Theorem supsubc 39569

Description: The supremum function distributes over subtraction in a sense similar to that in supaddc 10990. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
supsubc.a1	|- ( ph -> A C_ RR )
supsubc.a2	|- ( ph -> A =/= (/) )
supsubc.a3	|- ( ph -> E. x e. RR A. y e. A y <_ x )
supsubc.b	|- ( ph -> B e. RR )
supsubc.c	|- C = { z | E. v e. A z = ( v - B ) }

Assertion

Ref	Expression
supsubc	|- ( ph -> ( sup ( A , RR , < ) - B ) = sup ( C , RR , < ) )

Distinct variable groups:   v, A, x, y, z   v, B, x, y, z   ph, v, z

Allowed substitution hints:   ph( x, y)   C( x, y, z, v)

Proof of Theorem supsubc

Step	Hyp	Ref	Expression
1		supsubc.c	. . . . 5 |- C = { z | E. v e. A z = ( v - B ) }
2	1	a1i 11	. . . 4 |- ( ph -> C = { z | E. v e. A z = ( v - B ) } )
3		supsubc.a1	. . . . . . . . . . 11 |- ( ph -> A C_ RR )
4	3	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ v e. A ) -> v e. RR )
5	4	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> v e. CC )
6		supsubc.b	. . . . . . . . . . 11 |- ( ph -> B e. RR )
7	6	recnd 10068	. . . . . . . . . 10 |- ( ph -> B e. CC )
8	7	adantr 481	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> B e. CC )
9	5, 8	negsubd 10398	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( v + -u B ) = ( v - B ) )
10	9	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( v - B ) = ( v + -u B ) )
11	10	eqeq2d 2632	. . . . . 6 |- ( ( ph /\ v e. A ) -> ( z = ( v - B ) <-> z = ( v + -u B ) ) )
12	11	rexbidva 3049	. . . . 5 |- ( ph -> ( E. v e. A z = ( v - B ) <-> E. v e. A z = ( v + -u B ) ) )
13	12	abbidv 2741	. . . 4 |- ( ph -> { z | E. v e. A z = ( v - B ) } = { z | E. v e. A z = ( v + -u B ) } )
14		eqidd 2623	. . . 4 |- ( ph -> { z | E. v e. A z = ( v + -u B ) } = { z | E. v e. A z = ( v + -u B ) } )
15	2, 13, 14	3eqtrd 2660	. . 3 |- ( ph -> C = { z | E. v e. A z = ( v + -u B ) } )
16	15	supeq1d 8352	. 2 |- ( ph -> sup ( C , RR , < ) = sup ( { z | E. v e. A z = ( v + -u B ) } , RR , < ) )
17		supsubc.a2	. . . 4 |- ( ph -> A =/= (/) )
18		supsubc.a3	. . . 4 |- ( ph -> E. x e. RR A. y e. A y <_ x )
19	6	renegcld 10457	. . . 4 |- ( ph -> -u B e. RR )
20		eqid 2622	. . . 4 |- { z | E. v e. A z = ( v + -u B ) } = { z | E. v e. A z = ( v + -u B ) }
21	3, 17, 18, 19, 20	supaddc 10990	. . 3 |- ( ph -> ( sup ( A , RR , < ) + -u B ) = sup ( { z | E. v e. A z = ( v + -u B ) } , RR , < ) )
22	21	eqcomd 2628	. 2 |- ( ph -> sup ( { z | E. v e. A z = ( v + -u B ) } , RR , < ) = ( sup ( A , RR , < ) + -u B ) )
23		suprcl 10983	. . . . 5 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR )
24	3, 17, 18, 23	syl3anc 1326	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
25	24	recnd 10068	. . 3 |- ( ph -> sup ( A , RR , < ) e. CC )
26	25, 7	negsubd 10398	. 2 |- ( ph -> ( sup ( A , RR , < ) + -u B ) = ( sup ( A , RR , < ) - B ) )
27	16, 22, 26	3eqtrrd 2661	1 |- ( ph -> ( sup ( A , RR , < ) - B ) = sup ( C , RR , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-nel 2898  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  hoidmvlelem1  40809