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Theorem supaddc 10596
Description: The supremum function distributes over addition in a sense similar to that in supmul1 10598. (Contributed by Brendan Leahy, 25-Sep-2017.)
Hypotheses
Ref Expression
supadd.a1  |-  ( ph  ->  A  C_  RR )
supadd.a2  |-  ( ph  ->  A  =/=  (/) )
supadd.a3  |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )
supaddc.b  |-  ( ph  ->  B  e.  RR )
supaddc.c  |-  C  =  { z  |  E. v  e.  A  z  =  ( v  +  B ) }
Assertion
Ref Expression
supaddc  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  =  sup ( C ,  RR ,  <  ) )
Distinct variable groups:    x, y,
z, v, A    x, B, y, z, v    x, C    ph, z, v
Allowed substitution hints:    ph( x, y)    C( y, z, v)

Proof of Theorem supaddc
Dummy variables  w  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3034 . . . . . . 7  |-  w  e. 
_V
2 oveq1 6315 . . . . . . . . . 10  |-  ( v  =  a  ->  (
v  +  B )  =  ( a  +  B ) )
32eqeq2d 2481 . . . . . . . . 9  |-  ( v  =  a  ->  (
z  =  ( v  +  B )  <->  z  =  ( a  +  B
) ) )
43cbvrexv 3006 . . . . . . . 8  |-  ( E. v  e.  A  z  =  ( v  +  B )  <->  E. a  e.  A  z  =  ( a  +  B
) )
5 eqeq1 2475 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  =  ( a  +  B )  <->  w  =  ( a  +  B
) ) )
65rexbidv 2892 . . . . . . . 8  |-  ( z  =  w  ->  ( E. a  e.  A  z  =  ( a  +  B )  <->  E. a  e.  A  w  =  ( a  +  B
) ) )
74, 6syl5bb 265 . . . . . . 7  |-  ( z  =  w  ->  ( E. v  e.  A  z  =  ( v  +  B )  <->  E. a  e.  A  w  =  ( a  +  B
) ) )
8 supaddc.c . . . . . . 7  |-  C  =  { z  |  E. v  e.  A  z  =  ( v  +  B ) }
91, 7, 8elab2 3176 . . . . . 6  |-  ( w  e.  C  <->  E. a  e.  A  w  =  ( a  +  B
) )
10 supadd.a1 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
1110sselda 3418 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  RR )
12 supadd.a2 . . . . . . . . . . 11  |-  ( ph  ->  A  =/=  (/) )
13 supadd.a3 . . . . . . . . . . 11  |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )
14 suprcl 10591 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
1510, 12, 13, 14syl3anc 1292 . . . . . . . . . 10  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
1615adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
17 supaddc.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
1817adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  B  e.  RR )
1910, 12, 133jca 1210 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
20 suprub 10592 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2119, 20sylan 479 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2211, 16, 18, 21leadd1dd 10248 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
23 breq1 4398 . . . . . . . 8  |-  ( w  =  ( a  +  B )  ->  (
w  <_  ( sup ( A ,  RR ,  <  )  +  B )  <-> 
( a  +  B
)  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2422, 23syl5ibrcom 230 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2524rexlimdva 2871 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  w  =  ( a  +  B )  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B
) ) )
269, 25syl5bi 225 . . . . 5  |-  ( ph  ->  ( w  e.  C  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2726ralrimiv 2808 . . . 4  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
2811, 18readdcld 9688 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
29 eleq1a 2544 . . . . . . . . 9  |-  ( ( a  +  B )  e.  RR  ->  (
w  =  ( a  +  B )  ->  w  e.  RR )
)
3028, 29syl 17 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  ->  w  e.  RR )
)
3130rexlimdva 2871 . . . . . . 7  |-  ( ph  ->  ( E. a  e.  A  w  =  ( a  +  B )  ->  w  e.  RR ) )
329, 31syl5bi 225 . . . . . 6  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
3332ssrdv 3424 . . . . 5  |-  ( ph  ->  C  C_  RR )
34 ovex 6336 . . . . . . . . 9  |-  ( a  +  B )  e. 
