Theorem xaddval 10226

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 8362	. . . . . 6 |- 0 e. RR*
2	1	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> 0 e. RR* )
3		pnfxr 8368	. . . . . 6 |- +oo e. RR*
4	3	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> +oo e. RR* )
5		xrmnfdc 10224	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
6	5	adantl 277	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = -oo )
7	2, 4, 6	ifcldcd 3675	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = -oo , 0 , +oo ) e. RR* )
8	7	adantr 276	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = +oo ) -> if ( B = -oo , 0 , +oo ) e. RR* )
9		mnfxr 8372	. . . . . . 7 |- -oo e. RR*
10	9	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo e. RR* )
11		xrpnfdc 10223	. . . . . . 7 |- ( B e. RR* -> DECID B = +oo )
12	11	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = +oo )
13	2, 10, 12	ifcldcd 3675	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , 0 , -oo ) e. RR* )
14	13	ad2antrr 492	. . . 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 548	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = +oo )
18		simpl 109	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
19	18	ad4antr 498	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR* )
20		simpllr 540	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. A = -oo )
21	20	neqned 2427	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A =/= -oo )
22		xrnemnf 10158	. . . . . . . . . . 11 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
23	22	biimpi 120	. . . . . . . . . 10 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A e. RR \/ A = +oo ) )
24	19, 21, 23	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A e. RR \/ A = +oo ) )
25	17, 24	ecased 1390	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> A e. RR )
26		simplr 533	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = +oo )
27		simpr 110	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
28	27	ad4antr 498	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR* )
29		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
30	29	neqned 2427	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B =/= -oo )
31		xrnemnf 10158	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
32	31	biimpi 120	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B e. RR \/ B = +oo ) )
33	28, 30, 32	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( B e. RR \/ B = +oo ) )
34	26, 33	ecased 1390	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> B e. RR )
35	25, 34	readdcld 8345	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR )
36	35	rexrd 8365	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A + B ) e. RR* )
37	6	ad3antrrr 496	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> DECID B = -oo )
38	16, 36, 37	ifcldadc 3667	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) /\ -. B = +oo ) -> if ( B = -oo , -oo , ( A + B ) ) e. RR* )
39	12	ad2antrr 492	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) /\ -. A = -oo ) -> DECID B = +oo )
40	15, 38, 39	ifcldadc 3667	. . . 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 10224	. . . . 5 |- ( A e. RR* -> DECID A = -oo )
42	41	ad2antrr 492	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A = +oo ) -> DECID A = -oo )
43	14, 40, 42	ifcldadc 3667	. . 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 10223	. . . 4 |- ( A e. RR* -> DECID A = +oo )
45	44	adantr 276	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> DECID A = +oo )
46	8, 43, 45	ifcldadc 3667	. 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 109	. . . . 5 |- ( ( x = A /\ y = B ) -> x = A )
48	47	eqeq1d 2247	. . . 4 |- ( ( x = A /\ y = B ) -> ( x = +oo <-> A = +oo ) )
49		simpr 110	. . . . . 6 |- ( ( x = A /\ y = B ) -> y = B )
50	49	eqeq1d 2247	. . . . 5 |- ( ( x = A /\ y = B ) -> ( y = -oo <-> B = -oo ) )
51	50	ifbid 3659	. . . 4 |- ( ( x = A /\ y = B ) -> if ( y = -oo , 0 , +oo ) = if ( B = -oo , 0 , +oo ) )
52	47	eqeq1d 2247	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x = -oo <-> A = -oo ) )
53	49	eqeq1d 2247	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( y = +oo <-> B = +oo ) )
54	53	ifbid 3659	. . . . 5 |- ( ( x = A /\ y = B ) -> if ( y = +oo , 0 , -oo ) = if ( B = +oo , 0 , -oo ) )
55		oveq12 6084	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( x + y ) = ( A + B ) )
56	50, 55	ifbieq2d 3662	. . . . . 6 |- ( ( x = A /\ y = B ) -> if ( y = -oo , -oo , ( x + y ) ) = if ( B = -oo , -oo , ( A + B ) ) )
57	53, 56	ifbieq2d 3662	. . . . 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 3664	. . . 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 3664	. . 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 10154	. . 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 6208	. 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 1379	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 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   ifcif 3635  (class class class)co 6075   RRcr 8168   0cc0 8169   + caddc 8172   +oocpnf 8347   -oocmnf 8348   RR*cxr 8349   +ecxad 10151

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-rnegex 8278

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-xadd 10154

This theorem is referenced by:  xaddpnf1  10227  xaddpnf2  10228  xaddmnf1  10229  xaddmnf2  10230  pnfaddmnf  10231  mnfaddpnf  10232  rexadd  10233