Theorem xpncan 9946

Description: Extended real version of pncan 8232. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xpncan	|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )

Proof of Theorem xpncan

Step	Hyp	Ref	Expression
1		rexneg 9905	. . . 4 |- ( B e. RR -> -e B = -u B )
2	1	adantl 277	. . 3 |- ( ( A e. RR* /\ B e. RR ) -> -e B = -u B )
3	2	oveq2d 5938	. 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = ( ( A +e B ) +e -u B ) )
4		renegcl 8287	. . . . . 6 |- ( B e. RR -> -u B e. RR )
5	4	ad2antlr 489	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> -u B e. RR )
6		rexr 8072	. . . . . 6 |- ( -u B e. RR -> -u B e. RR* )
7		renepnf 8074	. . . . . 6 |- ( -u B e. RR -> -u B =/= +oo )
8		xaddmnf2 9924	. . . . . 6 |- ( ( -u B e. RR* /\ -u B =/= +oo ) -> ( -oo +e -u B ) = -oo )
9	6, 7, 8	syl2anc 411	. . . . 5 |- ( -u B e. RR -> ( -oo +e -u B ) = -oo )
10	5, 9	syl 14	. . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( -oo +e -u B ) = -oo )
11		oveq1 5929	. . . . . 6 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
12		rexr 8072	. . . . . . . 8 |- ( B e. RR -> B e. RR* )
13		renepnf 8074	. . . . . . . 8 |- ( B e. RR -> B =/= +oo )
14		xaddmnf2 9924	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
15	12, 13, 14	syl2anc 411	. . . . . . 7 |- ( B e. RR -> ( -oo +e B ) = -oo )
16	15	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> ( -oo +e B ) = -oo )
17	11, 16	sylan9eqr 2251	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( A +e B ) = -oo )
18	17	oveq1d 5937	. . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( ( A +e B ) +e -u B ) = ( -oo +e -u B ) )
19		simpr 110	. . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> A = -oo )
20	10, 18, 19	3eqtr4d 2239	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( ( A +e B ) +e -u B ) = A )
21		simpll 527	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> A e. RR* )
22		simpr 110	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> A =/= -oo )
23	12	ad2antlr 489	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR* )
24		renemnf 8075	. . . . . 6 |- ( B e. RR -> B =/= -oo )
25	24	ad2antlr 489	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B =/= -oo )
26	4	ad2antlr 489	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> -u B e. RR )
27	26, 6	syl 14	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> -u B e. RR* )
28		renemnf 8075	. . . . . 6 |- ( -u B e. RR -> -u B =/= -oo )
29	26, 28	syl 14	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> -u B =/= -oo )
30		xaddass 9944	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( -u B e. RR* /\ -u B =/= -oo ) ) -> ( ( A +e B ) +e -u B ) = ( A +e ( B +e -u B ) ) )
31	21, 22, 23, 25, 27, 29, 30	syl222anc 1265	. . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = ( A +e ( B +e -u B ) ) )
32		simplr 528	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR )
33	32, 26	rexaddd 9929	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = ( B + -u B ) )
34	32	recnd 8055	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. CC )
35	34	negidd 8327	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B + -u B ) = 0 )
36	33, 35	eqtrd 2229	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = 0 )
37	36	oveq2d 5938	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = ( A +e 0 ) )
38		xaddid1 9937	. . . . . 6 |- ( A e. RR* -> ( A +e 0 ) = A )
39	38	ad2antrr 488	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e 0 ) = A )
40	37, 39	eqtrd 2229	. . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = A )
41	31, 40	eqtrd 2229	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = A )
42		xrmnfdc 9918	. . . . . 6 |- ( A e. RR* -> DECID A = -oo )
43		exmiddc 837	. . . . . 6 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
44	42, 43	syl 14	. . . . 5 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
45		df-ne 2368	. . . . . 6 |- ( A =/= -oo <-> -. A = -oo )
46	45	orbi2i 763	. . . . 5 |- ( ( A = -oo \/ A =/= -oo ) <-> ( A = -oo \/ -. A = -oo ) )
47	44, 46	sylibr 134	. . . 4 |- ( A e. RR* -> ( A = -oo \/ A =/= -oo ) )
48	47	adantr 276	. . 3 |- ( ( A e. RR* /\ B e. RR ) -> ( A = -oo \/ A =/= -oo ) )
49	20, 41, 48	mpjaodan 799	. 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -u B ) = A )
50	3, 49	eqtrd 2229	1 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367  (class class class)co 5922   RRcr 7878   0cc0 7879   + caddc 7882   +oocpnf 8058   -oocmnf 8059   RR*cxr 8060   -ucneg 8198   -ecxne 9844   +ecxad 9845

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-xr 8065  df-sub 8199  df-neg 8200  df-xneg 9847  df-xadd 9848

This theorem is referenced by:  xnpcan  9947  xleadd1  9950  xrmaxaddlem  11425