Theorem nn0min 32901

Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12587. (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 3260	. . . . . . . . . 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 3743	. . . . . . . . . 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 3502	. . . . . . . . . 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 3502	. . . . . . . . . 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 2259	. . . . . . . . . 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 12416	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	11, 12	sbiev 2319	. . . . . . . . . . . . . 14 ⊢ ([𝑚 / 𝑛]𝜓 ↔ 𝜃)
27		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝜒
28		nn0min.0	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒))
29	27, 28	sbhypf 3502	. . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33		0p1e1 12262	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
34	32, 33	eqtrdi 2787	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
35		1nn 12156	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
36		eleq1 2824	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
37	35, 36	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
38	17	sbcieg 3780	. . . . . . . . . . . . . . 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 3745	. . . . . . . . . . . . . 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 3572	. . . . . . . . . 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 3287	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48		nnnn0 12408	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49		sbequ12r 2259	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
50	49	rspcv 3572	. . . . . . . . . . . 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 32900	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓))
56	55	rgen 3053	. . . . . 6 ⊢ ∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57		r19.21v 3161	. . . . . 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 3062	. . . . 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 3082	. . 3 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
64	61, 63	sylnib 328	. 2 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
65		dfrex2 3063	. 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 3051  ∃wrex 3060  [wsbc 3740  (class class class)co 7358  0cc0 11026  1c1 11027   + caddc 11029  ℕcn 12145  ℕ₀cn0 12401

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-nn 12146  df-n0 12402

This theorem is referenced by:  archirng  33270