Theorem uzind2 9367

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 9309	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
2		peano2z 9291	. . . . . . 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 9309	. . . . . . . . . . . . . . 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 1205	. . . . . . . . . . . . . . 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 1015	. . . . . . . . . 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 9366	. . . . . . . 8 |- ( ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) -> ( M e. ZZ -> ta ) )
23	22	3exp 1202	. . . . . . 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 1200	1 |- ( ( M e. ZZ /\ N e. ZZ /\ M < N ) -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   1c1 7814   + caddc 7816   < clt 7994   <_ cle 7995   ZZcz 9255

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256

This theorem is referenced by: (None)