Theorem nn0min 29567

Description: Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 11472. (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 481	. . . 4 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ∃𝑛 ∈ ℕ 𝜓)
3		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝜑
4		nfra1 2941	. . . . . . . . . 10 ⊢ Ⅎ𝑚∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)
5	3, 4	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏))
6		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑚 ¬ [𝑘 / 𝑛]𝜓
7	5, 6	nfim 1825	. . . . . . . 8 ⊢ Ⅎ𝑚((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓)
8		dfsbcq2 3438	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ([𝑘 / 𝑛]𝜓 ↔ [1 / 𝑛]𝜓))
9	8	notbid 308	. . . . . . . . 9 ⊢ (𝑘 = 1 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ [1 / 𝑛]𝜓))
10	9	imbi2d 330	. . . . . . . 8 ⊢ (𝑘 = 1 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)))
11		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜃
12		nn0min.1	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃))
13	11, 12	sbhypf 3253	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ([𝑘 / 𝑛]𝜓 ↔ 𝜃))
14	13	notbid 308	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜃))
15	14	imbi2d 330	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜃)))
16		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜏
17		nn0min.2	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏))
18	16, 17	sbhypf 3253	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + 1) → ([𝑘 / 𝑛]𝜓 ↔ 𝜏))
19	18	notbid 308	. . . . . . . . 9 ⊢ (𝑘 = (𝑚 + 1) → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜏))
20	19	imbi2d 330	. . . . . . . 8 ⊢ (𝑘 = (𝑚 + 1) → (((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [𝑘 / 𝑛]𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜏)))
21		sbequ12r 2112	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → ([𝑘 / 𝑛]𝜓 ↔ 𝜓))
22	21	notbid 308	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (¬ [𝑘 / 𝑛]𝜓 ↔ ¬ 𝜓))
23	22	imbi2d 330	. . . . . . . 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 11307	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	11, 12	sbie 2408	. . . . . . . . . . . . . 14 ⊢ ([𝑚 / 𝑛]𝜓 ↔ 𝜃)
27		nfv 1843	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝜒
28		nn0min.0	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒))
29	27, 28	sbhypf 3253	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → ([𝑚 / 𝑛]𝜓 ↔ 𝜒))
30	26, 29	syl5bbr 274	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → (𝜃 ↔ 𝜒))
31	30	notbid 308	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (¬ 𝜃 ↔ ¬ 𝜒))
32		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
33		0p1e1 11132	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
34	32, 33	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
35		1nn 11031	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
36		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 + 1) = 1 → ((𝑚 + 1) ∈ ℕ ↔ 1 ∈ ℕ))
37	35, 36	mpbiri 248	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 + 1) = 1 → (𝑚 + 1) ∈ ℕ)
38	17	sbcieg 3468	. . . . . . . . . . . . . . 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 3440	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → ([(𝑚 + 1) / 𝑛]𝜓 ↔ [1 / 𝑛]𝜓))
41	39, 40	bitr3d 270	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → (𝜏 ↔ [1 / 𝑛]𝜓))
42	41	notbid 308	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (¬ 𝜏 ↔ ¬ [1 / 𝑛]𝜓))
43	31, 42	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → ((¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜒 → ¬ [1 / 𝑛]𝜓)))
44	43	rspcv 3305	. . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ [1 / 𝑛]𝜓)
47		cbvralsv 3182	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑘 ∈ ℕ0 [𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏))
48		nnnn0 11299	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
49		sbequ12r 2112	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ([𝑘 / 𝑚](¬ 𝜃 → ¬ 𝜏) ↔ (¬ 𝜃 → ¬ 𝜏)))
50	49	rspcv 3305	. . . . . . . . . . . 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	syl5bi 232	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) → (¬ 𝜃 → ¬ 𝜏)))
53	52	adantld 483	. . . . . . . . 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 29565	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓))
56	55	rgen 2922	. . . . . 6 ⊢ ∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓)
57		r19.21v 2960	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ 𝜓) ↔ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓))
58	56, 57	mpbi 220	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ∀𝑛 ∈ ℕ ¬ 𝜓)
59		ralnex 2992	. . . . 5 ⊢ (∀𝑛 ∈ ℕ ¬ 𝜓 ↔ ¬ ∃𝑛 ∈ ℕ 𝜓)
60	58, 59	sylib 208	. . . 4 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏)) → ¬ ∃𝑛 ∈ ℕ 𝜓)
61	2, 60	pm2.65da 600	. . 3 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏))
62		imnan 438	. . . 4 ⊢ ((¬ 𝜃 → ¬ 𝜏) ↔ ¬ (¬ 𝜃 ∧ 𝜏))
63	62	ralbii 2980	. . 3 ⊢ (∀𝑚 ∈ ℕ0 (¬ 𝜃 → ¬ 𝜏) ↔ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
64	61, 63	sylnib 318	. 2 ⊢ (𝜑 → ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
65		dfrex2 2996	. 2 ⊢ (∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏) ↔ ¬ ∀𝑚 ∈ ℕ0 ¬ (¬ 𝜃 ∧ 𝜏))
66	64, 65	sylibr 224	1 ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  [wsb 1880   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  [wsbc 3435  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  ℕ₀cn0 11292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-nn 11021  df-n0 11293

This theorem is referenced by:  archirng  29742