Theorem inffz 16676

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 531	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
2	1	zred 9601	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. RR )
3		simprr 533	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
4	3	zred 9601	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. RR )
5	2, 4	lttri3d 8293	. 2 |- ( ( N e. ( ZZ>= ` M ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x = y <-> ( -. x < y /\ -. y < x ) ) )
6		eluzel2 9759	. 2 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7		eluzfz1 10265	. 2 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
8		elfzle1 10261	. . . 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 9601	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. RR )
11		elfzelz 10259	. . . . 5 |- ( z e. ( M ... N ) -> z e. ZZ )
12	11	zred 9601	. . . 4 |- ( z e. ( M ... N ) -> z e. RR )
13		lenlt 8254	. . . 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 7225	1 |- ( N e. ( ZZ>= ` M ) -> inf ( ( M ... N ) , ZZ , < ) = M )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017  infcinf 7181   RRcr 8030   < clt 8213   <_ cle 8214   ZZcz 9478   ZZ>=cuz 9754   ...cfz 10242

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143  ax-pre-apti 8146

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-neg 8352  df-z 9479  df-uz 9755  df-fz 10243

This theorem is referenced by: (None)