Theorem xaddid1 10096

Description: Extended real version of addrid 8316. (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 10010	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		0re 8178	. . . . 5 |- 0 e. RR
3		rexadd 10086	. . . . 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 8164	. . . . 5 |- ( A e. RR -> A e. CC )
6	5	addridd 8327	. . . 4 |- ( A e. RR -> ( A + 0 ) = A )
7	4, 6	eqtrd 2264	. . 3 |- ( A e. RR -> ( A +e 0 ) = A )
8		0xr 8225	. . . . 5 |- 0 e. RR*
9		renemnf 8227	. . . . . 6 |- ( 0 e. RR -> 0 =/= -oo )
10	2, 9	ax-mp 5	. . . . 5 |- 0 =/= -oo
11		xaddpnf2 10081	. . . . 5 |- ( ( 0 e. RR* /\ 0 =/= -oo ) -> ( +oo +e 0 ) = +oo )
12	8, 10, 11	mp2an 426	. . . 4 |- ( +oo +e 0 ) = +oo
13		oveq1 6024	. . . 4 |- ( A = +oo -> ( A +e 0 ) = ( +oo +e 0 ) )
14		id 19	. . . 4 |- ( A = +oo -> A = +oo )
15	12, 13, 14	3eqtr4a 2290	. . 3 |- ( A = +oo -> ( A +e 0 ) = A )
16		renepnf 8226	. . . . . 6 |- ( 0 e. RR -> 0 =/= +oo )
17	2, 16	ax-mp 5	. . . . 5 |- 0 =/= +oo
18		xaddmnf2 10083	. . . . 5 |- ( ( 0 e. RR* /\ 0 =/= +oo ) -> ( -oo +e 0 ) = -oo )
19	8, 17, 18	mp2an 426	. . . 4 |- ( -oo +e 0 ) = -oo
20		oveq1 6024	. . . 4 |- ( A = -oo -> ( A +e 0 ) = ( -oo +e 0 ) )
21		id 19	. . . 4 |- ( A = -oo -> A = -oo )
22	19, 20, 21	3eqtr4a 2290	. . 3 |- ( A = -oo -> ( A +e 0 ) = A )
23	7, 15, 22	3jaoi 1339	. 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 1003   = wceq 1397   e. wcel 2202   =/= wne 2402  (class class class)co 6017   RRcr 8030   0cc0 8031   + caddc 8034   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   +ecxad 10004

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128  ax-0id 8139  ax-rnegex 8140

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-xadd 10007

This theorem is referenced by:  xaddid2  10097  xaddid1d  10098  xnn0xadd0  10101  xpncan  10105  psmetsym  15052  psmetge0  15054  xmetge0  15088  xmetsym  15091