Theorem uzinfmiOLD 11239

Description: Extract the lower bound of an upper set of integers as its infimum. Note that the " `' < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.) Obsolete version of uzinfi 11238 as of 4-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
uzinfi.1	|- M e. ZZ

Assertion

Ref	Expression
uzinfmiOLD	|- sup ( ( ZZ>= ` M ) , RR , `' < ) = M

Proof of Theorem uzinfmiOLD

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzssz 11178	. . . 4 |- ( ZZ>= ` M ) C_ ZZ
2		zssre 10944	. . . 4 |- ZZ C_ RR
3	1, 2	sstri 3441	. . 3 |- ( ZZ>= ` M ) C_ RR
4		uzinfi.1	. . . . 5 |- M e. ZZ
5		uzid 11173	. . . . 5 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
6	4, 5	ax-mp 5	. . . 4 |- M e. ( ZZ>= ` M )
7		eluzle 11171	. . . . 5 |- ( k e. ( ZZ>= ` M ) -> M <_ k )
8	7	rgen 2747	. . . 4 |- A. k e. ( ZZ>= ` M ) M <_ k
9		breq1 4405	. . . . . 6 |- ( j = M -> ( j <_ k <-> M <_ k ) )
10	9	ralbidv 2827	. . . . 5 |- ( j = M -> ( A. k e. ( ZZ>= ` M ) j <_ k <-> A. k e. ( ZZ>= ` M ) M <_ k ) )
11	10	rspcev 3150	. . . 4 |- ( ( M e. ( ZZ>= ` M ) /\ A. k e. ( ZZ>= ` M ) M <_ k ) -> E. j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k )
12	6, 8, 11	mp2an 678	. . 3 |- E. j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k
13		lbinfmOLD 10560	. . 3 |- ( ( ( ZZ>= ` M ) C_ RR /\ E. j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k ) -> sup ( ( ZZ>= ` M ) , RR , `' < ) = ( iota_ j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k ) )
14	3, 12, 13	mp2an 678	. 2 |- sup ( ( ZZ>= ` M ) , RR , `' < ) = ( iota_ j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k )
15		lbreu 10556	. . . . 5 |- ( ( ( ZZ>= ` M ) C_ RR /\ E. j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k ) -> E! j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k )
16	3, 12, 15	mp2an 678	. . . 4 |- E! j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k
17	10	riota2 6274	. . . 4 |- ( ( M e. ( ZZ>= ` M ) /\ E! j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k ) -> ( A. k e. ( ZZ>= ` M ) M <_ k <-> ( iota_ j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k ) = M ) )
18	6, 16, 17	mp2an 678	. . 3 |- ( A. k e. ( ZZ>= ` M ) M <_ k <-> ( iota_ j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k ) = M )
19	8, 18	mpbi 212	. 2 |- ( iota_ j e. ( ZZ>= ` M ) A. k e. ( ZZ>= ` M ) j <_ k ) = M
20	14, 19	eqtri 2473	1 |- sup ( ( ZZ>= ` M ) , RR , `' < ) = M

Syntax hints:   <-> wb 188   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   E!wreu 2739   C_ wss 3404   class class class wbr 4402   `'ccnv 4833   ` cfv 5582   iota_crio 6251   supcsup 7954   RRcr 9538   < clt 9675   <_ cle 9676   ZZcz 10937   ZZ>=cuz 11159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-pre-lttri 9613  ax-pre-lttrn 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-neg 9863  df-z 10938  df-uz 11160

This theorem is referenced by:  nninfmOLD  11242  nn0infmOLD  11243