Theorem xaddid1 10195

Description: Extended real version of addrid 8411. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddid1	|- ( A e. RR* -> ( A +e 0 ) = A )

Proof of Theorem xaddid1

Step	Hyp	Ref	Expression
1		elxr 10109	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		0re 8274	. . . . 5 |- 0 e. RR
3		rexadd 10185	. . . . 5 |- ( ( A e. RR /\ 0 e. RR ) -> ( A +e 0 ) = ( A + 0 ) )
4	2, 3	mpan2 425	. . . 4 |- ( A e. RR -> ( A +e 0 ) = ( A + 0 ) )
5		recn 8260	. . . . 5 |- ( A e. RR -> A e. CC )
6	5	addridd 8422	. . . 4 |- ( A e. RR -> ( A + 0 ) = A )
7	4, 6	eqtrd 2265	. . 3 |- ( A e. RR -> ( A +e 0 ) = A )
8		0xr 8320	. . . . 5 |- 0 e. RR*
9		renemnf 8322	. . . . . 6 |- ( 0 e. RR -> 0 =/= -oo )
10	2, 9	ax-mp 5	. . . . 5 |- 0 =/= -oo
11		xaddpnf2 10180	. . . . 5 |- ( ( 0 e. RR* /\ 0 =/= -oo ) -> ( +oo +e 0 ) = +oo )
12	8, 10, 11	mp2an 426	. . . 4 |- ( +oo +e 0 ) = +oo
13		oveq1 6057	. . . 4 |- ( A = +oo -> ( A +e 0 ) = ( +oo +e 0 ) )
14		id 19	. . . 4 |- ( A = +oo -> A = +oo )
15	12, 13, 14	3eqtr4a 2291	. . 3 |- ( A = +oo -> ( A +e 0 ) = A )
16		renepnf 8321	. . . . . 6 |- ( 0 e. RR -> 0 =/= +oo )
17	2, 16	ax-mp 5	. . . . 5 |- 0 =/= +oo
18		xaddmnf2 10182	. . . . 5 |- ( ( 0 e. RR* /\ 0 =/= +oo ) -> ( -oo +e 0 ) = -oo )
19	8, 17, 18	mp2an 426	. . . 4 |- ( -oo +e 0 ) = -oo
20		oveq1 6057	. . . 4 |- ( A = -oo -> ( A +e 0 ) = ( -oo +e 0 ) )
21		id 19	. . . 4 |- ( A = -oo -> A = -oo )
22	19, 20, 21	3eqtr4a 2291	. . 3 |- ( A = -oo -> ( A +e 0 ) = A )
23	7, 15, 22	3jaoi 1340	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A +e 0 ) = A )
24	1, 23	sylbi 121	1 |- ( A e. RR* -> ( A +e 0 ) = A )

Syntax hints:   -> wi 4   \/ w3o 1004   = wceq 1398   e. wcel 2203   =/= wne 2412  (class class class)co 6050   RRcr 8126   0cc0 8127   + caddc 8130   +oocpnf 8305   -oocmnf 8306   RR*cxr 8307   +ecxad 10103

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-0id 8235  ax-rnegex 8236

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-xadd 10106

This theorem is referenced by:  xaddid2  10196  xaddid1d  10197  xnn0xadd0  10200  xpncan  10204  psmetsym  15194  psmetge0  15196  xmetge0  15230  xmetsym  15233