Theorem xaddid1 9819

Description: Extended real version of addid1 8057. (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 9733	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		0re 7920	. . . . 5 |- 0 e. RR
3		rexadd 9809	. . . . 5 |- ( ( A e. RR /\ 0 e. RR ) -> ( A +e 0 ) = ( A + 0 ) )
4	2, 3	mpan2 423	. . . 4 |- ( A e. RR -> ( A +e 0 ) = ( A + 0 ) )
5		recn 7907	. . . . 5 |- ( A e. RR -> A e. CC )
6	5	addid1d 8068	. . . 4 |- ( A e. RR -> ( A + 0 ) = A )
7	4, 6	eqtrd 2203	. . 3 |- ( A e. RR -> ( A +e 0 ) = A )
8		0xr 7966	. . . . 5 |- 0 e. RR*
9		renemnf 7968	. . . . . 6 |- ( 0 e. RR -> 0 =/= -oo )
10	2, 9	ax-mp 5	. . . . 5 |- 0 =/= -oo
11		xaddpnf2 9804	. . . . 5 |- ( ( 0 e. RR* /\ 0 =/= -oo ) -> ( +oo +e 0 ) = +oo )
12	8, 10, 11	mp2an 424	. . . 4 |- ( +oo +e 0 ) = +oo
13		oveq1 5860	. . . 4 |- ( A = +oo -> ( A +e 0 ) = ( +oo +e 0 ) )
14		id 19	. . . 4 |- ( A = +oo -> A = +oo )
15	12, 13, 14	3eqtr4a 2229	. . 3 |- ( A = +oo -> ( A +e 0 ) = A )
16		renepnf 7967	. . . . . 6 |- ( 0 e. RR -> 0 =/= +oo )
17	2, 16	ax-mp 5	. . . . 5 |- 0 =/= +oo
18		xaddmnf2 9806	. . . . 5 |- ( ( 0 e. RR* /\ 0 =/= +oo ) -> ( -oo +e 0 ) = -oo )
19	8, 17, 18	mp2an 424	. . . 4 |- ( -oo +e 0 ) = -oo
20		oveq1 5860	. . . 4 |- ( A = -oo -> ( A +e 0 ) = ( -oo +e 0 ) )
21		id 19	. . . 4 |- ( A = -oo -> A = -oo )
22	19, 20, 21	3eqtr4a 2229	. . 3 |- ( A = -oo -> ( A +e 0 ) = A )
23	7, 15, 22	3jaoi 1298	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A +e 0 ) = A )
24	1, 23	sylbi 120	1 |- ( A e. RR* -> ( A +e 0 ) = A )

Syntax hints:   -> wi 4   \/ w3o 972   = wceq 1348   e. wcel 2141   =/= wne 2340  (class class class)co 5853   RRcr 7773   0cc0 7774   + caddc 7777   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   +ecxad 9727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-0id 7882  ax-rnegex 7883

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-xadd 9730

This theorem is referenced by:  xaddid2  9820  xaddid1d  9821  xnn0xadd0  9824  xpncan  9828  psmetsym  13123  psmetge0  13125  xmetge0  13159  xmetsym  13162