xrge0adddir - Mathbox for Thierry Arnoux

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Theorem xrge0adddir 29692

Description: Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)

**Assertion**
Ref	Expression
xrge0adddir	$/\$ $/\$

**Proof of Theorem xrge0adddir**
Step	Hyp	Ref	Expression
1		iccssxr 12256	. . . 4
2		simpl1 1064	. . . 4 $/\$ $/\$ $/\$
3	1, 2	sseldi 3601	. . 3 $/\$ $/\$ $/\$
4		simpl2 1065	. . . 4 $/\$ $/\$ $/\$
5	1, 4	sseldi 3601	. . 3 $/\$ $/\$ $/\$
6		rge0ssre 12280	. . . 4
7		simpr 477	. . . 4 $/\$ $/\$ $/\$
8	6, 7	sseldi 3601	. . 3 $/\$ $/\$ $/\$
9		xadddir 12126	. . 3 $/\$ $/\$
10	3, 5, 8, 9	syl3anc 1326	. 2 $/\$ $/\$ $/\$
11		simpll1 1100	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
12	1, 11	sseldi 3601	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
13		simpll2 1101	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14	1, 13	sseldi 3601	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
15	12, 14	xaddcld 12131	. . . . . 6 $/\$ $/\$ $/\$ $/\$
16		simpr 477	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
17		xrge0addgt0 29691	. . . . . . 7 $/\$ $/\$
18	11, 13, 16, 17	syl21anc 1325	. . . . . 6 $/\$ $/\$ $/\$ $/\$
19		xmulpnf1 12104	. . . . . 6 $/\$
20	15, 18, 19	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$
21		oveq2 6658	. . . . . 6
22	21	ad2antlr 763	. . . . 5 $/\$ $/\$ $/\$ $/\$
23		simpll3 1102	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
24		ge0xmulcl 12287	. . . . . . . 8 $/\$
25	13, 23, 24	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	1, 25	sseldi 3601	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		xrge0neqmnf 12276	. . . . . . 7
28	25, 27	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$
29		xaddpnf2 12058	. . . . . 6 $/\$
30	26, 28, 29	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$
31	20, 22, 30	3eqtr4d 2666	. . . 4 $/\$ $/\$ $/\$ $/\$
32		oveq2 6658	. . . . . . 7
33	32	ad2antlr 763	. . . . . 6 $/\$ $/\$ $/\$ $/\$
34		xmulpnf1 12104	. . . . . . 7 $/\$
35	12, 16, 34	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$
36	33, 35	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$
37	36	oveq1d 6665	. . . 4 $/\$ $/\$ $/\$ $/\$
38	31, 37	eqtr4d 2659	. . 3 $/\$ $/\$ $/\$ $/\$
39		simpll3 1102	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	1, 39	sseldi 3601	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41		xmul02 12098	. . . . . . 7
42	40, 41	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$
43	42	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$
44		oveq1 6657	. . . . . . 7
45	44	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$
46	45	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$
47		simpll2 1101	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	1, 47	sseldi 3601	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	48, 40	xmulcld 12132	. . . . . 6 $/\$ $/\$ $/\$ $/\$
50		xaddid2 12073	. . . . . 6
51	49, 50	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$
52	43, 46, 51	3eqtr3d 2664	. . . 4 $/\$ $/\$ $/\$ $/\$
53		xaddid2 12073	. . . . . 6
54	48, 53	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$
55	54	oveq1d 6665	. . . 4 $/\$ $/\$ $/\$ $/\$
56		oveq1 6657	. . . . . 6
57	56	oveq1d 6665	. . . . 5
58	57	adantl 482	. . . 4 $/\$ $/\$ $/\$ $/\$
59	52, 55, 58	3eqtr2rd 2663	. . 3 $/\$ $/\$ $/\$ $/\$
60		0xr 10086	. . . . 5
61	60	a1i 11	. . . 4 $/\$ $/\$ $/\$
62		simpl1 1064	. . . . 5 $/\$ $/\$ $/\$
63	1, 62	sseldi 3601	. . . 4 $/\$ $/\$ $/\$
64		pnfxr 10092	. . . . . 6
65	64	a1i 11	. . . . 5 $/\$ $/\$ $/\$
66		iccgelb 12230	. . . . 5 $/\$ $/\$
67	61, 65, 62, 66	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$
68		xrleloe 11977	. . . . 5 $/\$ $\/$
69	68	biimpa 501	. . . 4 $/\$ $/\$ $\/$
70	61, 63, 67, 69	syl21anc 1325	. . 3 $/\$ $/\$ $/\$ $\/$
71	38, 59, 70	mpjaodan 827	. 2 $/\$ $/\$ $/\$
72		0lepnf 11966	. . . . 5
73		eliccelico 29539	. . . . 5 $/\$ $/\$ $\/$
74	60, 64, 72, 73	mp3an 1424	. . . 4 $\/$
75	74	3anbi3i 1255	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
76	75	simp3bi 1078	. 2 $/\$ $/\$ $\/$
77	10, 71, 76	mpjaodan 827	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794 class class class wbr 4653 (class class class)co 6650

cr 9935

cc0 9936

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

cle 10075

cxad 11944

cxmu 11945

cico 12177

cicc 12178

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 ax-pre-mulgt0 10013

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 df-ico 12181 df-icc 12182

This theorem is referenced by: xrge0adddi 29693 xrge0slmod 29844 esummulc1 30143

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