Theorem uzind4ALT 9548

Description: Induction on the upper set of integers that starts at an integer M. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 9547 or uzind4ALT 9548 may be used; see comment for nnind 8894. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
uzind4ALT.5	|- ( M e. ZZ -> ps )
uzind4ALT.6	|- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )
uzind4ALT.1	|- ( j = M -> ( ph <-> ps ) )
uzind4ALT.2	|- ( j = k -> ( ph <-> ch ) )
uzind4ALT.3	|- ( j = ( k + 1 ) -> ( ph <-> th ) )
uzind4ALT.4	|- ( j = N -> ( ph <-> ta ) )

Assertion

Ref	Expression
uzind4ALT	|- ( 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 uzind4ALT

Step	Hyp	Ref	Expression
1		uzind4ALT.1	. 2 |- ( j = M -> ( ph <-> ps ) )
2		uzind4ALT.2	. 2 |- ( j = k -> ( ph <-> ch ) )
3		uzind4ALT.3	. 2 |- ( j = ( k + 1 ) -> ( ph <-> th ) )
4		uzind4ALT.4	. 2 |- ( j = N -> ( ph <-> ta ) )
5		uzind4ALT.5	. 2 |- ( M e. ZZ -> ps )
6		uzind4ALT.6	. 2 |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )
7	1, 2, 3, 4, 5, 6	uzind4 9547	1 |- ( N e. ( ZZ>= ` M ) -> ta )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   ` cfv 5198  (class class class)co 5853   1c1 7775   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487

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-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  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  df-uz 9488

This theorem is referenced by: (None)