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Theorem xrlexaddrp 37662
Description: If an extended real number  A can be approximated from above, adding positive reals to  B, then  A is smaller or equal than  B. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
xrlexaddrp.1  |-  ( ph  ->  A  e.  RR* )
xrlexaddrp.2  |-  ( ph  ->  B  e.  RR* )
xrlexaddrp.3  |-  ( (
ph  /\  x  e.  RR+ )  ->  A  <_  ( B +e x ) )
Assertion
Ref Expression
xrlexaddrp  |-  ( ph  ->  A  <_  B )
Distinct variable groups:    x, A    x, B    ph, x

Proof of Theorem xrlexaddrp
StepHypRef Expression
1 xrlexaddrp.1 . . . . 5  |-  ( ph  ->  A  e.  RR* )
2 pnfge 11455 . . . . 5  |-  ( A  e.  RR*  ->  A  <_ +oo )
31, 2syl 17 . . . 4  |-  ( ph  ->  A  <_ +oo )
43adantr 472 . . 3  |-  ( (
ph  /\  B  = +oo )  ->  A  <_ +oo )
5 id 22 . . . . 5  |-  ( B  = +oo  ->  B  = +oo )
65eqcomd 2477 . . . 4  |-  ( B  = +oo  -> +oo  =  B )
76adantl 473 . . 3  |-  ( (
ph  /\  B  = +oo )  -> +oo  =  B )
84, 7breqtrd 4420 . 2  |-  ( (
ph  /\  B  = +oo )  ->  A  <_  B )
9 simpl 464 . . 3  |-  ( (
ph  /\  -.  B  = +oo )  ->  ph )
10 neqne 2651 . . . 4  |-  ( -.  B  = +oo  ->  B  =/= +oo )
1110adantl 473 . . 3  |-  ( (
ph  /\  -.  B  = +oo )  ->  B  =/= +oo )
12 simpr 468 . . . . . 6  |-  ( (
ph  /\  A  = -oo )  ->  A  = -oo )
13 xrlexaddrp.2 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
14 mnfle 11458 . . . . . . . 8  |-  ( B  e.  RR*  -> -oo  <_  B )
1513, 14syl 17 . . . . . . 7  |-  ( ph  -> -oo  <_  B )
1615adantr 472 . . . . . 6  |-  ( (
ph  /\  A  = -oo )  -> -oo  <_  B )
1712, 16eqbrtrd 4416 . . . . 5  |-  ( (
ph  /\  A  = -oo )  ->  A  <_  B )
1817adantlr 729 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  A  = -oo )  ->  A  <_  B )
19 simpl 464 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  -.  A  = -oo )  ->  ( ph  /\  B  =/= +oo ) )
20 neqne 2651 . . . . . 6  |-  ( -.  A  = -oo  ->  A  =/= -oo )
2120adantl 473 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  -.  A  = -oo )  ->  A  =/= -oo )
22 simpll 768 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  A  =/= -oo )  ->  ph )
2313adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  =/= +oo )  ->  B  e.  RR* )
24 simpr 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  =/= +oo )  ->  B  =/= +oo )
2523, 24jca 541 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =/= +oo )  ->  ( B  e.  RR*  /\  B  =/= +oo ) )
26 xrnepnf 11443 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
2725, 26sylib 201 . . . . . . . . . 10  |-  ( (
ph  /\  B  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo ) )
2827adantr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  -.  B  e.  RR )  ->  ( B  e.  RR  \/  B  = -oo ) )
29 simpr 468 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  -.  B  e.  RR )  ->  -.  B  e.  RR )
30 pm2.53 380 . . . . . . . . 9  |-  ( ( B  e.  RR  \/  B  = -oo )  ->  ( -.  B  e.  RR  ->  B  = -oo ) )
3128, 29, 30sylc 61 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  -.  B  e.  RR )  ->  B  = -oo )
3231adantlr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  B  =/= +oo )  /\  A  =/= -oo )  /\  -.  B  e.  RR )  ->  B  = -oo )
33 id 22 . . . . . . . . . . . . 13  |-  ( ph  ->  ph )
34 1rp 11329 . . . . . . . . . . . . . 14  |-  1  e.  RR+
3534a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  RR+ )
36 1re 9660 . . . . . . . . . . . . . . 15  |-  1  e.  RR
3736elexi 3041 . . . . . . . . . . . . . 14  |-  1  e.  _V
38 eleq1 2537 . . . . . . . . . . . . . . . 16  |-  ( x  =  1  ->  (
x  e.  RR+  <->  1  e.  RR+ ) )
3938anbi2d 718 . . . . . . . . . . . . . . 15  |-  ( x  =  1  ->  (
( ph  /\  x  e.  RR+ )  <->  ( ph  /\  1  e.  RR+ )
) )
40 oveq2 6316 . . . . . . . . . . . . . . . 16  |-  ( x  =  1  ->  ( B +e x )  =  ( B +e 1 ) )
4140breq2d 4407 . . . . . . . . . . . . . . 15  |-  ( x  =  1  ->  ( A  <_  ( B +e x )  <->  A  <_  ( B +e 1 ) ) )
4239, 41imbi12d 327 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  (
( ( ph  /\  x  e.  RR+ )  ->  A  <_  ( B +e x ) )  <-> 
( ( ph  /\  1  e.  RR+ )  ->  A  <_  ( B +e 1 ) ) ) )
43 xrlexaddrp.3 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR+ )  ->  A  <_  ( B +e x ) )
4437, 42, 43vtocl 3086 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  e.  RR+ )  ->  A  <_  ( B +e 1 ) )
4533, 35, 44syl2anc 673 . . . . . . . . . . . 12  |-  ( ph  ->  A  <_  ( B +e 1 ) )
4645ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  A  <_  ( B +e 1 ) )
47 oveq1 6315 . . . . . . . . . . . . . . . 16  |-  ( B  = -oo  ->  ( B +e 1 )  =  ( -oo +e 1 ) )
4836rexri 9711 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR*
49 ltpnf 11445 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1  e.  RR  ->  1  < +oo )
5036, 49ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  1  < +oo
5136, 50ltneii 9765 . . . . . . . . . . . . . . . . . 18  |-  1  =/= +oo
