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Theorem supaddc 28342
Description: The supremum function distributes over addition in a sense similar to that in supmul1 10291. (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 2973 . . . . . . 7  |-  w  e. 
_V
2 oveq1 6097 . . . . . . . . . 10  |-  ( v  =  a  ->  (
v  +  B )  =  ( a  +  B ) )
32eqeq2d 2452 . . . . . . . . 9  |-  ( v  =  a  ->  (
z  =  ( v  +  B )  <->  z  =  ( a  +  B
) ) )
43cbvrexv 2946 . . . . . . . 8  |-  ( E. v  e.  A  z  =  ( v  +  B )  <->  E. a  e.  A  z  =  ( a  +  B
) )
5 eqeq1 2447 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  =  ( a  +  B )  <->  w  =  ( a  +  B
) ) )
65rexbidv 2734 . . . . . . . 8  |-  ( z  =  w  ->  ( E. a  e.  A  z  =  ( a  +  B )  <->  E. a  e.  A  w  =  ( a  +  B
) ) )
74, 6syl5bb 257 . . . . . . 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 3106 . . . . . 6  |-  ( w  e.  C  <->  E. a  e.  A  w  =  ( a  +  B
) )
10 supadd.a1 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
1110sselda 3353 . . . . . . . . 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 10286 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
1510, 12, 13, 14syl3anc 1213 . . . . . . . . . 10  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
1615adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
17 supaddc.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
1817adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  B  e.  RR )
1910, 12, 133jca 1163 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
20 suprub 10287 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2119, 20sylan 468 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2211, 16, 18, 21leadd1dd 9949 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
23 breq1 4292 . . . . . . . 8  |-  ( w  =  ( a  +  B )  ->  (
w  <_  ( sup ( A ,  RR ,  <  )  +  B )  <-> 
( a  +  B
)  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2422, 23syl5ibrcom 222 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2524rexlimdva 2839 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  w  =  ( a  +  B )  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B
) ) )
269, 25syl5bi 217 . . . . 5  |-  ( ph  ->  ( w  e.  C  ->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
2726ralrimiv 2796 . . . 4  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
2811, 18readdcld 9409 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
29 eleq1a 2510 . . . . . . . . 9  |-  ( ( a  +  B )  e.  RR  ->  (
w  =  ( a  +  B )  ->  w  e.  RR )
)
3028, 29syl 16 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  ->  w  e.  RR )
)
3130rexlimdva 2839 . . . . . . 7  |-  ( ph  ->  ( E. a  e.  A  w  =  ( a  +  B )  ->  w  e.  RR ) )
329, 31syl5bi 217 . . . . . 6  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
3332ssrdv 3359 . . . . 5  |-  ( ph  ->  C  C_  RR )
34 ovex 6115 . . . . . . . . 9  |-  ( a  +  B )  e. 
