Theorem nn0min 32827

Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12711. (Contributed by Thierry Arnoux, 6-May-2018.)

Hypotheses

Ref	Expression
nn0min.0	⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒))
nn0min.1	⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃))
nn0min.2	⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏))
nn0min.3	⊢ (𝜑 → ¬ 𝜒)
nn0min.4	⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝜓)

Assertion

Ref	Expression
nn0min	⊢ (𝜑 → ∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏))

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.

Step	Hyp	Ref	Expression
1		nn0min.4	. . . . 5 ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝜓)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ∃𝑛 ∈ ℕ 𝜓)
3		nfv 1912	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝜑
4		nfra1 3282	. . . . . . . . . 10 ⊢ Ⅎ𝑚∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)
5	3, 4	nfan 1897	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏))
6		nfv 1912	. . . . . . . . 9 ⊢ Ⅎ𝑚 ¬ [𝑘 / 𝑛]𝜓
7	5, 6	nfim 1894	. . . . . . . 8 ⊢ Ⅎ𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8		dfsbcq2 3794	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ([𝑘 / 𝑛]𝜓 ↔ [1 / 𝑛]𝜓))
9	8	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = 1 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ [1 / 𝑛]𝜓))
10	9	imbi2d 340	. . . . . . . 8 ⊢ (𝑘 = 1 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)))
11		nfv 1912	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜃
12		nn0min.1	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃))
13	11, 12	sbhypf 3544	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ([𝑘 / 𝑛]𝜓 ↔ 𝜃))
14	13	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜃))
15	14	imbi2d 340	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜃)))
16		nfv 1912	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜏
17		nn0min.2	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏))
18	16, 17	sbhypf 3544	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + 1) → ([𝑘 / 𝑛]𝜓 ↔ 𝜏))
19	18	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = (𝑚 + 1) → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜏))
20	19	imbi2d 340	. . . . . . . 8 ⊢ (𝑘 = (𝑚 + 1) → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
21		sbequ12r 2250	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → ([𝑘 / 𝑛]𝜓 ↔ 𝜓))
22	21	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜓))
23	22	imbi2d 340	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓)))
24		nn0min.3	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝜒)
25		0nn0 12539	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	11, 12	sbiev 2313	. . . . . . . . . . . . . 14 ⊢ ([𝑚 / 𝑛]𝜓 ↔ 𝜃)
27		nfv 1912	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝜒
28		nn0min.0	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒))
29	27, 28	sbhypf 3544	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → ([𝑚 / 𝑛]𝜓 ↔ 𝜒))
30	26, 29	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → (𝜃 ↔ 𝜒))
31	30	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (¬ 𝜃 ↔ ¬ 𝜒))
32		oveq1 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33		0p1e1 12386	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
34	32, 33	eqtrdi 2791	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
35		1nn 12275	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
36		eleq1 2827	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
37	35, 36	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
38	17	sbcieg 3832	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 + 1) ∈ ℕ → ([(𝑚 + 1) / 𝑛]𝜓 ↔ 𝜏))
39	34, 37, 38	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓 ↔ 𝜏))
40	34	sbceq1d 3796	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓 ↔ [1 / 𝑛]𝜓))
41	39, 40	bitr3d 281	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → (𝜏 ↔ [1 / 𝑛]𝜓))
42	41	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (¬ 𝜏 ↔ ¬ [1 / 𝑛]𝜓))
43	31, 42	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → ((¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
44	43	rspcv 3618	. . . . . . . . . 10 ⊢ (0 ∈ ℕ0 → (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
45	25, 44	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) → (¬ 𝜒 → ¬ [1 / 𝑛]𝜓))
46	24, 45	mpan9 506	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)
47		cbvralsvw 3315	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48		nnnn0 12531	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49		sbequ12r 2250	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
50	49	rspcv 3618	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
51	48, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
52	47, 51	biimtrid 242	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
53	52	adantld 490	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → (¬ 𝜃 → ¬ 𝜏)))
54	53	a2d 29	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜃) → ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
55	7, 10, 15, 20, 23, 46, 54	nnindf 32826	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓))
56	55	rgen 3061	. . . . . 6 ⊢ ∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57		r19.21v 3178	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓))
58	56, 57	mpbi 230	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓)
59		ralnex 3070	. . . . 5 ⊢ (∀𝑛 ∈ ℕ ¬ 𝜓 ↔ ¬ ∃𝑛 ∈ ℕ 𝜓)
60	58, 59	sylib 218	. . . 4 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ ∃𝑛 ∈ ℕ 𝜓)
61	2, 60	pm2.65da 817	. . 3 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏))
62		imnan 399	. . . 4 ⊢ ((¬ 𝜃 → ¬ 𝜏) ↔ ¬ (¬ 𝜃 ∧ 𝜏))
63	62	ralbii 3091	. . 3 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
64	61, 63	sylnib 328	. 2 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
65		dfrex2 3071	. 2 ⊢ (∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏) ↔ ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
66	64, 65	sylibr 234	1 ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  [wsb 2062   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  [wsbc 3791  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156  ℕcn 12264  ℕ₀cn0 12524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-ltxr 11298  df-nn 12265  df-n0 12525

This theorem is referenced by:  archirng  33178