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Theorem nn0min 32722
Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12701. (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 479 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ∃𝑛 ∈ ℕ 𝜓)
3 nfv 1910 . . . . . . . . . 10 𝑚𝜑
4 nfra1 3272 . . . . . . . . . 10 𝑚𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)
53, 4nfan 1895 . . . . . . . . 9 𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
6 nfv 1910 . . . . . . . . 9 𝑚 ¬ [𝑘 / 𝑛]𝜓
75, 6nfim 1892 . . . . . . . 8 𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8 dfsbcq2 3779 . . . . . . . . . 10 (𝑘 = 1 → ([𝑘 / 𝑛]𝜓[1 / 𝑛]𝜓))
98notbid 317 . . . . . . . . 9 (𝑘 = 1 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ [1 / 𝑛]𝜓))
109imbi2d 339 . . . . . . . 8 (𝑘 = 1 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)))
11 nfv 1910 . . . . . . . . . . 11 𝑛𝜃
12 nn0min.1 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝜓𝜃))
1311, 12sbhypf 3530 . . . . . . . . . 10 (𝑘 = 𝑚 → ([𝑘 / 𝑛]𝜓𝜃))
1413notbid 317 . . . . . . . . 9 (𝑘 = 𝑚 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜃))
1514imbi2d 339 . . . . . . . 8 (𝑘 = 𝑚 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜃)))
16 nfv 1910 . . . . . . . . . . 11 𝑛𝜏
17 nn0min.2 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (𝜓𝜏))
1816, 17sbhypf 3530 . . . . . . . . . 10 (𝑘 = (𝑚 + 1) → ([𝑘 / 𝑛]𝜓𝜏))
1918notbid 317 . . . . . . . . 9 (𝑘 = (𝑚 + 1) → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜏))
2019imbi2d 339 . . . . . . . 8 (𝑘 = (𝑚 + 1) → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
21 sbequ12r 2240 . . . . . . . . . 10 (𝑘 = 𝑛 → ([𝑘 / 𝑛]𝜓𝜓))
2221notbid 317 . . . . . . . . 9 (𝑘 = 𝑛 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜓))
2322imbi2d 339 . . . . . . . 8 (𝑘 = 𝑛 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)))
24 nn0min.3 . . . . . . . . 9 (𝜑 → ¬ 𝜒)
25 0nn0 12531 . . . . . . . . . 10 0 ∈ ℕ0
2611, 12sbiev 2304 . . . . . . . . . . . . . 14 ([𝑚 / 𝑛]𝜓𝜃)
27 nfv 1910 . . . . . . . . . . . . . . 15 𝑛𝜒
28 nn0min.0 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝜓𝜒))
2927, 28sbhypf 3530 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([𝑚 / 𝑛]𝜓𝜒))
3026, 29bitr3id 284 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜃𝜒))
3130notbid 317 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜃 ↔ ¬ 𝜒))
32 oveq1 7421 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33 0p1e1 12378 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
3432, 33eqtrdi 2782 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝑚 + 1) = 1)
35 1nn 12267 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
36 eleq1 2814 . . . . . . . . . . . . . . . 16 ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
3735, 36mpbiri 257 . . . . . . . . . . . . . . 15 ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
3817sbcieg 3817 . . . . . . . . . . . . . . 15 ((𝑚 + 1) ∈ ℕ → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
3934, 37, 383syl 18 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓𝜏))
4034sbceq1d 3781 . . . . . . . . . . . . . 14 (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓[1 / 𝑛]𝜓))
4139, 40bitr3d 280 . . . . . . . . . . . . 13 (𝑚 = 0 → (𝜏[1 / 𝑛]𝜓))
4241notbid 317 . . . . . . . . . . . 12 (𝑚 = 0 → (¬ 𝜏 ↔ ¬ [1 / 𝑛]𝜓))
4331, 42imbi12d 343 . . . . . . . . . . 11 (𝑚 = 0 → ((¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4443rspcv 3604 . . . . . . . . . 10 (0 ∈ ℕ0 → (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
4525, 44ax-mp 5 . . . . . . . . 9 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓))
4624, 45mpan9 505 . . . . . . . 8 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)
47 cbvralsvw 3305 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48 nnnn0 12523 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49 sbequ12r 2240 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
5049rspcv 3604 . . . . . . . . . . . 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 489 . . . . . . . . 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 32721 . . . . . . 7 (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓))
5655rgen 3053 . . . . . 6 𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57 r19.21v 3170 . . . . . 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 3062 . . . . 5 (∀𝑛 ∈ ℕ ¬ 𝜓 ↔ ¬ ∃𝑛 ∈ ℕ 𝜓)
6058, 59sylib 217 . . . 4 ((𝜑 ∧ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏)) → ¬ ∃𝑛 ∈ ℕ 𝜓)
612, 60pm2.65da 815 . . 3 (𝜑 → ¬ ∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏))
62 imnan 398 . . . 4 ((¬ 𝜃 → ¬ 𝜏) ↔ ¬ (¬ 𝜃𝜏))
6362ralbii 3083 . . 3 (∀𝑚 ∈ ℕ0𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
6461, 63sylnib 327 . 2 (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃𝜏))
65 dfrex2 3063 . 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 394   = wceq 1534  [wsb 2060  wcel 2099  wral 3051  wrex 3060  [wsbc 3776  (class class class)co 7414  0cc0 11147  1c1 11148   + caddc 11150  cn 12256  0cn0 12516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  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 7417  df-om 7867  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11289  df-mnf 11290  df-ltxr 11292  df-nn 12257  df-n0 12517
This theorem is referenced by:  archirng  33055
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