Theorem uzind4i 9657

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 9653 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 9597). (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 9653	1 |- ( N e. ( ZZ>= ` M ) -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   1c1 7873   + caddc 7875   ZZcz 9317   ZZ>=cuz 9592

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593

This theorem is referenced by:  rebtwn2zlemshrink  10322  seqfveq2g  10548  seqhomog  10601