Theorem xadddi 12125

Description: Distributive property for extended real addition and multiplication. Like xaddass 12079, this has an unusual domain of correctness due to counterexamples like ( +oo x. ( 2 - 1 ) ) = -oo =/= ( ( +oo x. 2 ) - ( +oo x. 1 ) ) = ( +oo - +oo ) = 0. In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

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

Proof of Theorem xadddi

Step	Hyp	Ref	Expression
1		xadddilem 12124	. 2 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 < A ) -> ( A xe ( B +e C ) ) = ( ( A xe B ) +e ( A xe C ) ) )
2		simpl2 1065	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> B e. RR* )
3		simpl3 1066	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> C e. RR* )
4		xaddcl 12070	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
5	2, 3, 4	syl2anc 693	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( B +e C ) e. RR* )
6		xmul02 12098	. . . . 5 |- ( ( B +e C ) e. RR* -> ( 0 xe ( B +e C ) ) = 0 )
7	5, 6	syl 17	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 xe ( B +e C ) ) = 0 )
8		0xr 10086	. . . . 5 |- 0 e. RR*
9		xaddid1 12072	. . . . 5 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
10	8, 9	ax-mp 5	. . . 4 |- ( 0 +e 0 ) = 0
11	7, 10	syl6eqr 2674	. . 3 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 xe ( B +e C ) ) = ( 0 +e 0 ) )
12		simpr 477	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> 0 = A )
13	12	oveq1d 6665	. . 3 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 xe ( B +e C ) ) = ( A xe ( B +e C ) ) )
14		xmul02 12098	. . . . . 6 |- ( B e. RR* -> ( 0 xe B ) = 0 )
15	2, 14	syl 17	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 xe B ) = 0 )
16	12	oveq1d 6665	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 xe B ) = ( A xe B ) )
17	15, 16	eqtr3d 2658	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> 0 = ( A xe B ) )
18		xmul02 12098	. . . . . 6 |- ( C e. RR* -> ( 0 xe C ) = 0 )
19	3, 18	syl 17	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 xe C ) = 0 )
20	12	oveq1d 6665	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 xe C ) = ( A xe C ) )
21	19, 20	eqtr3d 2658	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> 0 = ( A xe C ) )
22	17, 21	oveq12d 6668	. . 3 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( 0 +e 0 ) = ( ( A xe B ) +e ( A xe C ) ) )
23	11, 13, 22	3eqtr3d 2664	. 2 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 = A ) -> ( A xe ( B +e C ) ) = ( ( A xe B ) +e ( A xe C ) ) )
24		simp1 1061	. . . . . . 7 |- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> A e. RR )
25	24	adantr 481	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> A e. RR )
26		rexneg 12042	. . . . . . 7 |- ( A e. RR -> -e A = -u A )
27		renegcl 10344	. . . . . . 7 |- ( A e. RR -> -u A e. RR )
28	26, 27	eqeltrd 2701	. . . . . 6 |- ( A e. RR -> -e A e. RR )
29	25, 28	syl 17	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> -e A e. RR )
30		simpl2 1065	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> B e. RR* )
31		simpl3 1066	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> C e. RR* )
32	24	rexrd 10089	. . . . . . 7 |- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
33		xlt0neg1 12050	. . . . . . 7 |- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) )
34	32, 33	syl 17	. . . . . 6 |- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> ( A < 0 <-> 0 < -e A ) )
35	34	biimpa 501	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> 0 < -e A )
36		xadddilem 12124	. . . . 5 |- ( ( ( -e A e. RR /\ B e. RR* /\ C e. RR* ) /\ 0 < -e A ) -> ( -e A xe ( B +e C ) ) = ( ( -e A xe B ) +e ( -e A xe C ) ) )
37	29, 30, 31, 35, 36	syl31anc 1329	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( -e A xe ( B +e C ) ) = ( ( -e A xe B ) +e ( -e A xe C ) ) )
