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Theorem supaddc 28429
Description: The supremum function distributes over addition in a sense similar to that in supmul1 10307. (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 2987 . . . . . . 7  |-  w  e. 
_V
2 oveq1 6110 . . . . . . . . . 10  |-  ( v  =  a  ->  (
v  +  B )  =  ( a  +  B ) )
32eqeq2d 2454 . . . . . . . . 9  |-  ( v  =  a  ->  (
z  =  ( v  +  B )  <->  z  =  ( a  +  B
) ) )
43cbvrexv 2960 . . . . . . . 8  |-  ( E. v  e.  A  z  =  ( v  +  B )  <->  E. a  e.  A  z  =  ( a  +  B
) )
5 eqeq1 2449 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  =  ( a  +  B )  <->  w  =  ( a  +  B
) ) )
65rexbidv 2748 . . . . . . . 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 3121 . . . . . 6  |-  ( w  e.  C  <->  E. a  e.  A  w  =  ( a  +  B
) )
10 supadd.a1 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
1110sselda 3368 . . . . . . . . 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 10302 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
1510, 12, 13, 14syl3anc 1218 . . . . . . . . . 10  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
1615adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
17 supaddc.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
1817adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  B  e.  RR )
1910, 12, 133jca 1168 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
20 suprub 10303 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2119, 20sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  a  <_  sup ( A ,  RR ,  <  ) )
2211, 16, 18, 21leadd1dd 9965 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
23 breq1 4307 . . . . . . . 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 2853 . . . . . 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 2810 . . . 4  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) )
2811, 18readdcld 9425 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
29 eleq1a 2512 . . . . . . . . 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 2853 . . . . . . 7  |-  ( ph  ->  ( E. a  e.  A  w  =  ( a  +  B )  ->  w  e.  RR ) )
329, 31syl5bi 217 . . . . . 6  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
3332ssrdv 3374 . . . . 5  |-  ( ph  ->  C  C_  RR )
34 ovex 6128 . . . . . . . . 9  |-  ( a  +  B )  e. 
_V
3534isseti 2990 . . . . . . . 8  |-  E. w  w  =  ( a  +  B )
3635rgenw 2795 . . . . . . 7  |-  A. a  e.  A  E. w  w  =  ( a  +  B )
37 r19.2z 3781 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A. a  e.  A  E. w  w  =  (
a  +  B ) )  ->  E. a  e.  A  E. w  w  =  ( a  +  B ) )
3812, 36, 37sylancl 662 . . . . . 6  |-  ( ph  ->  E. a  e.  A  E. w  w  =  ( a  +  B
) )
399exbii 1634 . . . . . . 7  |-  ( E. w  w  e.  C  <->  E. w E. a  e.  A  w  =  ( a  +  B ) )
40 n0 3658 . . . . . . 7  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
41 rexcom4 3004 . . . . . . 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 9425 . . . . . 6  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  e.  RR )
45 breq2 4308 . . . . . . . 8  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( w  <_  x 
<->  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4645ralbidv 2747 . . . . . . 7  |-  ( x  =  ( sup ( A ,  RR ,  <  )  +  B )  ->  ( A. w  e.  C  w  <_  x  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  +  B ) ) )
4746rspcev 3085 . . . . . 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 661 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
49 suprleub 10306 . . . . 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 1221 . . . 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 10302 . . . . . . . 8  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
5333, 43, 48, 52syl3anc 1218 . . . . . . 7  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
5453, 17, 15ltsubaddd 9947 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) ) )
5554biimpar 485 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  ( sup ( C ,  RR ,  <  )  -  B
)  <  sup ( A ,  RR ,  <  ) )
5653, 17resubcld 9788 . . . . . . 7  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  -  B )  e.  RR )
57 suprlub 10304 . . . . . . 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 1221 . . . . . 6  |-  ( ph  ->  ( ( sup ( C ,  RR ,  <  )  -  B )  <  sup ( A ,  RR ,  <  )  <->  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a ) )
5958adantr 465 . . . . 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 2789 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  w  e.  C
)
64 simplrr 760 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  =  ( a  +  B
) )
6533, 43, 483jca 1168 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
66 suprub 10303 . . . . . . . . . . . . . . 15  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6765, 66sylan 471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
6867adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  w  <_  sup ( C ,  RR ,  <  ) )
6964, 68eqbrtrrd 4326 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  A  /\  w  =  ( a  +  B ) ) )  /\  w  e.  C
)  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7063, 69mpdan 668 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  A  /\  w  =  ( a  +  B ) ) )  ->  ( a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7170expr 615 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( a  +  B )  -> 
( a  +  B
)  <_  sup ( C ,  RR ,  <  ) ) )
7271exlimdv 1690 . . . . . . . . 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 714 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  <_  sup ( C ,  RR ,  <  ) )
7528adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  (
a  +  B )  e.  RR )
7653ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  sup ( C ,  RR ,  <  )  e.  RR )
7775, 76lenltd 9532 . . . . . . 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 725 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  B  e.  RR )
8011adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  /\  a  e.  A )  ->  a  e.  RR )
8176, 79, 80ltsubaddd 9947 . . . . . 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 2831 . . . 4  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B
) )  ->  -.  E. a  e.  A  ( sup ( C ,  RR ,  <  )  -  B )  <  a
)
8460, 83pm2.65da 576 . . 3  |-  ( ph  ->  -.  sup ( C ,  RR ,  <  )  <  ( sup ( A ,  RR ,  <  )  +  B ) )
8553, 44eqleltd 9530 . . 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 913 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  =  ( sup ( A ,  RR ,  <  )  +  B ) )
8786eqcomd 2448 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2618   A.wral 2727   E.wrex 2728    C_ wss 3340   (/)c0 3649   class class class wbr 4304  (class class class)co 6103   supcsup 7702   RRcr 9293    + caddc 9297    < clt 9430    <_ cle 9431    - cmin 9607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610
This theorem is referenced by:  supadd  28430
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