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Theorem xaddid1 9862
Description: Extended real version of addid1 8095. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddid1  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )

Proof of Theorem xaddid1
StepHypRef Expression
1 elxr 9776 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 0re 7957 . . . . 5  |-  0  e.  RR
3 rexadd 9852 . . . . 5  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A +e 0 )  =  ( A  +  0 ) )
42, 3mpan2 425 . . . 4  |-  ( A  e.  RR  ->  ( A +e 0 )  =  ( A  + 
0 ) )
5 recn 7944 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
65addid1d 8106 . . . 4  |-  ( A  e.  RR  ->  ( A  +  0 )  =  A )
74, 6eqtrd 2210 . . 3  |-  ( A  e.  RR  ->  ( A +e 0 )  =  A )
8 0xr 8004 . . . . 5  |-  0  e.  RR*
9 renemnf 8006 . . . . . 6  |-  ( 0  e.  RR  ->  0  =/= -oo )
102, 9ax-mp 5 . . . . 5  |-  0  =/= -oo
11 xaddpnf2 9847 . . . . 5  |-  ( ( 0  e.  RR*  /\  0  =/= -oo )  ->  ( +oo +e 0 )  = +oo )
128, 10, 11mp2an 426 . . . 4  |-  ( +oo +e 0 )  = +oo
13 oveq1 5882 . . . 4  |-  ( A  = +oo  ->  ( A +e 0 )  =  ( +oo +e 0 ) )
14 id 19 . . . 4  |-  ( A  = +oo  ->  A  = +oo )
1512, 13, 143eqtr4a 2236 . . 3  |-  ( A  = +oo  ->  ( A +e 0 )  =  A )
16 renepnf 8005 . . . . . 6  |-  ( 0  e.  RR  ->  0  =/= +oo )
172, 16ax-mp 5 . . . . 5  |-  0  =/= +oo
18 xaddmnf2 9849 . . . . 5  |-  ( ( 0  e.  RR*  /\  0  =/= +oo )  ->  ( -oo +e 0 )  = -oo )
198, 17, 18mp2an 426 . . . 4  |-  ( -oo +e 0 )  = -oo
20 oveq1 5882 . . . 4  |-  ( A  = -oo  ->  ( A +e 0 )  =  ( -oo +e 0 ) )
21 id 19 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
2219, 20, 213eqtr4a 2236 . . 3  |-  ( A  = -oo  ->  ( A +e 0 )  =  A )
237, 15, 223jaoi 1303 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A +e 0 )  =  A )
241, 23sylbi 121 1  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 977    = wceq 1353    e. wcel 2148    =/= wne 2347  (class class class)co 5875   RRcr 7810   0cc0 7811    + caddc 7814   +oocpnf 7989   -oocmnf 7990   RR*cxr 7991   +ecxad 9770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908  ax-0id 7919  ax-rnegex 7920
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-xadd 9773
This theorem is referenced by:  xaddid2  9863  xaddid1d  9864  xnn0xadd0  9867  xpncan  9871  psmetsym  13832  psmetge0  13834  xmetge0  13868  xmetsym  13871
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