Theorem xaddval 9802

Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddval	|- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )

Proof of Theorem xaddval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0xr 7966	. . . . . 6 |- 0 e. RR*
2	1	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> 0 e. RR* )
3		pnfxr 7972	. . . . . 6 |- +oo e. RR*
4	3	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> +oo e. RR* )
5		xrmnfdc 9800	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
6	5	adantl 275	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = -oo )
7	2, 4, 6	ifcldcd 3561	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = -oo , 0 , +oo ) e. RR* )
8	7	adantr 274	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = +oo ) -> if ( B = -oo , 0 , +oo ) e. RR* )
9		mnfxr 7976	. . . . . . 7 |- -oo e. RR*
10	9	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo e. RR* )
11		xrpnfdc 9799	. . . . . . 7 |- ( B e. RR* -> DECID B = +oo )
12	11	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = +oo )
13	2, 10, 12	ifcldcd 3561	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , 0 , -oo ) e. RR* )
14	13	ad2antrr 485	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ A = -oo ) -> if ( B = +oo , 0 , -oo ) e. RR* )
15	3	a1i 9	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ B = +oo ) -> +oo e. RR* )
16	9	a1i 9	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ B = -oo ) -> -oo e. RR* )
17		simp-4r 537	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = +oo )
18		simpl 108	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
19	18	ad4antr 491	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR* )
20		simpllr 529	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = -oo )
21	20	neqned 2347	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A =/= -oo )
22		xrnemnf 9734	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
23	22	biimpi 119	. . . . . . . . . 10 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A e. RR \/ A = +oo ) )
24	19, 21, 23	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A e. RR \/ A = +oo ) )
25	17, 24	ecased 1344	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR )
26		simplr 525	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = +oo )
27		simpr 109	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
28	27	ad4antr 491	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR* )
29		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
30	29	neqned 2347	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B =/= -oo )
31		xrnemnf 9734	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
32	31	biimpi 119	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B e. RR \/ B = +oo ) )
33	28, 30, 32	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( B e. RR \/ B = +oo ) )
34	26, 33	ecased 1344	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR )
35	25, 34	readdcld 7949	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR )
36	35	rexrd 7969	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR* )
37	6	ad3antrrr 489	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> DECID B = -oo )
38	16, 36, 37	ifcldadc 3555	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> if ( B = -oo , -oo , ( A + B ) ) e. RR* )
39	12	ad2antrr 485	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) -> DECID B = +oo )
40	15, 38, 39	ifcldadc 3555	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) -> if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) e. RR* )
41		xrmnfdc 9800	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
42	41	ad2antrr 485	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) -> DECID A = -oo )
43	14, 40, 42	ifcldadc 3555	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) -> if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) e. RR* )
44		xrpnfdc 9799	. . . 4 |- ( A e. RR* -> DECID A = +oo )
45	44	adantr 274	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> DECID A = +oo )
46	8, 43, 45	ifcldadc 3555	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) e. RR* )
47		simpl 108	. . . . 5 |- ( ( x = A /\ y = B ) -> x = A )
48	47	eqeq1d 2179	. . . 4 |- ( ( x = A /\ y = B ) -> ( x = +oo <-> A = +oo ) )
49		simpr 109	. . . . . 6 |- ( ( x = A /\ y = B ) -> y = B )
50	49	eqeq1d 2179	. . . . 5 |- ( ( x = A /\ y = B ) -> ( y = -oo <-> B = -oo ) )
51	50	ifbid 3547	. . . 4 |- ( ( x = A /\ y = B ) -> if ( y = -oo , 0 , +oo ) = if ( B = -oo , 0 , +oo ) )
52	47	eqeq1d 2179	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x = -oo <-> A = -oo ) )
53	49	eqeq1d 2179	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( y = +oo <-> B = +oo ) )
54	53	ifbid 3547	. . . . 5 |- ( ( x = A /\ y = B ) -> if ( y = +oo , 0 , -oo ) = if ( B = +oo , 0 , -oo ) )
55		oveq12 5862	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( x + y ) = ( A + B ) )
56	50, 55	ifbieq2d 3550	. . . . . 6 |- ( ( x = A /\ y = B ) -> if ( y = -oo , -oo , ( x + y ) ) = if ( B = -oo , -oo , ( A + B ) ) )
57	53, 56	ifbieq2d 3550	. . . . 5 |- ( ( x = A /\ y = B ) -> if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) = if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) )
58	52, 54, 57	ifbieq12d 3552	. . . 4 |- ( ( x = A /\ y = B ) -> if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) = if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) )
59	48, 51, 58	ifbieq12d 3552	. . 3 |- ( ( x = A /\ y = B ) -> if ( x = +oo , if ( y = -oo , 0 , +oo ) , if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )
60		df-xadd 9730	. . 3 |- +e = ( x e. RR* , y e. RR* |-> if ( x = +oo , if ( y = -oo , 0 , +oo ) , if ( x = -oo , if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo , if ( y = -oo , -oo , ( x + y ) ) ) ) ) )
61	59, 60	ovmpoga 5982	. 2 |- ( ( A e. RR* /\ B e. RR* /\ if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) e. RR* ) -> ( A +e B ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )
62	46, 61	mpd3an3 1333	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = if ( A = +oo , if ( B = -oo , 0 , +oo ) , if ( A = -oo , if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   ifcif 3526  (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-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:  xaddpnf1  9803  xaddpnf2  9804  xaddmnf1  9805  xaddmnf2  9806  pnfaddmnf  9807  mnfaddpnf  9808  rexadd  9809