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Theorem nn0min 30016
Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 11719. (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 472 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∃𝑛 ∈ ℕ 𝜓)
3 nfv 2009 . . . . . . . . . 10 𝑚𝜑
4 nfra1 3088 . . . . . . . . . 10 𝑚𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)
53, 4nfan 1998 . . . . . . . . 9 𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
6 nfv 2009 . . . . . . . . 9 𝑚 ¬ [𝑘 / 𝑛]𝜓
75, 6nfim 1995 . . . . . . . 8 𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8 dfsbcq2 3599 . . . . . . . . . 10 (𝑘 = 1 → ([𝑘 / 𝑛]𝜓[1 / 𝑛]𝜓))
98notbid 309 . . . . . . . . 9 (𝑘 = 1 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ [1 / 𝑛]𝜓))
109imbi2d 331 . . . . . . . 8 (𝑘 = 1 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)))
11 nfv 2009 . . . . . . . . . . 11 𝑛𝜃
12 nn0min.1 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝜓𝜃))
1311, 12sbhypf 3406 . . . . . . . . . 10 (𝑘 = 𝑚 → ([𝑘 / 𝑛]𝜓𝜃))
1413notbid 309 . . . . . . . . 9 (𝑘 = 𝑚 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜃))
1514imbi2d 331 . . . . . . . 8 (𝑘 = 𝑚 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜃)))
16 nfv 2009 . . . . . . . . . . 11 𝑛𝜏
17 nn0min.2 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (𝜓𝜏))
1816, 17sbhypf 3406 . . . . . . . . . 10 (𝑘 = (𝑚 + 1) → ([𝑘 / 𝑛]𝜓𝜏))
1918notbid 309 . . . . . . . . 9 (𝑘 = (𝑚 + 1) → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜏))
2019imbi2d 331 . . . . . . . 8 (𝑘 = (𝑚 + 1) → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
21 sbequ12r 2279 . . . . . . . . . 10 (𝑘 = 𝑛 → ([𝑘 / 𝑛]𝜓𝜓))
2221notbid 309 . . . . . . . . 9 (𝑘 = 𝑛 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜓))
2322imbi2d 331 . . . . . . . 8 (𝑘 = 𝑛 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)))
24 nn0min.3 . . . . . . . . 9 (𝜑 → ¬ 𝜒)
25 0nn0 11555 . . . . . . . . . 10 0 ∈ ℕ0
2611, 12sbie 2499 . . . . . . . . . . . . . 14 ([𝑚 / 𝑛]𝜓𝜃)
27 nfv 2009 . . . . . . . . . . . . . . 15 𝑛𝜒
28 nn0min.0 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝜓𝜒))
2927, 28sbhypf 3406 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([𝑚 / 𝑛]𝜓𝜒))
3026, 29syl5bbr 276 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜃𝜒))
3130notbid 309 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜃 ↔ ¬ 𝜒))
32 oveq1 6849 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33 0p1e1 11401 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
3432, 33syl6eq 2815 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝑚 + 1) = 1)
35 1nn 11287 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
36 eleq1 2832 . . . . . . . . . . . . . . . 16 ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
3735, 36mpbiri 249 . . . . . . . . . . . . . . 15 ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
3817sbcieg 3629 . . . . . . . . . . . . . . 15 ((𝑚 + 1) ∈ ℕ → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
3934, 37, 383syl 18 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
4034sbceq1d 3601 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓[1 / 𝑛]𝜓))
4139, 40bitr3d 272 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜏[1 / 𝑛]𝜓))
4241notbid 309 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜏 ↔ ¬ [1 / 𝑛]𝜓))
4331, 42imbi12d 335 . . . . . . . . . . 11 (𝑚 = 0 → ((¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4443rspcv 3457 . . . . . . . . . 10 (0 ∈ ℕ0 → (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4525, 44ax-mp 5 . . . . . . . . 9 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓))
4624, 45mpan9 502 . . . . . . . 8 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)
47 cbvralsv 3330 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48 nnnn0 11546 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49 sbequ12r 2279 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
5049rspcv 3457 . . . . . . . . . . . 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 233 . . . . . . . . . 10 (𝑚 ∈ ℕ → (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
5352adantld 484 . . . . . . . . 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 30014 . . . . . . 7 (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓))
5655rgen 3069 . . . . . 6 𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57 r19.21v 3107 . . . . . 6 (∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓))
5856, 57mpbi 221 . . . . 5 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓)
59 ralnex 3139 . . . . 5 (∀𝑛 ∈ ℕ ¬ 𝜓 ↔ ¬ ∃𝑛 ∈ ℕ 𝜓)
6058, 59sylib 209 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ ∃𝑛 ∈ ℕ 𝜓)
612, 60pm2.65da 851 . . 3 (𝜑 → ¬ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
62 imnan 388 . . . 4 ((¬ 𝜃 → ¬ 𝜏) ↔ ¬ (¬ 𝜃𝜏))
6362ralbii 3127 . . 3 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
6461, 63sylnib 319 . 2 (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
65 dfrex2 3142 . 2 (∃𝑚 ∈ ℕ0𝜃𝜏) ↔ ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
6664, 65sylibr 225 1 (𝜑 → ∃𝑚 ∈ ℕ0𝜃𝜏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  [wsb 2062  wcel 2155  wral 3055  wrex 3056  [wsbc 3596  (class class class)co 6842  0cc0 10189  1c1 10190   + caddc 10192  cn 11274  0cn0 11538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-ltxr 10333  df-nn 11275  df-n0 11539
This theorem is referenced by:  archirng  30189
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