Theorem toslub 29668

Description: In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup ( A , B , .< ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)

Hypotheses

Ref	Expression
toslub.b	|- B = ( Base ` K )
toslub.l	|- .< = ( lt ` K )
toslub.1	|- ( ph -> K e. Toset )
toslub.2	|- ( ph -> A C_ B )

Assertion

Ref	Expression
toslub	|- ( ph -> ( ( lub ` K ) ` A ) = sup ( A , B , .< ) )

Proof of Theorem toslub

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		toslub.b	. . . 4 |- B = ( Base ` K )
2		toslub.l	. . . 4 |- .< = ( lt ` K )
3		toslub.1	. . . 4 |- ( ph -> K e. Toset )
4		toslub.2	. . . 4 |- ( ph -> A C_ B )
5		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
6	1, 2, 3, 4, 5	toslublem 29667	. . 3 |- ( ( ph /\ a e. B ) -> ( ( A. b e. A b ( le ` K ) a /\ A. c e. B ( A. b e. A b ( le ` K ) c -> a ( le ` K ) c ) ) <-> ( A. b e. A -. a .< b /\ A. b e. B ( b .< a -> E. d e. A b .< d ) ) ) )
7	6	riotabidva 6627	. 2 |- ( ph -> ( iota_ a e. B ( A. b e. A b ( le ` K ) a /\ A. c e. B ( A. b e. A b ( le ` K ) c -> a ( le ` K ) c ) ) ) = ( iota_ a e. B ( A. b e. A -. a .< b /\ A. b e. B ( b .< a -> E. d e. A b .< d ) ) ) )
8		eqid 2622	. . 3 |- ( lub ` K ) = ( lub ` K )
9		biid 251	. . 3 |- ( ( A. b e. A b ( le ` K ) a /\ A. c e. B ( A. b e. A b ( le ` K ) c -> a ( le ` K ) c ) ) <-> ( A. b e. A b ( le ` K ) a /\ A. c e. B ( A. b e. A b ( le ` K ) c -> a ( le ` K ) c ) ) )
10	1, 5, 8, 9, 3, 4	lubval 16984	. 2 |- ( ph -> ( ( lub ` K ) ` A ) = ( iota_ a e. B ( A. b e. A b ( le ` K ) a /\ A. c e. B ( A. b e. A b ( le ` K ) c -> a ( le ` K ) c ) ) ) )
11	1, 5, 2	tosso 17036	. . . . 5 |- ( K e. Toset -> ( K e. Toset <-> ( .< Or B /\ ( _I |` B ) C_ ( le ` K ) ) ) )
12	11	ibi 256	. . . 4 |- ( K e. Toset -> ( .< Or B /\ ( _I |` B ) C_ ( le ` K ) ) )
13	12	simpld 475	. . 3 |- ( K e. Toset -> .< Or B )
14		id 22	. . . 4 |- ( .< Or B -> .< Or B )
15	14	supval2 8361	. . 3 |- ( .< Or B -> sup ( A , B , .< ) = ( iota_ a e. B ( A. b e. A -. a .< b /\ A. b e. B ( b .< a -> E. d e. A b .< d ) ) ) )
16	3, 13, 15	3syl 18	. 2 |- ( ph -> sup ( A , B , .< ) = ( iota_ a e. B ( A. b e. A -. a .< b /\ A. b e. B ( b .< a -> E. d e. A b .< d ) ) ) )
17	7, 10, 16	3eqtr4d 2666	1 |- ( ph -> ( ( lub ` K ) ` A ) = sup ( A , B , .< ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   _I cid 5023   Or wor 5034   |` cres 5116   ` cfv 5888   iota_crio 6610   supcsup 8346   Basecbs 15857   lecple 15948   ltcplt 16941   lubclub 16942  Tosetctos 17033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-sup 8348  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-toset 17034

This theorem is referenced by:  xrsp1  29682