Theorem nn0min 29567

Description: Extracting the minimum positive integer for which a property ch does not hold. This uses substitutions similar to nn0ind 11472. (Contributed by Thierry Arnoux, 6-May-2018.)

Hypotheses

Ref	Expression
nn0min.0	|- ( n = 0 -> ( ps <-> ch ) )
nn0min.1	|- ( n = m -> ( ps <-> th ) )
nn0min.2	|- ( n = ( m + 1 ) -> ( ps <-> ta ) )
nn0min.3	|- ( ph -> -. ch )
nn0min.4	|- ( ph -> E. n e. NN ps )

Assertion

Ref	Expression
nn0min	|- ( ph -> E. m e. NN0 ( -. th /\ ta ) )

Distinct variable groups:   m, n, ph   ps, m   ta, n   th, n   ch, m, n

Allowed substitution hints:   ps( n)   th( m)   ta( m)

Proof of Theorem nn0min

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0min.4	. . . . 5 |- ( ph -> E. n e. NN ps )
2	1	adantr 481	. . . 4 |- ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> E. n e. NN ps )
3		nfv 1843	. . . . . . . . . 10 |- F/ m ph
4		nfra1 2941	. . . . . . . . . 10 |- F/ m A. m e. NN0 ( -. th -> -. ta )
5	3, 4	nfan 1828	. . . . . . . . 9 |- F/ m ( ph /\ A. m e. NN0 ( -. th -> -. ta ) )
6		nfv 1843	. . . . . . . . 9 |- F/ m -. [ k / n ] ps
7	5, 6	nfim 1825	. . . . . . . 8 |- F/ m ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. [ k / n ] ps )
8		dfsbcq2 3438	. . . . . . . . . 10 |- ( k = 1 -> ( [ k / n ] ps <-> [. 1 / n ]. ps ) )
9	8	notbid 308	. . . . . . . . 9 |- ( k = 1 -> ( -. [ k / n ] ps <-> -. [. 1 / n ]. ps ) )
10	9	imbi2d 330	. . . . . . . 8 |- ( k = 1 -> ( ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. [ k / n ] ps ) <-> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. [. 1 / n ]. ps ) ) )
11		nfv 1843	. . . . . . . . . . 11 |- F/ n th
12		nn0min.1	. . . . . . . . . . 11 |- ( n = m -> ( ps <-> th ) )
13	11, 12	sbhypf 3253	. . . . . . . . . 10 |- ( k = m -> ( [ k / n ] ps <-> th ) )
14	13	notbid 308	. . . . . . . . 9 |- ( k = m -> ( -. [ k / n ] ps <-> -. th ) )
15	14	imbi2d 330	. . . . . . . 8 |- ( k = m -> ( ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. [ k / n ] ps ) <-> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. th ) ) )
16		nfv 1843	. . . . . . . . . . 11 |- F/ n ta
17		nn0min.2	. . . . . . . . . . 11 |- ( n = ( m + 1 ) -> ( ps <-> ta ) )
18	16, 17	sbhypf 3253	. . . . . . . . . 10 |- ( k = ( m + 1 ) -> ( [ k / n ] ps <-> ta ) )
19	18	notbid 308	. . . . . . . . 9 |- ( k = ( m + 1 ) -> ( -. [ k / n ] ps <-> -. ta ) )
20	19	imbi2d 330	. . . . . . . 8 |- ( k = ( m + 1 ) -> ( ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. [ k / n ] ps ) <-> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. ta ) ) )
21		sbequ12r 2112	. . . . . . . . . 10 |- ( k = n -> ( [ k / n ] ps <-> ps ) )
22	21	notbid 308	. . . . . . . . 9 |- ( k = n -> ( -. [ k / n ] ps <-> -. ps ) )
23	22	imbi2d 330	. . . . . . . 8 |- ( k = n -> ( ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. [ k / n ] ps ) <-> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. ps ) ) )
24		nn0min.3	. . . . . . . . 9 |- ( ph -> -. ch )
25		0nn0 11307	. . . . . . . . . 10 |- 0 e. NN0
26	11, 12	sbie 2408	. . . . . . . . . . . . . 14 |- ( [ m / n ] ps <-> th )
27		nfv 1843	. . . . . . . . . . . . . . 15 |- F/ n ch
28		nn0min.0	. . . . . . . . . . . . . . 15 |- ( n = 0 -> ( ps <-> ch ) )
29	27, 28	sbhypf 3253	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( [ m / n ] ps <-> ch ) )
30	26, 29	syl5bbr 274	. . . . . . . . . . . . 13 |- ( m = 0 -> ( th <-> ch ) )
31	30	notbid 308	. . . . . . . . . . . 12 |- ( m = 0 -> ( -. th <-> -. ch ) )
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 e. NN
