Theorem xrge0slmod 29844

Description: The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.)

Hypotheses

Ref	Expression
xrge0slmod.1	|- G = ( RR*s ↾s ( 0 [,] +oo ) )
xrge0slmod.2	|- W = ( G ↾v ( 0 [,) +oo ) )

Assertion

Ref	Expression
xrge0slmod	|- W e. SLMod

Proof of Theorem xrge0slmod

Dummy variables r q w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrge0slmod.1	. . . 4 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
2		xrge0cmn 19788	. . . 4 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
3	1, 2	eqeltri 2697	. . 3 |- G e. CMnd
4		ovex 6678	. . . 4 |- ( 0 [,) +oo ) e. _V
5		xrge0slmod.2	. . . . 5 |- W = ( G ↾v ( 0 [,) +oo ) )
6	5	resvcmn 29838	. . . 4 |- ( ( 0 [,) +oo ) e. _V -> ( G e. CMnd <-> W e. CMnd ) )
7	4, 6	ax-mp 5	. . 3 |- ( G e. CMnd <-> W e. CMnd )
8	3, 7	mpbi 220	. 2 |- W e. CMnd
9		rge0srg 19817	. 2 |- (ℂfld ↾s ( 0 [,) +oo ) ) e. SRing
10		icossicc 12260	. . . . . . . 8 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
11		simplr 792	. . . . . . . 8 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> r e. ( 0 [,) +oo ) )
12	10, 11	sseldi 3601	. . . . . . 7 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> r e. ( 0 [,] +oo ) )
13		simprr 796	. . . . . . 7 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> w e. ( 0 [,] +oo ) )
14		ge0xmulcl 12287	. . . . . . 7 |- ( ( r e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) -> ( r xe w ) e. ( 0 [,] +oo ) )
15	12, 13, 14	syl2anc 693	. . . . . 6 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( r xe w ) e. ( 0 [,] +oo ) )
16		simprl 794	. . . . . . 7 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> x e. ( 0 [,] +oo ) )
17		xrge0adddi 29693	. . . . . . 7 |- ( ( w e. ( 0 [,] +oo ) /\ x e. ( 0 [,] +oo ) /\ r e. ( 0 [,] +oo ) ) -> ( r xe ( w +e x ) ) = ( ( r xe w ) +e ( r xe x ) ) )
18	13, 16, 12, 17	syl3anc 1326	. . . . . 6 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( r xe ( w +e x ) ) = ( ( r xe w ) +e ( r xe x ) ) )
19		rge0ssre 12280	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
20		simpll 790	. . . . . . . . . 10 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> q e. ( 0 [,) +oo ) )
21	19, 20	sseldi 3601	. . . . . . . . 9 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> q e. RR )
22	19, 11	sseldi 3601	. . . . . . . . 9 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> r e. RR )
23		rexadd 12063	. . . . . . . . 9 |- ( ( q e. RR /\ r e. RR ) -> ( q +e r ) = ( q + r ) )
24	21, 22, 23	syl2anc 693	. . . . . . . 8 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( q +e r ) = ( q + r ) )
25	24	oveq1d 6665	. . . . . . 7 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( q +e r ) xe w ) = ( ( q + r ) xe w ) )
26	10, 20	sseldi 3601	. . . . . . . 8 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> q e. ( 0 [,] +oo ) )
27		xrge0adddir 29692	. . . . . . . 8 |- ( ( q e. ( 0 [,] +oo ) /\ r e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) -> ( ( q +e r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) )
28	26, 12, 13, 27	syl3anc 1326	. . . . . . 7 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( q +e r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) )
29	25, 28	eqtr3d 2658	. . . . . 6 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( q + r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) )
