Theorem nn0min 32829

Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12578. (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 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝜑
4		nfra1 3257	. . . . . . . . . 10 ⊢ Ⅎ𝑚∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)
5	3, 4	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏))
6		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑚 ¬ [𝑘 / 𝑛]𝜓
7	5, 6	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8		dfsbcq2 3740	. . . . . . . . . 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 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜃
12		nn0min.1	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃))
13	11, 12	sbhypf 3499	. . . . . . . . . 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 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜏
17		nn0min.2	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏))
18	16, 17	sbhypf 3499	. . . . . . . . . 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 2257	. . . . . . . . . 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 12407	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	11, 12	sbiev 2317	. . . . . . . . . . . . . 14 ⊢ ([𝑚 / 𝑛]𝜓 ↔ 𝜃)
27		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝜒
28		nn0min.0	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒))
29	27, 28	sbhypf 3499	. . . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33		0p1e1 12253	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
34	32, 33	eqtrdi 2784	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
35		1nn 12147	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
36		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
37	35, 36	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
38	17	sbcieg 3777	. . . . . . . . . . . . . . 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 3742	. . . . . . . . . . . . . 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 3569	. . . . . . . . . 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 3284	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48		nnnn0 12399	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49		sbequ12r 2257	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
50	49	rspcv 3569	. . . . . . . . . . . 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 32828	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓))
56	55	rgen 3050	. . . . . 6 ⊢ ∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57		r19.21v 3158	. . . . . 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 3059	. . . . 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 816	. . 3 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏))
62		imnan 399	. . . 4 ⊢ ((¬ 𝜃 → ¬ 𝜏) ↔ ¬ (¬ 𝜃 ∧ 𝜏))
63	62	ralbii 3079	. . 3 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
64	61, 63	sylnib 328	. 2 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
65		dfrex2 3060	. 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 1541  [wsb 2067   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  [wsbc 3737  (class class class)co 7355  0cc0 11017  1c1 11018   + caddc 11020  ℕcn 12136  ℕ₀cn0 12392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-ltxr 11162  df-nn 12137  df-n0 12393

This theorem is referenced by:  archirng  33198