Theorem uzind2 9324

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 9266	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
2		peano2z 9248	. . . . . . 7 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
3		uzind2.1	. . . . . . . . . 10 |- ( j = ( M + 1 ) -> ( ph <-> ps ) )
4	3	imbi2d 229	. . . . . . . . 9 |- ( j = ( M + 1 ) -> ( ( M e. ZZ -> ph ) <-> ( M e. ZZ -> ps ) ) )
5		uzind2.2	. . . . . . . . . 10 |- ( j = k -> ( ph <-> ch ) )
6	5	imbi2d 229	. . . . . . . . 9 |- ( j = k -> ( ( M e. ZZ -> ph ) <-> ( M e. ZZ -> ch ) ) )
7		uzind2.3	. . . . . . . . . 10 |- ( j = ( k + 1 ) -> ( ph <-> th ) )
8	7	imbi2d 229	. . . . . . . . 9 |- ( j = ( k + 1 ) -> ( ( M e. ZZ -> ph ) <-> ( M e. ZZ -> th ) ) )
9		uzind2.4	. . . . . . . . . 10 |- ( j = N -> ( ph <-> ta ) )
10	9	imbi2d 229	. . . . . . . . 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 9266	. . . . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ k e. ZZ ) -> ( M < k -> ( ch -> th ) ) )
16	13, 15	sylbird 169	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ k e. ZZ ) -> ( ( M + 1 ) <_ k -> ( ch -> th ) ) )
17	16	ex 114	. . . . . . . . . . . . 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 123	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ ( M + 1 ) <_ k ) -> ( M e. ZZ -> ( ch -> th ) ) )
20	19	3adant1 1010	. . . . . . . . . 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 9323	. . . . . . . 8 |- ( ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) -> ( M e. ZZ -> ta ) )
23	22	3exp 1197	. . . . . . 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 123	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M + 1 ) <_ N -> ta ) )
28	1, 27	sylbid 149	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N -> ta ) )
29	28	3impia 1195	1 |- ( ( M e. ZZ /\ N e. ZZ /\ M < N ) -> ta )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   1c1 7775   + caddc 7777   < clt 7954   <_ cle 7955   ZZcz 9212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213

This theorem is referenced by: (None)