36		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( ( m + 1 ) = 1 -> ( ( m + 1 ) e. NN <-> 1 e. NN ) )
37	35, 36	mpbiri 248	. . . . . . . . . . . . . . 15 |- ( ( m + 1 ) = 1 -> ( m + 1 ) e. NN )
38	17	sbcieg 3468	. . . . . . . . . . . . . . 15 |- ( ( m + 1 ) e. NN -> ( [. ( m + 1 ) / n ]. ps <-> ta ) )
39	34, 37, 38	3syl 18	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( [. ( m + 1 ) / n ]. ps <-> ta ) )
40	34	sbceq1d 3440	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( [. ( m + 1 ) / n ]. ps <-> [. 1 / n ]. ps ) )
41	39, 40	bitr3d 270	. . . . . . . . . . . . 13 |- ( m = 0 -> ( ta <-> [. 1 / n ]. ps ) )
42	41	notbid 308	. . . . . . . . . . . 12 |- ( m = 0 -> ( -. ta <-> -. [. 1 / n ]. ps ) )
43	31, 42	imbi12d 334	. . . . . . . . . . 11 |- ( m = 0 -> ( ( -. th -> -. ta ) <-> ( -. ch -> -. [. 1 / n ]. ps ) ) )
44	43	rspcv 3305	. . . . . . . . . 10 |- ( 0 e. NN0 -> ( A. m e. NN0 ( -. th -> -. ta ) -> ( -. ch -> -. [. 1 / n ]. ps ) ) )
45	25, 44	ax-mp 5	. . . . . . . . 9 |- ( A. m e. NN0 ( -. th -> -. ta ) -> ( -. ch -> -. [. 1 / n ]. ps ) )
46	24, 45	mpan9 486	. . . . . . . 8 |- ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. [. 1 / n ]. ps )
47		cbvralsv 3182	. . . . . . . . . . 11 |- ( A. m e. NN0 ( -. th -> -. ta ) <-> A. k e. NN0 [ k / m ] ( -. th -> -. ta ) )
48		nnnn0 11299	. . . . . . . . . . . 12 |- ( m e. NN -> m e. NN0 )
49		sbequ12r 2112	. . . . . . . . . . . . 13 |- ( k = m -> ( [ k / m ] ( -. th -> -. ta ) <-> ( -. th -> -. ta ) ) )
50	49	rspcv 3305	. . . . . . . . . . . 12 |- ( m e. NN0 -> ( A. k e. NN0 [ k / m ] ( -. th -> -. ta ) -> ( -. th -> -. ta ) ) )
51	48, 50	syl 17	. . . . . . . . . . 11 |- ( m e. NN -> ( A. k e. NN0 [ k / m ] ( -. th -> -. ta ) -> ( -. th -> -. ta ) ) )
52	47, 51	syl5bi 232	. . . . . . . . . 10 |- ( m e. NN -> ( A. m e. NN0 ( -. th -> -. ta ) -> ( -. th -> -. ta ) ) )
53	52	adantld 483	. . . . . . . . 9 |- ( m e. NN -> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> ( -. th -> -. ta ) ) )
54	53	a2d 29	. . . . . . . 8 |- ( m e. NN -> ( ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. th ) -> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. ta ) ) )
55	7, 10, 15, 20, 23, 46, 54	nnindf 29565	. . . . . . 7 |- ( n e. NN -> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. ps ) )
56	55	rgen 2922	. . . . . 6 |- A. n e. NN ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. ps )
57		r19.21v 2960	. . . . . 6 |- ( A. n e. NN ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. ps ) <-> ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> A. n e. NN -. ps ) )
58	56, 57	mpbi 220	. . . . 5 |- ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> A. n e. NN -. ps )
59		ralnex 2992	. . . . 5 |- ( A. n e. NN -. ps <-> -. E. n e. NN ps )
60	58, 59	sylib 208	. . . 4 |- ( ( ph /\ A. m e. NN0 ( -. th -> -. ta ) ) -> -. E. n e. NN ps )
61	2, 60	pm2.65da 600	. . 3 |- ( ph -> -. A. m e. NN0 ( -. th -> -. ta ) )
62		imnan 438	. . . 4 |- ( ( -. th -> -. ta ) <-> -. ( -. th /\ ta ) )
63	62	ralbii 2980	. . 3 |- ( A. m e. NN0 ( -. th -> -. ta ) <-> A. m e. NN0 -. ( -. th /\ ta ) )
64	61, 63	sylnib 318	. 2 |- ( ph -> -. A. m e. NN0 -. ( -. th /\ ta ) )
65		dfrex2 2996	. 2 |- ( E. m e. NN0 ( -. th /\ ta ) <-> -. A. m e. NN0 -. ( -. th /\ ta ) )
66	64, 65	sylibr 224	1 |- ( ph -> E. m e. NN0 ( -. th /\ ta ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   [wsb 1880   e. wcel 1990   A.wral 2912   E.wrex 2913   [.wsbc 3435  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 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