Theorem nn0min 32822

Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12713. (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 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝜑
4		nfra1 3284	. . . . . . . . . 10 ⊢ Ⅎ𝑚∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)
5	3, 4	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏))
6		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑚 ¬ [𝑘 / 𝑛]𝜓
7	5, 6	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8		dfsbcq2 3791	. . . . . . . . . 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 1914	. . . . . . . . . . 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 1914	. . . . . . . . . . 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 2252	. . . . . . . . . 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 12541	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	11, 12	sbiev 2314	. . . . . . . . . . . . . 14 ⊢ ([𝑚 / 𝑛]𝜓 ↔ 𝜃)
27		nfv 1914	. . . . . . . . . . . . . . 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 12388	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
34	32, 33	eqtrdi 2793	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
35		1nn 12277	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
36		eleq1 2829	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
37	35, 36	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
38	17	sbcieg 3828	. . . . . . . . . . . . . . 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 3793	. . . . . . . . . . . . . 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 3317	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48		nnnn0 12533	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49		sbequ12r 2252	. . . . . . . . . . . . 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 32821	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓))
56	55	rgen 3063	. . . . . 6 ⊢ ∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57		r19.21v 3180	. . . . . 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 3072	. . . . 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 3093	. . 3 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
64	61, 63	sylnib 328	. 2 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
65		dfrex2 3073	. 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 1540  [wsb 2064   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  [wsbc 3788  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  ℕcn 12266  ℕ₀cn0 12526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-nn 12267  df-n0 12527

This theorem is referenced by:  archirng  33195