝑁 ∈ (ℤ𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ ∃𝑛 ∈ {𝑀}𝜓))
39 rexsns 3527 . . . . . . . . 9 (∃𝑛 ∈ {𝑀}𝜓[𝑀 / 𝑛]𝜓)
4038, 39syl6bb 195 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓[𝑀 / 𝑛]𝜓))
4140dcbid 806 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ𝑀)) → (DECID𝑛 ∈ (𝑀...𝑀)𝜓DECID [𝑀 / 𝑛]𝜓))
4234, 41mpbird 166 . . . . . 6 ((𝜑𝑁 ∈ (ℤ𝑀)) → DECID𝑛 ∈ (𝑀...𝑀)𝜓)
4342expcom 115 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (𝜑DECID𝑛 ∈ (𝑀...𝑀)𝜓))
44 simpr 109 . . . . . . . . . 10 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ (𝑀...𝑦)𝜓)
45 fzofzp1 9891 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑀..^𝑁) → (𝑦 + 1) ∈ (𝑀...𝑁))
4645ad2antrr 477 . . . . . . . . . . . 12 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑦 + 1) ∈ (𝑀...𝑁))
4727ad2antlr 478 . . . . . . . . . . . 12 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
48 nfsbc1v 2894 . . . . . . . . . . . . . 14 𝑛[(𝑦 + 1) / 𝑛]𝜓
4948nfdc 1618 . . . . . . . . . . . . 13 𝑛DECID [(𝑦 + 1) / 𝑛]𝜓
50 sbceq1a 2885 . . . . . . . . . . . . . 14 (𝑛 = (𝑦 + 1) → (𝜓[(𝑦 + 1) / 𝑛]𝜓))
5150dcbid 806 . . . . . . . . . . . . 13 (𝑛 = (𝑦 + 1) → (DECID 𝜓DECID [(𝑦 + 1) / 𝑛]𝜓))
5249, 51rspc 2752 . . . . . . . . . . . 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 3527 . . . . . . . . . . . 12 (∃𝑛 ∈ {(𝑦 + 1)}𝜓[(𝑦 + 1) / 𝑛]𝜓)
5554dcbii 808 . . . . . . . . . . 11 (DECID𝑛 ∈ {(𝑦 + 1)}𝜓DECID [(𝑦 + 1) / 𝑛]𝜓)
5653, 55sylibr 133 . . . . . . . . . 10 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ {(𝑦 + 1)}𝜓)
57 dcor 900 . . . . . . . . . 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 3220 . . . . . . . . . 10 (∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
6059dcbii 808 . . . . . . . . 9 (DECID𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
6158, 60sylibr 133 . . . . . . . 8 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓)
62 elfzouz 9815 . . . . . . . . . . . 12 (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ𝑀))
6362ad2antrr 477 . . . . . . . . . . 11 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → 𝑦 ∈ (ℤ𝑀))
64 fzsuc 9736 . . . . . . . . . . 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 2605 . . . . . . . . 9 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
6766dcbid 806 . . . . . . . 8 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓DECID𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
6861, 67mpbird 166 . . . . . . 7 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → DECID𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)
6968exp31 359 . . . . . 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 9903 . . . 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 3344 . . . . 5 ¬ ∃𝑛 ∈ ∅ 𝜓
75 zltnle 8998 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀𝑁))
762, 1, 75syl2anc 406 . . . . . . . 8 (𝜑 → (𝑁 < 𝑀 ↔ ¬ 𝑀𝑁))
7776biimpar 293 . . . . . . 7 ((𝜑 ∧ ¬ 𝑀𝑁) → 𝑁 < 𝑀)
78 fzn 9709 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
791, 2, 78syl2anc 406 . . . . . . . 8 (𝜑 → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
8079adantr 272 . . . . . . 7 ((𝜑 ∧ ¬ 𝑀𝑁) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
8177, 80mpbid 146 . . . . . 6 ((𝜑 ∧ ¬ 𝑀𝑁) → (𝑀...𝑁) = ∅)
8281rexeqdv 2605 . . . . 5 ((𝜑 ∧ ¬ 𝑀𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ ∃𝑛 ∈ ∅ 𝜓))
8374, 82mtbiri 647 . . . 4 ((𝜑 ∧ ¬ 𝑀𝑁) → ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓)
8483olcd 706 . . 3 ((𝜑 ∧ ¬ 𝑀𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
85 df-dc 803 . . 3 (DECID𝑛 ∈ (𝑀...𝑁)𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
8684, 85sylibr 133 . 2 ((𝜑 ∧ ¬ 𝑀𝑁) → DECID𝑛 ∈ (𝑀...𝑁)𝜓)
87 zdcle 9025 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀𝑁)
88 exmiddc 804 . . . 4 (DECID 𝑀𝑁 → (𝑀𝑁 ∨ ¬ 𝑀𝑁))
8987, 88syl 14 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ∨ ¬ 𝑀𝑁))
901, 2, 89syl2anc 406 . 2 (𝜑 → (𝑀𝑁 ∨ ¬ 𝑀𝑁))
9173, 86, 90mpjaodan 770 1 (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802   = wceq 1312  wcel 1461  wral 2388  wrex 2389  [wsbc 2876  cun 3033  c0 3327  {csn 3491   class class class wbr 3893  cfv 5079  (class class class)co 5726  1c1 7542   + caddc 7544   < clt 7718  cle 7719  cz 8952  cuz 9222  ...cfz 9677  ..^cfzo 9806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953  df-uz 9223  df-fz 9678  df-fzo 9807
This theorem is referenced by:  prmind2  11641
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