_V
3534isseti 2976 . . . . . . . 8  |-  E. w  w  =  ( a  +  B )
3635rgenw 2781 . . . . . . 7  |-  A. a  e.  A  E. w  w  =  ( a  +  B )
37 r19.2z 3766 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A. a  e.  A  E. w  w  =  (
a  +  B ) )  ->  E. a  e.  A  E. w  w  =  ( a  +  B ) )
3812, 36, 37sylancl 657 . . . . . 6  |-  ( ph  ->  E. a  e.  A  E. w  w  =  ( a  +  B
) )
399exbii 1639 . . . . . . 7  |-  ( E. w  w  e.  C  <->  E. w E. a  e.  A  w  =  ( a  +  B ) )
40 n0 3643 . . . . . . 7  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
41 rexcom4 2990 . . . . . . 7  |-  ( E. a  e.  A  E. w  w  =  (
a  +  B )  <->  E. w E. a  e.  A  w  =  ( a  +  B ) )
4239, 40, 413bitr4i 277 . . . . . 6  |-  ( C  =/=  (/)  <->  E. a  e.  A  E. w  w  =  ( a  +  B
) )
4338, 42sylibr 212 . . . . 5  |-  ( ph  ->  C  =/=  (/) )
4415, 17readdcld 9409 . . . . . 6  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  e.  RR )
45 breq2 4293 . . . . . . . 8  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( w  <_  x 
<->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4645ralbidv 2733 . . . . . . 7  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( A. w  e.  C  w  <_  x  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4746rspcev 3070 . . . . . 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 656 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
49 suprleub 10290 . . . . 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 1216 . . . 4  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  +  B )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
5127, 50mpbird 232 . . 3  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( sup ( A ,  RR ,  <  )  +  B ) )
52 suprcl 10286 . . . . . . . 8  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
5333, 43, 48, 52syl3anc 1213 . . . . . . 7  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
5453, 17, 15ltsubaddd 9931 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) ) )
5554biimpar 482 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  ( sup ( C ,  RR ,  <  )  -  B
)  <  sup ( A ,  RR ,  <  ) )
5653, 17resubcld 9772 . . . . . . 7  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  -  B )  e.  RR )
57 suprlub 10288 . . . . . . 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 1216 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a ) )
5958adantr 462 . . . . 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 210 . . . 4  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a )
61 rspe 2775 . . . . . . . . . . . . . 14  |-  ( ( a  e.  A  /\  w  =  ( a  +  B ) )  ->  E. a  e.  A  w  =  ( a  +  B ) )
6261, 9sylibr 212 . . . . . . . . . . . . 13  |-  ( ( a  e.  A  /\  w  =  ( a  +  B ) )  ->  w  e.  C )
6362adantl 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  w  e.  C
)
64 simplrr 755 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  =  ( a  +  B
) )
6533, 43, 483jca 1163 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
66 suprub 10287 . . . . . . . . . . . . . . 15  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6765, 66sylan 468 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6867adantlr 709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  <_  sup ( C ,  RR ,  <  ) )
6964, 68eqbrtrrd 4311 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7063, 69mpdan 663 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7170expr 612 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  -> 
( a  +  B
)  <_  sup ( C ,  RR ,  <  ) ) )
7271exlimdv 1695 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  ( E. w  w  =  ( a  +  B
)  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) ) )
7335, 72mpi 17 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7473adantlr 709 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7528adantlr 709 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
7653ad2antrr 720 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  sup ( C ,  RR ,  <  )  e.  RR )
7775, 76lenltd 9516 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
( a  +  B
)  <_  sup ( C ,  RR ,  <  )  <->  -.  sup ( C ,  RR ,  <  )  <  ( a  +  B ) ) )
7874, 77mpbid 210 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  -.  sup ( C ,  RR ,  <  )  <  (
a  +  B ) )
7917ad2antrr 720 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  B  e.  RR )
8011adantlr 709 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  a  e.  RR )
8176, 79, 80ltsubaddd 9931 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
( sup ( C ,  RR ,  <  )  -  B )  < 
a  <->  sup ( C ,  RR ,  <  )  < 
( a  +  B
) ) )
8278, 81mtbird 301 . . . . 5  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  -.  ( sup ( C ,  RR ,  <  )  -  B )  <  a
)
8382nrexdv 2817 . . . 4  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  -.  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a
)
8460, 83pm2.65da 573 . . 3  |-  ( ph  ->  -.  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B ) )
8553, 44eqleltd 9514 . . 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 908 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  =  ( sup ( A ,  RR ,  <  )  +  B ) )
8786eqcomd 2446 1  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364   E.wex 1591    e. wcel 1761   {cab 2427    =/= wne 2604   A.wral 2713   E.wrex 2714    C_ wss 3325   (/)c0 3634   class class class wbr 4289  (class class class)co 6090   supcsup 7686   RRcr 9277    + caddc 9281    < clt 9414    <_ cle 9415    - cmin 9591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594
This theorem is referenced by:  supadd  28343
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