Theorem uzind4i 9668

Description: Induction on the upper integers that start at M. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 9664 assuming that ps holds unconditionally. Notice that N e. ( ZZ>= ` M ) implies that the lower bound M is an integer ( M e. ZZ, see eluzel2 9608). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.)

Hypotheses

Ref	Expression
uzind4i.1	|- ( j = M -> ( ph <-> ps ) )
uzind4i.2	|- ( j = k -> ( ph <-> ch ) )
uzind4i.3	|- ( j = ( k + 1 ) -> ( ph <-> th ) )
uzind4i.4	|- ( j = N -> ( ph <-> ta ) )
uzind4i.5	|- ps
uzind4i.6	|- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )

Assertion

Ref	Expression
uzind4i	|- ( N e. ( ZZ>= ` M ) -> 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 uzind4i

Step	Hyp	Ref	Expression
1		uzind4i.1	. 2 |- ( j = M -> ( ph <-> ps ) )
2		uzind4i.2	. 2 |- ( j = k -> ( ph <-> ch ) )
3		uzind4i.3	. 2 |- ( j = ( k + 1 ) -> ( ph <-> th ) )
4		uzind4i.4	. 2 |- ( j = N -> ( ph <-> ta ) )
5		uzind4i.5	. . 3 |- ps
6	5	a1i 9	. 2 |- ( M e. ZZ -> ps )
7		uzind4i.6	. 2 |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )
8	1, 2, 3, 4, 6, 7	uzind4 9664	1 |- ( N e. ( ZZ>= ` M ) -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5923   1c1 7882   + caddc 7884   ZZcz 9328   ZZ>=cuz 9603

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-0id 7989  ax-rnegex 7990  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-ltadd 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-inn 8993  df-n0 9252  df-z 9329  df-uz 9604

This theorem is referenced by:  rebtwn2zlemshrink  10345  seqfveq2g  10571  seqhomog  10624  2expltfac  12618