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Theorem xrge0npcan 23335
Description: Extended non-negative real version of npcan 9062. (Contributed by Thierry Arnoux, 9-Jun-2017.)
Assertion
Ref Expression
xrge0npcan  |-  ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  B  <_  A
)  ->  ( ( A + e  - e B ) + e B )  =  A )

Proof of Theorem xrge0npcan
StepHypRef Expression
1 simpl1 958 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  A  e.  ( 0 [,]  +oo ) )
2 iccssxr 10734 . . . . . . . . . 10  |-  ( 0 [,]  +oo )  C_  RR*
32sseli 3178 . . . . . . . . 9  |-  ( A  e.  ( 0 [,] 
+oo )  ->  A  e.  RR* )
41, 3syl 15 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  A  e.  RR* )
5 simpr 447 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  B  =  +oo )
6 simpl3 960 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  B  <_  A )
75, 6eqbrtrrd 4047 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  +oo  <_  A )
8 xgtpnf 23116 . . . . . . . . 9  |-  ( A  e.  RR*  ->  (  +oo  <_  A  <->  A  =  +oo ) )
98biimpa 470 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  +oo  <_  A )  ->  A  =  +oo )
104, 7, 9syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  A  =  +oo )
11 xnegeq 10536 . . . . . . . 8  |-  ( B  =  +oo  ->  - e B  =  - e  +oo )
125, 11syl 15 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  - e B  =  - e  +oo )
1310, 12oveq12d 5878 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( A + e  - e B )  =  (  +oo + e  - e  +oo ) )
14 pnfxr 10457 . . . . . . 7  |-  +oo  e.  RR*
15 xnegid 10565 . . . . . . 7  |-  (  +oo  e.  RR*  ->  (  +oo + e  - e  +oo )  =  0 )
1614, 15ax-mp 8 . . . . . 6  |-  (  +oo + e  - e  +oo )  =  0
1713, 16syl6eq 2333 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( A + e  - e B )  =  0 )
1817oveq1d 5875 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  ( 0 + e B ) )
195oveq2d 5876 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( 0 + e B )  =  ( 0 + e  +oo ) )
20 xaddid2 10569 . . . . 5  |-  (  +oo  e.  RR*  ->  ( 0 + e  +oo )  =  +oo )
2114, 20mp1i 11 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( 0 + e  +oo )  =  +oo )
2218, 19, 213eqtrd 2321 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  +oo )
2322, 10eqtr4d 2320 . 2  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  A )
24 simpl1 958 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  A  e.  ( 0 [,]  +oo )
)
2524, 3syl 15 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  A  e.  RR* )
26 xrge0neqmnf 23332 . . . . 5  |-  ( A  e.  ( 0 [,] 
+oo )  ->  A  =/=  -oo )
2724, 26syl 15 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  A  =/=  -oo )
28 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  B  e.  ( 0 [,]  +oo )
)
292sseli 3178 . . . . . 6  |-  ( B  e.  ( 0 [,] 
+oo )  ->  B  e.  RR* )
3028, 29syl 15 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  B  e.  RR* )
3130xnegcld 10622 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  - e B  e. 
RR* )
32 simpr 447 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  -.  B  =  +oo )
33 xnegneg 10543 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  - e  - e B  =  B )
3433adantr 451 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  - e B  =  -oo )  ->  - e  - e B  =  B )
35 xnegeq 10536 . . . . . . . . . . . 12  |-  (  - e B  =  -oo  -> 
- e  - e B  =  - e  -oo )
3635adantl 452 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  - e B  =  -oo )  ->  - e  - e B  =  - e  -oo )
3734, 36eqtr3d 2319 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  - e B  =  -oo )  ->  B  =  - e  -oo )
38 xnegmnf 10539 . . . . . . . . . 10  |-  - e  -oo  =  +oo
3937, 38syl6eq 2333 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  - e B  =  -oo )  ->  B  =  +oo )
4039ex 423 . . . . . . . 8  |-  ( B  e.  RR*  ->  (  - e B  =  -oo  ->  B  =  +oo )
)
4140con3d 125 . . . . . . 7  |-  ( B  e.  RR*  ->  ( -.  B  =  +oo  ->  -.  - e B  =  -oo ) )
4241imp 418 . . . . . 6  |-  ( ( B  e.  RR*  /\  -.  B  =  +oo )  ->  -.  - e B  = 
-oo )
4342neneqad 2518 . . . . 5  |-  ( ( B  e.  RR*  /\  -.  B  =  +oo )  ->  - e B  =/=  -oo )
4430, 32, 43syl2anc 642 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  - e B  =/= 
-oo )
45 xrge0neqmnf 23332 . . . . 5  |-  ( B  e.  ( 0 [,] 
+oo )  ->  B  =/=  -oo )
4628, 45syl 15 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  B  =/=  -oo )
47 xaddass 10571 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/=  -oo )  /\  (  - e B  e.  RR*  /\  - e B  =/=  -oo )  /\  ( B  e.  RR*  /\  B  =/=  -oo ) )  -> 
( ( A + e  - e B ) + e B )  =  ( A + e (  - e B + e B ) ) )
4825, 27, 31, 44, 30, 46, 47syl222anc 1198 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  ( ( A + e  - e B ) + e B )  =  ( A + e ( 
- e B + e B ) ) )
49 xnegcl 10542 . . . . . . . . 9  |-  ( B  e.  RR*  ->  - e B  e.  RR* )
50 xaddcom 10567 . . . . . . . . 9  |-  ( ( 
- e B  e. 
RR*  /\  B  e.  RR* )  ->  (  - e B + e B )  =  ( B + e  - e B ) )
5149, 50mpancom 650 . . . . . . . 8  |-  ( B  e.  RR*  ->  (  - e B + e B )  =  ( B + e  - e B ) )
52 xnegid 10565 . . . . . . . 8  |-  ( B  e.  RR*  ->  ( B + e  - e B )  =  0 )
5351, 52eqtrd 2317 . . . . . . 7  |-  ( B  e.  RR*  ->  (  - e B + e B )  =  0 )
5453oveq2d 5876 . . . . . 6  |-  ( B  e.  RR*  ->  ( A + e (  - e B + e B ) )  =  ( A + e 0 ) )
5554adantl 452 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e (  - e B + e B ) )  =  ( A + e 0 ) )
56 xaddid1 10568 . . . . . 6  |-  ( A  e.  RR*  ->  ( A + e 0 )  =  A )
5756adantr 451 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e 0 )  =  A )
5855, 57eqtrd 2317 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e (  - e B + e B ) )  =  A )
5925, 30, 58syl2anc 642 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  ( A + e (  - e B + e B ) )  =  A )
6048, 59eqtrd 2317 . 2  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  ( ( A + e  - e B ) + e B )  =  A )
6123, 60pm2.61dan 766 1  |-  ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  B  <_  A
)  ->  ( ( A + e  - e B ) + e B )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025  (class class class)co 5860   0cc0 8739    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868    <_ cle 8870    - ecxne 10451   + ecxad 10452   [,]cicc 10661
This theorem is referenced by:  esumle  23435  esumlef  23437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-xneg 10454  df-xadd 10455  df-icc 10665
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