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Theorem exfzdc 10272
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 9572 . . . . 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 10064 . . . 4 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
8 oveq2 5905 . . . . . . . 8 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
98rexeqdv 2693 . . . . . . 7 (𝑤 = 𝑀 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑀)𝜓))
109dcbid 839 . . . . . 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 5905 . . . . . . . 8 (𝑤 = 𝑦 → (𝑀...𝑤) = (𝑀...𝑦))
1312rexeqdv 2693 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑦)𝜓))
1413dcbid 839 . . . . . 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 5905 . . . . . . . 8 (𝑤 = (𝑦 + 1) → (𝑀...𝑤) = (𝑀...(𝑦 + 1)))
1716rexeqdv 2693 . . . . . . 7 (𝑤 = (𝑦 + 1) → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
1817dcbid 839 . . . . . 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 5905 . . . . . . . 8 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
2120rexeqdv 2693 . . . . . . 7 (𝑤 = 𝑁 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
2221dcbid 839 . . . . . 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 10063 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
2524adantl 277 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → 𝑀 ∈ (𝑀...𝑁))
26 exfzdc.3 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)
2726ralrimiva 2563 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
2827adantr 276 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
29 nfsbc1v 2996 . . . . . . . . . 10 𝑛[𝑀 / 𝑛]𝜓
3029nfdc 1670 . . . . . . . . 9 𝑛DECID [𝑀 / 𝑛]𝜓
31 sbceq1a 2987 . . . . . . . . . 10 (𝑛 = 𝑀 → (𝜓[𝑀 / 𝑛]𝜓))
3231dcbid 839 . . . . . . . . 9 (𝑛 = 𝑀 → (DECID 𝜓DECID [𝑀 / 𝑛]𝜓))
3330, 32rspc 2850 . . . . . . . 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 10098 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3735, 36syl 14 . . . . . . . . . 10 ((𝜑𝑁 ∈ (ℤ𝑀)) → (𝑀...𝑀) = {𝑀})
3837rexeqdv 2693 . . . . . . . . 9 ((𝜑𝑁 ∈ (ℤ𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ ∃𝑛 ∈ {𝑀}𝜓))
39 rexsns 3646 . . . . . . . . 9 (∃𝑛 ∈ {𝑀}𝜓[𝑀 / 𝑛]𝜓)
4038, 39bitrdi 196 . . . . . . . 8 ((𝜑𝑁 ∈ (ℤ𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓[𝑀 / 𝑛]𝜓))
4140dcbid 839 . . . . . . 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 10259 . . . . . . . . . . . . 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 2996 . . . . . . . . . . . . . 14 𝑛[(𝑦 + 1) / 𝑛]𝜓
4948nfdc 1670 . . . . . . . . . . . . 13 𝑛DECID [(𝑦 + 1) / 𝑛]𝜓
50 sbceq1a 2987 . . . . . . . . . . . . . 14 (𝑛 = (𝑦 + 1) → (𝜓[(𝑦 + 1) / 𝑛]𝜓))
5150dcbid 839 . . . . . . . . . . . . 13 (𝑛 = (𝑦 + 1) → (DECID 𝜓DECID [(𝑦 + 1) / 𝑛]𝜓))
5249, 51rspc 2850 . . . . . . . . . . . 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 3646 . . . . . . . . . . . 12 (∃𝑛 ∈ {(𝑦 + 1)}𝜓[(𝑦 + 1) / 𝑛]𝜓)
5554dcbii 841 . . . . . . . . . . 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 937 . . . . . . . . . 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 3330 . . . . . . . . . 10 (∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
6059dcbii 841 . . . . . . . . 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 10183 . . . . . . . . . . . 12 (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ𝑀))
6362ad2antrr 488 . . . . . . . . . . 11 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → 𝑦 ∈ (ℤ𝑀))
64 fzsuc 10101 . . . . . . . . . . 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 2693 . . . . . . . . 9 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID𝑛 ∈ (𝑀...𝑦)𝜓) → (∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
6766dcbid 839 . . . . . . . 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 10271 . . . 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 3455 . . . . 5 ¬ ∃𝑛 ∈ ∅ 𝜓
75 zltnle 9330 . . . . . . . . 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 10074 . . . . . . . . 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 2693 . . . . 5 ((𝜑 ∧ ¬ 𝑀𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ ∃𝑛 ∈ ∅ 𝜓))
8374, 82mtbiri 676 . . . 4 ((𝜑 ∧ ¬ 𝑀𝑁) → ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓)
8483olcd 735 . . 3 ((𝜑 ∧ ¬ 𝑀𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
85 df-dc 836 . . 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 9360 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀𝑁)
88 exmiddc 837 . . . 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 799 1 (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835   = wceq 1364  wcel 2160  wral 2468  wrex 2469  [wsbc 2977  cun 3142  c0 3437  {csn 3607   class class class wbr 4018  cfv 5235  (class class class)co 5897  1c1 7843   + caddc 7845   < clt 8023  cle 8024  cz 9284  cuz 9559  ...cfz 10040  ..^cfzo 10174
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285  df-uz 9560  df-fz 10041  df-fzo 10175
This theorem is referenced by:  prmind2  12155  4sqlemafi  12430  4sqexercise1  12433  4sqexercise2  12434  4sqlemsdc  12435
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