38	32	adantr 481	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> A e. RR* )
39	30, 31, 4	syl2anc 693	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( B +e C ) e. RR* )
40		xmulneg1 12099	. . . . 5 |- ( ( A e. RR* /\ ( B +e C ) e. RR* ) -> ( -e A xe ( B +e C ) ) = -e ( A xe ( B +e C ) ) )
41	38, 39, 40	syl2anc 693	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( -e A xe ( B +e C ) ) = -e ( A xe ( B +e C ) ) )
42		xmulneg1 12099	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( -e A xe B ) = -e ( A xe B ) )
43	38, 30, 42	syl2anc 693	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( -e A xe B ) = -e ( A xe B ) )
44		xmulneg1 12099	. . . . . . 7 |- ( ( A e. RR* /\ C e. RR* ) -> ( -e A xe C ) = -e ( A xe C ) )
45	38, 31, 44	syl2anc 693	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( -e A xe C ) = -e ( A xe C ) )
46	43, 45	oveq12d 6668	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( ( -e A xe B ) +e ( -e A xe C ) ) = ( -e ( A xe B ) +e -e ( A xe C ) ) )
47		xmulcl 12103	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A xe B ) e. RR* )
48	38, 30, 47	syl2anc 693	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( A xe B ) e. RR* )
49		xmulcl 12103	. . . . . . 7 |- ( ( A e. RR* /\ C e. RR* ) -> ( A xe C ) e. RR* )
50	38, 31, 49	syl2anc 693	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( A xe C ) e. RR* )
51		xnegdi 12078	. . . . . 6 |- ( ( ( A xe B ) e. RR* /\ ( A xe C ) e. RR* ) -> -e ( ( A xe B ) +e ( A xe C ) ) = ( -e ( A xe B ) +e -e ( A xe C ) ) )
52	48, 50, 51	syl2anc 693	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> -e ( ( A xe B ) +e ( A xe C ) ) = ( -e ( A xe B ) +e -e ( A xe C ) ) )
53	46, 52	eqtr4d 2659	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( ( -e A xe B ) +e ( -e A xe C ) ) = -e ( ( A xe B ) +e ( A xe C ) ) )
54	37, 41, 53	3eqtr3d 2664	. . 3 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> -e ( A xe ( B +e C ) ) = -e ( ( A xe B ) +e ( A xe C ) ) )
55		xmulcl 12103	. . . . 5 |- ( ( A e. RR* /\ ( B +e C ) e. RR* ) -> ( A xe ( B +e C ) ) e. RR* )
56	38, 39, 55	syl2anc 693	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( A xe ( B +e C ) ) e. RR* )
57		xaddcl 12070	. . . . 5 |- ( ( ( A xe B ) e. RR* /\ ( A xe C ) e. RR* ) -> ( ( A xe B ) +e ( A xe C ) ) e. RR* )
58	48, 50, 57	syl2anc 693	. . . 4 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( ( A xe B ) +e ( A xe C ) ) e. RR* )
59		xneg11 12046	. . . 4 |- ( ( ( A xe ( B +e C ) ) e. RR* /\ ( ( A xe B ) +e ( A xe C ) ) e. RR* ) -> ( -e ( A xe ( B +e C ) ) = -e ( ( A xe B ) +e ( A xe C ) ) <-> ( A xe ( B +e C ) ) = ( ( A xe B ) +e ( A xe C ) ) ) )
60	56, 58, 59	syl2anc 693	. . 3 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( -e ( A xe ( B +e C ) ) = -e ( ( A xe B ) +e ( A xe C ) ) <-> ( A xe ( B +e C ) ) = ( ( A xe B ) +e ( A xe C ) ) ) )
61	54, 60	mpbid 222	. 2 |- ( ( ( A e. RR /\ B e. RR* /\ C e. RR* ) /\ A < 0 ) -> ( A xe ( B +e C ) ) = ( ( A xe B ) +e ( A xe C ) ) )
62		0re 10040	. . 3 |- 0 e. RR
63		lttri4 10122	. . 3 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A \/ 0 = A \/ A < 0 ) )
64	62, 24, 63	sylancr 695	. 2 |- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> ( 0 < A \/ 0 = A \/ A < 0 ) )
65	1, 23, 61, 64	mpjao3dan 1395	1 |- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> ( A xe ( B +e C ) ) = ( ( A xe B ) +e ( A xe C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   RR*cxr 10073   < clt 10074   -ucneg 10267   -ecxne 11943   +ecxad 11944   xecxmu 11945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-xneg 11946  df-xadd 11947  df-xmul 11948

This theorem is referenced by:  xadddir  12126  xadddi2  12127