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Theorem nn0min 30566
Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12069. (Contributed by Thierry Arnoux, 6-May-2018.)
Hypotheses
Ref Expression
nn0min.0 (𝑛 = 0 → (𝜓𝜒))
nn0min.1 (𝑛 = 𝑚 → (𝜓𝜃))
nn0min.2 (𝑛 = (𝑚 + 1) → (𝜓𝜏))
nn0min.3 (𝜑 → ¬ 𝜒)
nn0min.4 (𝜑 → ∃𝑛 ∈ ℕ 𝜓)
Assertion
Ref Expression
nn0min (𝜑 → ∃𝑚 ∈ ℕ0𝜃𝜏))
Distinct variable groups:   𝑚,𝑛,𝜑   𝜓,𝑚   𝜏,𝑛   𝜃,𝑛   𝜒,𝑚,𝑛
Allowed substitution hints:   𝜓(𝑛)   𝜃(𝑚)   𝜏(𝑚)

Proof of Theorem nn0min
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 nn0min.4 . . . . 5 (𝜑 → ∃𝑛 ∈ ℕ 𝜓)
21adantr 484 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∃𝑛 ∈ ℕ 𝜓)
3 nfv 1915 . . . . . . . . . 10 𝑚𝜑
4 nfra1 3186 . . . . . . . . . 10 𝑚𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)
53, 4nfan 1900 . . . . . . . . 9 𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
6 nfv 1915 . . . . . . . . 9 𝑚 ¬ [𝑘 / 𝑛]𝜓
75, 6nfim 1897 . . . . . . . 8 𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8 dfsbcq2 3726 . . . . . . . . . 10 (𝑘 = 1 → ([𝑘 / 𝑛]𝜓[1 / 𝑛]𝜓))
98notbid 321 . . . . . . . . 9 (𝑘 = 1 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ [1 / 𝑛]𝜓))
109imbi2d 344 . . . . . . . 8 (𝑘 = 1 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)))
11 nfv 1915 . . . . . . . . . . 11 𝑛𝜃
12 nn0min.1 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝜓𝜃))
1311, 12sbhypf 3503 . . . . . . . . . 10 (𝑘 = 𝑚 → ([𝑘 / 𝑛]𝜓𝜃))
1413notbid 321 . . . . . . . . 9 (𝑘 = 𝑚 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜃))
1514imbi2d 344 . . . . . . . 8 (𝑘 = 𝑚 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜃)))
16 nfv 1915 . . . . . . . . . . 11 𝑛𝜏
17 nn0min.2 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (𝜓𝜏))
1816, 17sbhypf 3503 . . . . . . . . . 10 (𝑘 = (𝑚 + 1) → ([𝑘 / 𝑛]𝜓𝜏))
1918notbid 321 . . . . . . . . 9 (𝑘 = (𝑚 + 1) → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜏))
2019imbi2d 344 . . . . . . . 8 (𝑘 = (𝑚 + 1) → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
21 sbequ12r 2253 . . . . . . . . . 10 (𝑘 = 𝑛 → ([𝑘 / 𝑛]𝜓𝜓))
2221notbid 321 . . . . . . . . 9 (𝑘 = 𝑛 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜓))
2322imbi2d 344 . . . . . . . 8 (𝑘 = 𝑛 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)))
24 nn0min.3 . . . . . . . . 9 (𝜑 → ¬ 𝜒)
25 0nn0 11904 . . . . . . . . . 10 0 ∈ ℕ0
2611, 12sbiev 2324 . . . . . . . . . . . . . 14 ([𝑚 / 𝑛]𝜓𝜃)
27 nfv 1915 . . . . . . . . . . . . . . 15 𝑛𝜒
28 nn0min.0 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝜓𝜒))
2927, 28sbhypf 3503 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([𝑚 / 𝑛]𝜓𝜒))
3026, 29bitr3id 288 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜃𝜒))
3130notbid 321 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜃 ↔ ¬ 𝜒))
32 oveq1 7146 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33 0p1e1 11751 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
3432, 33eqtrdi 2852 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝑚 + 1) = 1)
35 1nn 11640 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
36 eleq1 2880 . . . . . . . . . . . . . . . 16 ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
3735, 36mpbiri 261 . . . . . . . . . . . . . . 15 ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
3817sbcieg 3761 . . . . . . . . . . . . . . 15 ((𝑚 + 1) ∈ ℕ → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
3934, 37, 383syl 18 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
4034sbceq1d 3728 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓[1 / 𝑛]𝜓))
4139, 40bitr3d 284 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜏[1 / 𝑛]𝜓))
4241notbid 321 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜏 ↔ ¬ [1 / 𝑛]𝜓))
4331, 42imbi12d 348 . . . . . . . . . . 11 (𝑚 = 0 → ((¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4443rspcv 3569 . . . . . . . . . 10 (0 ∈ ℕ0 → (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4525, 44ax-mp 5 . . . . . . . . 9 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓))
4624, 45mpan9 510 . . . . . . . 8 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)
47 cbvralsvw 3417 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48 nnnn0 11896 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49 sbequ12r 2253 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
5049rspcv 3569 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
5148, 50syl 17 . . . . . . . . . . 11 (𝑚 ∈ ℕ → (∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
5247, 51syl5bi 245 . . . . . . . . . 10 (𝑚 ∈ ℕ → (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
5352adantld 494 . . . . . . . . 9 (𝑚 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → (¬ 𝜃 → ¬ 𝜏)))
5453a2d 29 . . . . . . . 8 (𝑚 ∈ ℕ → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜃) → ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
557, 10, 15, 20, 23, 46, 54nnindf 30565 . . . . . . 7 (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓))
5655rgen 3119 . . . . . 6 𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57 r19.21v 3145 . . . . . 6 (∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓))
5856, 57mpbi 233 . . . . 5 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓)
59 ralnex 3202 . . . . 5 (∀𝑛 ∈ ℕ ¬ 𝜓 ↔ ¬ ∃𝑛 ∈ ℕ 𝜓)
6058, 59sylib 221 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ ∃𝑛 ∈ ℕ 𝜓)
612, 60pm2.65da 816 . . 3 (𝜑 → ¬ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
62 imnan 403 . . . 4 ((¬ 𝜃 → ¬ 𝜏) ↔ ¬ (¬ 𝜃𝜏))
6362ralbii 3136 . . 3 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
6461, 63sylnib 331 . 2 (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
65 dfrex2 3205 . 2 (∃𝑚 ∈ ℕ0𝜃𝜏) ↔ ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
6664, 65sylibr 237 1 (𝜑 → ∃𝑚 ∈ ℕ0𝜃𝜏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  [wsb 2069  wcel 2112  wral 3109  wrex 3110  [wsbc 3723  (class class class)co 7139  0cc0 10530  1c1 10531   + caddc 10533  cn 11629  0cn0 11889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-ltxr 10673  df-nn 11630  df-n0 11890
This theorem is referenced by:  archirng  30871
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