Theorem uzind4i 9647

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 9643 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 9587). (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 9643	1 |- ( N e. ( ZZ>= ` M ) -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   ` cfv 5246  (class class class)co 5910   1c1 7863   + caddc 7865   ZZcz 9307   ZZ>=cuz 9582

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 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-ltadd 7978

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 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-inn 8973  df-n0 9231  df-z 9308  df-uz 9583

This theorem is referenced by:  rebtwn2zlemshrink  10312  seqfveq2g  10538  seqhomog  10591