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Theorem nn0min 32026
Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12657. (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 482 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∃𝑛 ∈ ℕ 𝜓)
3 nfv 1918 . . . . . . . . . 10 𝑚𝜑
4 nfra1 3282 . . . . . . . . . 10 𝑚𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)
53, 4nfan 1903 . . . . . . . . 9 𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
6 nfv 1918 . . . . . . . . 9 𝑚 ¬ [𝑘 / 𝑛]𝜓
75, 6nfim 1900 . . . . . . . 8 𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8 dfsbcq2 3781 . . . . . . . . . 10 (𝑘 = 1 → ([𝑘 / 𝑛]𝜓[1 / 𝑛]𝜓))
98notbid 318 . . . . . . . . 9 (𝑘 = 1 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ [1 / 𝑛]𝜓))
109imbi2d 341 . . . . . . . 8 (𝑘 = 1 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)))
11 nfv 1918 . . . . . . . . . . 11 𝑛𝜃
12 nn0min.1 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝜓𝜃))
1311, 12sbhypf 3539 . . . . . . . . . 10 (𝑘 = 𝑚 → ([𝑘 / 𝑛]𝜓𝜃))
1413notbid 318 . . . . . . . . 9 (𝑘 = 𝑚 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜃))
1514imbi2d 341 . . . . . . . 8 (𝑘 = 𝑚 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜃)))
16 nfv 1918 . . . . . . . . . . 11 𝑛𝜏
17 nn0min.2 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (𝜓𝜏))
1816, 17sbhypf 3539 . . . . . . . . . 10 (𝑘 = (𝑚 + 1) → ([𝑘 / 𝑛]𝜓𝜏))
1918notbid 318 . . . . . . . . 9 (𝑘 = (𝑚 + 1) → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜏))
2019imbi2d 341 . . . . . . . 8 (𝑘 = (𝑚 + 1) → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
21 sbequ12r 2245 . . . . . . . . . 10 (𝑘 = 𝑛 → ([𝑘 / 𝑛]𝜓𝜓))
2221notbid 318 . . . . . . . . 9 (𝑘 = 𝑛 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜓))
2322imbi2d 341 . . . . . . . 8 (𝑘 = 𝑛 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)))
24 nn0min.3 . . . . . . . . 9 (𝜑 → ¬ 𝜒)
25 0nn0 12487 . . . . . . . . . 10 0 ∈ ℕ0
2611, 12sbiev 2309 . . . . . . . . . . . . . 14 ([𝑚 / 𝑛]𝜓𝜃)
27 nfv 1918 . . . . . . . . . . . . . . 15 𝑛𝜒
28 nn0min.0 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝜓𝜒))
2927, 28sbhypf 3539 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([𝑚 / 𝑛]𝜓𝜒))
3026, 29bitr3id 285 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜃𝜒))
3130notbid 318 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜃 ↔ ¬ 𝜒))
32 oveq1 7416 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33 0p1e1 12334 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
3432, 33eqtrdi 2789 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝑚 + 1) = 1)
35 1nn 12223 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
36 eleq1 2822 . . . . . . . . . . . . . . . 16 ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
3735, 36mpbiri 258 . . . . . . . . . . . . . . 15 ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
3817sbcieg 3818 . . . . . . . . . . . . . . 15 ((𝑚 + 1) ∈ ℕ → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
3934, 37, 383syl 18 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
4034sbceq1d 3783 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓[1 / 𝑛]𝜓))
4139, 40bitr3d 281 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜏[1 / 𝑛]𝜓))
4241notbid 318 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜏 ↔ ¬ [1 / 𝑛]𝜓))
4331, 42imbi12d 345 . . . . . . . . . . 11 (𝑚 = 0 → ((¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4443rspcv 3609 . . . . . . . . . 10 (0 ∈ ℕ0 → (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4525, 44ax-mp 5 . . . . . . . . 9 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓))
4624, 45mpan9 508 . . . . . . . 8 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)
47 cbvralsvw 3315 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48 nnnn0 12479 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49 sbequ12r 2245 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
5049rspcv 3609 . . . . . . . . . . . 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, 51biimtrid 241 . . . . . . . . . 10 (𝑚 ∈ ℕ → (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
5352adantld 492 . . . . . . . . 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 32025 . . . . . . 7 (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓))
5655rgen 3064 . . . . . 6 𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57 r19.21v 3180 . . . . . 6 (∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓))
5856, 57mpbi 229 . . . . 5 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓)
59 ralnex 3073 . . . . 5 (∀𝑛 ∈ ℕ ¬ 𝜓 ↔ ¬ ∃𝑛 ∈ ℕ 𝜓)
6058, 59sylib 217 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ ∃𝑛 ∈ ℕ 𝜓)
612, 60pm2.65da 816 . . 3 (𝜑 → ¬ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
62 imnan 401 . . . 4 ((¬ 𝜃 → ¬ 𝜏) ↔ ¬ (¬ 𝜃𝜏))
6362ralbii 3094 . . 3 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
6461, 63sylnib 328 . 2 (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
65 dfrex2 3074 . 2 (∃𝑚 ∈ ℕ0𝜃𝜏) ↔ ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
6664, 65sylibr 233 1 (𝜑 → ∃𝑚 ∈ ℕ0𝜃𝜏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  [wsb 2068  wcel 2107  wral 3062  wrex 3071  [wsbc 3778  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  cn 12212  0cn0 12472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-ltxr 11253  df-nn 12213  df-n0 12473
This theorem is referenced by:  archirng  32334
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