Theorem uzind2 9690

Description: Induction on the upper integers that start after an integer M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)

Hypotheses

Ref	Expression
uzind2.1	|- ( j = ( M + 1 ) -> ( ph <-> ps ) )
uzind2.2	|- ( j = k -> ( ph <-> ch ) )
uzind2.3	|- ( j = ( k + 1 ) -> ( ph <-> th ) )
uzind2.4	|- ( j = N -> ( ph <-> ta ) )
uzind2.5	|- ( M e. ZZ -> ps )
uzind2.6	|- ( ( M e. ZZ /\ k e. ZZ /\ M < k ) -> ( ch -> th ) )

Assertion

Ref	Expression
uzind2	|- ( ( M e. ZZ /\ N e. ZZ /\ M < N ) -> ta )

Distinct variable groups:   j, N   ps, j   ch, j   th, j   ta, j   ph, k   j, k, M

Allowed substitution hints:   ph( j)   ps( k)   ch( k)   th( k)   ta( k)   N( k)

Proof of Theorem uzind2

Step	Hyp	Ref	Expression
1		zltp1le 9632	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
2		peano2z 9613	. . . . . . 7 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
3		uzind2.1	. . . . . . . . . 10 |- ( j = ( M + 1 ) -> ( ph <-> ps ) )
4	3	imbi2d 230	. . . . . . . . 9 |- ( j = ( M + 1 ) -> ( ( M e. ZZ -> ph ) <-> ( M e. ZZ -> ps ) ) )
5		uzind2.2	. . . . . . . . . 10 |- ( j = k -> ( ph <-> ch ) )
6	5	imbi2d 230	. . . . . . . . 9 |- ( j = k -> ( ( M e. ZZ -> ph ) <-> ( M e. ZZ -> ch ) ) )
7		uzind2.3	. . . . . . . . . 10 |- ( j = ( k + 1 ) -> ( ph <-> th ) )
8	7	imbi2d 230	. . . . . . . . 9 |- ( j = ( k + 1 ) -> ( ( M e. ZZ -> ph ) <-> ( M e. ZZ -> th ) ) )
9		uzind2.4	. . . . . . . . . 10 |- ( j = N -> ( ph <-> ta ) )
10	9	imbi2d 230	. . . . . . . . 9 |- ( j = N -> ( ( M e. ZZ -> ph ) <-> ( M e. ZZ -> ta ) ) )
11		uzind2.5	. . . . . . . . . 10 |- ( M e. ZZ -> ps )
12	11	a1i 9	. . . . . . . . 9 |- ( ( M + 1 ) e. ZZ -> ( M e. ZZ -> ps ) )
13		zltp1le 9632	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ k e. ZZ ) -> ( M < k <-> ( M + 1 ) <_ k ) )
14		uzind2.6	. . . . . . . . . . . . . . . 16 |- ( ( M e. ZZ /\ k e. ZZ /\ M < k ) -> ( ch -> th ) )
15	14	3expia 1232	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ k e. ZZ ) -> ( M < k -> ( ch -> th ) ) )
16	13, 15	sylbird 170	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ k e. ZZ ) -> ( ( M + 1 ) <_ k -> ( ch -> th ) ) )
17	16	ex 115	. . . . . . . . . . . . 13 |- ( M e. ZZ -> ( k e. ZZ -> ( ( M + 1 ) <_ k -> ( ch -> th ) ) ) )
18	17	com3l 81	. . . . . . . . . . . 12 |- ( k e. ZZ -> ( ( M + 1 ) <_ k -> ( M e. ZZ -> ( ch -> th ) ) ) )
19	18	imp 124	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ ( M + 1 ) <_ k ) -> ( M e. ZZ -> ( ch -> th ) ) )
20	19	3adant1 1042	. . . . . . . . . 10 |- ( ( ( M + 1 ) e. ZZ /\ k e. ZZ /\ ( M + 1 ) <_ k ) -> ( M e. ZZ -> ( ch -> th ) ) )
21	20	a2d 26	. . . . . . . . 9 |- ( ( ( M + 1 ) e. ZZ /\ k e. ZZ /\ ( M + 1 ) <_ k ) -> ( ( M e. ZZ -> ch ) -> ( M e. ZZ -> th ) ) )
22	4, 6, 8, 10, 12, 21	uzind 9689	. . . . . . . 8 |- ( ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) -> ( M e. ZZ -> ta ) )
23	22	3exp 1229	. . . . . . 7 |- ( ( M + 1 ) e. ZZ -> ( N e. ZZ -> ( ( M + 1 ) <_ N -> ( M e. ZZ -> ta ) ) ) )
24	2, 23	syl 14	. . . . . 6 |- ( M e. ZZ -> ( N e. ZZ -> ( ( M + 1 ) <_ N -> ( M e. ZZ -> ta ) ) ) )
25	24	com34 83	. . . . 5 |- ( M e. ZZ -> ( N e. ZZ -> ( M e. ZZ -> ( ( M + 1 ) <_ N -> ta ) ) ) )
26	25	pm2.43a 51	. . . 4 |- ( M e. ZZ -> ( N e. ZZ -> ( ( M + 1 ) <_ N -> ta ) ) )
27	26	imp 124	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M + 1 ) <_ N -> ta ) )
28	1, 27	sylbid 150	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N -> ta ) )
29	28	3impia 1227	1 |- ( ( M e. ZZ /\ N e. ZZ /\ M < N ) -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   1c1 8128   + caddc 8130   < clt 8308   <_ cle 8309   ZZcz 9577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578

This theorem is referenced by: (None)