52 xaddmnf2 11545 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  RR*  /\  1  =/= +oo )  ->  ( -oo +e 1 )  = -oo )
5348, 51, 52mp2an 686 . . . . . . . . . . . . . . . . 17  |-  ( -oo +e 1 )  = -oo
5453a1i 11 . . . . . . . . . . . . . . . 16  |-  ( B  = -oo  ->  ( -oo +e 1 )  = -oo )
5547, 54eqtr2d 2506 . . . . . . . . . . . . . . 15  |-  ( B  = -oo  -> -oo  =  ( B +e 1 ) )
5655adantl 473 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  -> -oo  =  ( B +e 1 ) )
5756eqcomd 2477 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  ( B +e 1 )  = -oo )
581adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  =/= -oo )  ->  A  e.  RR* )
59 simpr 468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  =/= -oo )  ->  A  =/= -oo )
60 nemnftgtmnft 37654 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  -> -oo  <  A )
6158, 59, 60syl2anc 673 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/= -oo )  -> -oo  <  A
)
6261adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  -> -oo  <  A )
6357, 62eqbrtrd 4416 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  ( B +e 1 )  <  A )
6413ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  B  e.  RR* )
6548a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  1  e.  RR* )
6664, 65xaddcld 11612 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  ( B +e 1 )  e.  RR* )
671ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  A  e.  RR* )
68 xrltnle 9719 . . . . . . . . . . . . 13  |-  ( ( ( B +e 1 )  e.  RR*  /\  A  e.  RR* )  ->  ( ( B +e 1 )  < 
A  <->  -.  A  <_  ( B +e 1 ) ) )
6966, 67, 68syl2anc 673 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  (
( B +e 1 )  <  A  <->  -.  A  <_  ( B +e 1 ) ) )
7063, 69mpbid 215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/= -oo )  /\  B  = -oo )  ->  -.  A  <_  ( B +e 1 ) )
7146, 70pm2.65da 586 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/= -oo )  ->  -.  B  = -oo )
7271neqned 2650 . . . . . . . . 9  |-  ( (
ph  /\  A  =/= -oo )  ->  B  =/= -oo )
7372ad4ant13 1258 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  =/= +oo )  /\  A  =/= -oo )  /\  -.  B  e.  RR )  ->  B  =/= -oo )
7473neneqd 2648 . . . . . . 7  |-  ( ( ( ( ph  /\  B  =/= +oo )  /\  A  =/= -oo )  /\  -.  B  e.  RR )  ->  -.  B  = -oo )
7532, 74condan 811 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  A  =/= -oo )  ->  B  e.  RR )
7643adantlr 729 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR )  /\  x  e.  RR+ )  ->  A  <_  ( B +e
x ) )
77 simpl 464 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  x  e.  RR+ )  ->  B  e.  RR )
78 rpre 11331 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  x  e.  RR )
7978adantl 473 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  x  e.  RR+ )  ->  x  e.  RR )
80 rexadd 11548 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( B +e
x )  =  ( B  +  x ) )
8177, 79, 80syl2anc 673 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  x  e.  RR+ )  -> 
( B +e
x )  =  ( B  +  x ) )
8281adantll 728 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR )  /\  x  e.  RR+ )  ->  ( B +e x )  =  ( B  +  x ) )
8376, 82breqtrd 4420 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR )  /\  x  e.  RR+ )  ->  A  <_  ( B  +  x
) )
8483ralrimiva 2809 . . . . . . 7  |-  ( (
ph  /\  B  e.  RR )  ->  A. x  e.  RR+  A  <_  ( B  +  x )
)
851adantr 472 . . . . . . . 8  |-  ( (
ph  /\  B  e.  RR )  ->  A  e. 
RR* )
86 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  B  e.  RR )  ->  B  e.  RR )
87 xralrple 11521 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  <->  A. x  e.  RR+  A  <_  ( B  +  x )
) )
8885, 86, 87syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  B  e.  RR )  ->  ( A  <_  B  <->  A. x  e.  RR+  A  <_  ( B  +  x )
) )
8984, 88mpbird 240 . . . . . 6  |-  ( (
ph  /\  B  e.  RR )  ->  A  <_  B )
9022, 75, 89syl2anc 673 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  A  =/= -oo )  ->  A  <_  B )
9119, 21, 90syl2anc 673 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  -.  A  = -oo )  ->  A  <_  B )
9218, 91pm2.61dan 808 . . 3  |-  ( (
ph  /\  B  =/= +oo )  ->  A  <_  B )
939, 11, 92syl2anc 673 . 2  |-  ( (
ph  /\  -.  B  = +oo )  ->  A  <_  B )
948, 93pm2.61dan 808 1  |-  ( ph  ->  A  <_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   class class class wbr 4395  (class class class)co 6308   RRcr 9556   1c1 9558    + caddc 9560   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694   RR+crp 11325   +ecxad 11430
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-cnex 9613  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-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  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-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433
This theorem is referenced by:  infleinf  37682  sge0xaddlem2  38390  ovnsubadd  38512
  Copyright terms: Public domain W3C validator