Theorem inffz 17122

Description: The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)

Assertion

Ref	Expression
inffz	|- ( N e. ( ZZ>= ` M ) -> inf ( ( M ... N ) , ZZ , < ) = M )

Proof of Theorem inffz

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 535	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
2	1	zred 9768	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. RR )
3		simprr 537	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
4	3	zred 9768	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. RR )
5	2, 4	lttri3d 8440	. 2 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x = y <-> ( -. x < y /\ -. y < x ) ) )
6		eluzel2 9926	. 2 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7		eluzfz1 10435	. 2 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
8		elfzle1 10431	. . . 4 |- ( z e. ( M ... N ) -> M <_ z )
9	8	adantl 277	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ z e. ( M ... N ) ) -> M <_ z )
10	6	zred 9768	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. RR )
11		elfzelz 10428	. . . . 5 |- ( z e. ( M ... N ) -> z e. ZZ )
12	11	zred 9768	. . . 4 |- ( z e. ( M ... N ) -> z e. RR )
13		lenlt 8401	. . . 4 |- ( ( M e. RR /\ z e. RR ) -> ( M <_ z <-> -. z < M ) )
14	10, 12, 13	syl2an 289	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ z e. ( M ... N ) ) -> ( M <_ z <-> -. z < M ) )
15	9, 14	mpbid 147	. 2 |- ( ( N e. ( ZZ>= ` M ) /\ z e. ( M ... N ) ) -> -. z < M )
16	5, 6, 7, 15	infminti 7367	1 |- ( N e. ( ZZ>= ` M ) -> inf ( ( M ... N ) , ZZ , < ) = M )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085  infcinf 7323   RRcr 8178   < clt 8360   <_ cle 8361   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-apti 8294

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-neg 8500  df-z 9645  df-uz 9922  df-fz 10412

This theorem is used by: (None)