_V
3534isseti 3037 . . . . . . . 8  |-  E. w  w  =  ( a  +  B )
3635rgenw 2768 . . . . . . 7  |-  A. a  e.  A  E. w  w  =  ( a  +  B )
37 r19.2z 3849 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A. a  e.  A  E. w  w  =  (
a  +  B ) )  ->  E. a  e.  A  E. w  w  =  ( a  +  B ) )
3812, 36, 37sylancl 675 . . . . . 6  |-  ( ph  ->  E. a  e.  A  E. w  w  =  ( a  +  B
) )
399exbii 1726 . . . . . . 7  |-  ( E. w  w  e.  C  <->  E. w E. a  e.  A  w  =  ( a  +  B ) )
40 n0 3732 . . . . . . 7  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
41 rexcom4 3053 . . . . . . 7  |-  ( E. a  e.  A  E. w  w  =  (
a  +  B )  <->  E. w E. a  e.  A  w  =  ( a  +  B ) )
4239, 40, 413bitr4i 285 . . . . . 6  |-  ( C  =/=  (/)  <->  E. a  e.  A  E. w  w  =  ( a  +  B
) )
4338, 42sylibr 217 . . . . 5  |-  ( ph  ->  C  =/=  (/) )
4415, 17readdcld 9688 . . . . . 6  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  e.  RR )
45 breq2 4399 . . . . . . . 8  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( w  <_  x 
<->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4645ralbidv 2829 . . . . . . 7  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( A. w  e.  C  w  <_  x  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4746rspcev 3136 . . . . . 6  |-  ( ( ( sup ( A ,  RR ,  <  )  +  B )  e.  RR  /\  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) )  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
4844, 27, 47syl2anc 673 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
49 suprleub 10595 . . . . 5  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  ( sup ( A ,  RR ,  <  )  +  B
)  e.  RR )  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  +  B )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
5033, 43, 48, 44, 49syl31anc 1295 . . . 4  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  +  B )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
5127, 50mpbird 240 . . 3  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( sup ( A ,  RR ,  <  )  +  B ) )
52 suprcl 10591 . . . . . . . 8  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
5333, 43, 48, 52syl3anc 1292 . . . . . . 7  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
5453, 17, 15ltsubaddd 10230 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) ) )
5554biimpar 493 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  ( sup ( C ,  RR ,  <  )  -  B
)  <  sup ( A ,  RR ,  <  ) )
5653, 17resubcld 10068 . . . . . . 7  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  -  B )  e.  RR )
57 suprlub 10593 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( sup ( C ,  RR ,  <  )  -  B
)  e.  RR )  ->  ( ( sup ( C ,  RR ,  <  )  -  B
)  <  sup ( A ,  RR ,  <  )  <->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a
) )
5810, 12, 13, 56, 57syl31anc 1295 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a ) )
5958adantr 472 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  (
( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a ) )
6055, 59mpbid 215 . . . 4  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a )
61 rspe 2844 . . . . . . . . . . . . . 14  |-  ( ( a  e.  A  /\  w  =  ( a  +  B ) )  ->  E. a  e.  A  w  =  ( a  +  B ) )
6261, 9sylibr 217 . . . . . . . . . . . . 13  |-  ( ( a  e.  A  /\  w  =  ( a  +  B ) )  ->  w  e.  C )
6362adantl 473 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  w  e.  C
)
64 simplrr 779 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  =  ( a  +  B
) )
6533, 43, 483jca 1210 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
66 suprub 10592 . . . . . . . . . . . . . . 15  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6765, 66sylan 479 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6867adantlr 729 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  <_  sup ( C ,  RR ,  <  ) )
6964, 68eqbrtrrd 4418 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7063, 69mpdan 681 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7170expr 626 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  -> 
( a  +  B
)  <_  sup ( C ,  RR ,  <  ) ) )
7271exlimdv 1787 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  ( E. w  w  =  ( a  +  B
)  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) ) )
7335, 72mpi 20 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7473adantlr 729 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7528adantlr 729 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
7653ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  sup ( C ,  RR ,  <  )  e.  RR )
7775, 76lenltd 9798 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
( a  +  B
)  <_  sup ( C ,  RR ,  <  )  <->  -.  sup ( C ,  RR ,  <  )  <  ( a  +  B ) ) )
7874, 77mpbid 215 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  -.  sup ( C ,  RR ,  <  )  <  (
a  +  B ) )
7917ad2antrr 740 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  B  e.  RR )
8011adantlr 729 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  a  e.  RR )
8176, 79, 80ltsubaddd 10230 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
( sup ( C ,  RR ,  <  )  -  B )  < 
a  <->  sup ( C ,  RR ,  <  )  < 
( a  +  B
) ) )
8278, 81mtbird 308 . . . . 5  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  -.  ( sup ( C ,  RR ,  <  )  -  B )  <  a
)
8382nrexdv 2842 . . . 4  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  -.  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a
)
8460, 83pm2.65da 586 . . 3  |-  ( ph  ->  -.  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B ) )
8553, 44eqleltd 9796 . . 3  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  =  ( sup ( A ,  RR ,  <  )  +  B )  <-> 
( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  +  B )  /\  -.  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) ) ) )
8651, 84, 85mpbir2and 936 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  =  ( sup ( A ,  RR ,  <  )  +  B ) )
8786eqcomd 2477 1  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757    C_ wss 3390   (/)c0 3722   class class class wbr 4395  (class class class)co 6308   supcsup 7972   RRcr 9556    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883
This theorem is referenced by:  supadd  10597  supsubc  37663
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