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Theorem supaddc 10571
Description: The supremum function distributes over addition in a sense similar to that in supmul1 10573. (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 3047 . . . . . . 7  |-  w  e. 
_V
2 oveq1 6295 . . . . . . . . . 10  |-  ( v  =  a  ->  (
v  +  B )  =  ( a  +  B ) )
32eqeq2d 2460 . . . . . . . . 9  |-  ( v  =  a  ->  (
z  =  ( v  +  B )  <->  z  =  ( a  +  B
) ) )
43cbvrexv 3019 . . . . . . . 8  |-  ( E. v  e.  A  z  =  ( v  +  B )  <->  E. a  e.  A  z  =  ( a  +  B
) )
5 eqeq1 2454 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  =  ( a  +  B )  <->  w  =  ( a  +  B
) ) )
65rexbidv 2900 . . . . . . . 8  |-  ( z  =  w  ->  ( E. a  e.  A  z  =  ( a  +  B )  <->  E. a  e.  A  w  =  ( a  +  B
) ) )
74, 6syl5bb 261 . . . . . . 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 3187 . . . . . 6  |-  ( w  e.  C  <->  E. a  e.  A  w  =  ( a  +  B
) )
10 supadd.a1 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
1110sselda 3431 . . . . . . . . 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 10566 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
1510, 12, 13, 14syl3anc 1267 . . . . . . . . . 10  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
1615adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
17 supaddc.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
1817adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  B  e.  RR )
1910, 12, 133jca 1187 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
20 suprub 10567 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2119, 20sylan 474 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2211, 16, 18, 21leadd1dd 10224 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
23 breq1 4404 . . . . . . . 8  |-  ( w  =  ( a  +  B )  ->  (
w  <_  ( sup ( A ,  RR ,  <  )  +  B )  <-> 
( a  +  B
)  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2422, 23syl5ibrcom 226 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2524rexlimdva 2878 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  w  =  ( a  +  B )  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B
) ) )
269, 25syl5bi 221 . . . . 5  |-  ( ph  ->  ( w  e.  C  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2726ralrimiv 2799 . . . 4  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
2811, 18readdcld 9667 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
29 eleq1a 2523 . . . . . . . . 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 2878 . . . . . . 7  |-  ( ph  ->  ( E. a  e.  A  w  =  ( a  +  B )  ->  w  e.  RR ) )
329, 31syl5bi 221 . . . . . 6  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
3332ssrdv 3437 . . . . 5  |-  ( ph  ->  C  C_  RR )
34 ovex 6316 . . . . . . . . 9  |-  ( a  +  B )  e. 
_V
3534isseti 3050 . . . . . . . 8  |-  E. w  w  =  ( a  +  B )
3635rgenw 2748 . . . . . . 7  |-  A. a  e.  A  E. w  w  =  ( a  +  B )
37 r19.2z 3857 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A. a  e.  A  E. w  w  =  (
a  +  B ) )  ->  E. a  e.  A  E. w  w  =  ( a  +  B ) )
3812, 36, 37sylancl 667 . . . . . 6  |-  ( ph  ->  E. a  e.  A  E. w  w  =  ( a  +  B
) )
399exbii 1717 . . . . . . 7  |-  ( E. w  w  e.  C  <->  E. w E. a  e.  A  w  =  ( a  +  B ) )
40 n0 3740 . . . . . . 7  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
41 rexcom4 3066 . . . . . . 7  |-  ( E. a  e.  A  E. w  w  =  (
a  +  B )  <->  E. w E. a  e.  A  w  =  ( a  +  B ) )
4239, 40, 413bitr4i 281 . . . . . 6  |-  ( C  =/=  (/)  <->  E. a  e.  A  E. w  w  =  ( a  +  B
) )
4338, 42sylibr 216 . . . . 5  |-  ( ph  ->  C  =/=  (/) )
4415, 17readdcld 9667 . . . . . 6  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  e.  RR )
45 breq2 4405 . . . . . . . 8  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( w  <_  x 
<->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4645ralbidv 2826 . . . . . . 7  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( A. w  e.  C  w  <_  x  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4746rspcev 3149 . . . . . 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 666 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
49 suprleub 10570 . . . . 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 1270 . . . 4  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  +  B )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
5127, 50mpbird 236 . . 3  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( sup ( A ,  RR ,  <  )  +  B ) )
52 suprcl 10566 . . . . . . . 8  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
5333, 43, 48, 52syl3anc 1267 . . . . . . 7  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
5453, 17, 15ltsubaddd 10206 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) ) )
5554biimpar 488 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  ( sup ( C ,  RR ,  <  )  -  B
)  <  sup ( A ,  RR ,  <  ) )
5653, 17resubcld 10044 . . . . . . 7  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  -  B )  e.  RR )
57 suprlub 10568 . . . . . . 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 1270 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a ) )
5958adantr 467 . . . . 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 214 . . . 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 216 . . . . . . . . . . . . 13  |-  ( ( a  e.  A  /\  w  =  ( a  +  B ) )  ->  w  e.  C )
6362adantl 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  w  e.  C
)
64 simplrr 770 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  =  ( a  +  B
) )
6533, 43, 483jca 1187 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
66 suprub 10567 . . . . . . . . . . . . . . 15  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6765, 66sylan 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6867adantlr 720 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  <_  sup ( C ,  RR ,  <  ) )
6964, 68eqbrtrrd 4424 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7063, 69mpdan 673 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7170expr 619 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  -> 
( a  +  B
)  <_  sup ( C ,  RR ,  <  ) ) )
7271exlimdv 1778 . . . . . . . . 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 720 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7528adantlr 720 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
7653ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  sup ( C ,  RR ,  <  )  e.  RR )
7775, 76lenltd 9778 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
( a  +  B
)  <_  sup ( C ,  RR ,  <  )  <->  -.  sup ( C ,  RR ,  <  )  <  ( a  +  B ) ) )
7874, 77mpbid 214 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  -.  sup ( C ,  RR ,  <  )  <  (
a  +  B ) )
7917ad2antrr 731 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  B  e.  RR )
8011adantlr 720 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  a  e.  RR )
8176, 79, 80ltsubaddd 10206 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
( sup ( C ,  RR ,  <  )  -  B )  < 
a  <->  sup ( C ,  RR ,  <  )  < 
( a  +  B
) ) )
8278, 81mtbird 303 . . . . 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 579 . . 3  |-  ( ph  ->  -.  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B ) )
8553, 44eqleltd 9776 . . 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 932 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  =  ( sup ( A ,  RR ,  <  )  +  B ) )
8786eqcomd 2456 1  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886   {cab 2436    =/= wne 2621   A.wral 2736   E.wrex 2737    C_ wss 3403   (/)c0 3730   class class class wbr 4401  (class class class)co 6288   supcsup 7951   RRcr 9535    + caddc 9539    < clt 9672    <_ cle 9673    - cmin 9857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860
This theorem is referenced by:  supadd  10572  supsubc  37570
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