30	15, 18, 29	3jca 1242	. . . . 5 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( r xe w ) e. ( 0 [,] +oo ) /\ ( r xe ( w +e x ) ) = ( ( r xe w ) +e ( r xe x ) ) /\ ( ( q + r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) ) )
31		rexmul 12101	. . . . . . . . 9 |- ( ( q e. RR /\ r e. RR ) -> ( q xe r ) = ( q x. r ) )
32	21, 22, 31	syl2anc 693	. . . . . . . 8 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( q xe r ) = ( q x. r ) )
33	32	oveq1d 6665	. . . . . . 7 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( q xe r ) xe w ) = ( ( q x. r ) xe w ) )
34	21	rexrd 10089	. . . . . . . 8 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> q e. RR* )
35	22	rexrd 10089	. . . . . . . 8 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> r e. RR* )
36		iccssxr 12256	. . . . . . . . 9 |- ( 0 [,] +oo ) C_ RR*
37	36, 13	sseldi 3601	. . . . . . . 8 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> w e. RR* )
38		xmulass 12117	. . . . . . . 8 |- ( ( q e. RR* /\ r e. RR* /\ w e. RR* ) -> ( ( q xe r ) xe w ) = ( q xe ( r xe w ) ) )
39	34, 35, 37, 38	syl3anc 1326	. . . . . . 7 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( q xe r ) xe w ) = ( q xe ( r xe w ) ) )
40	33, 39	eqtr3d 2658	. . . . . 6 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( q x. r ) xe w ) = ( q xe ( r xe w ) ) )
41		xmulid2 12110	. . . . . . 7 |- ( w e. RR* -> ( 1 xe w ) = w )
42	37, 41	syl 17	. . . . . 6 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( 1 xe w ) = w )
43		xmul02 12098	. . . . . . 7 |- ( w e. RR* -> ( 0 xe w ) = 0 )
44	37, 43	syl 17	. . . . . 6 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( 0 xe w ) = 0 )
45	40, 42, 44	3jca 1242	. . . . 5 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( ( q x. r ) xe w ) = ( q xe ( r xe w ) ) /\ ( 1 xe w ) = w /\ ( 0 xe w ) = 0 ) )
46	30, 45	jca 554	. . . 4 |- ( ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,] +oo ) /\ w e. ( 0 [,] +oo ) ) ) -> ( ( ( r xe w ) e. ( 0 [,] +oo ) /\ ( r xe ( w +e x ) ) = ( ( r xe w ) +e ( r xe x ) ) /\ ( ( q + r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) ) /\ ( ( ( q x. r ) xe w ) = ( q xe ( r xe w ) ) /\ ( 1 xe w ) = w /\ ( 0 xe w ) = 0 ) ) )
47	46	ralrimivva 2971	. . 3 |- ( ( q e. ( 0 [,) +oo ) /\ r e. ( 0 [,) +oo ) ) -> A. x e. ( 0 [,] +oo ) A. w e. ( 0 [,] +oo ) ( ( ( r xe w ) e. ( 0 [,] +oo ) /\ ( r xe ( w +e x ) ) = ( ( r xe w ) +e ( r xe x ) ) /\ ( ( q + r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) ) /\ ( ( ( q x. r ) xe w ) = ( q xe ( r xe w ) ) /\ ( 1 xe w ) = w /\ ( 0 xe w ) = 0 ) ) )
48	47	rgen2a 2977	. 2 |- A. q e. ( 0 [,) +oo ) A. r e. ( 0 [,) +oo ) A. x e. ( 0 [,] +oo ) A. w e. ( 0 [,] +oo ) ( ( ( r xe w ) e. ( 0 [,] +oo ) /\ ( r xe ( w +e x ) ) = ( ( r xe w ) +e ( r xe x ) ) /\ ( ( q + r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) ) /\ ( ( ( q x. r ) xe w ) = ( q xe ( r xe w ) ) /\ ( 1 xe w ) = w /\ ( 0 xe w ) = 0 ) )
49		xrge0base 29685	. . . . . 6 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
50	1	fveq2i 6194	. . . . . 6 |- ( Base ` G ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
51	49, 50	eqtr4i 2647	. . . . 5 |- ( 0 [,] +oo ) = ( Base ` G )
52	5, 51	resvbas 29832	. . . 4 |- ( ( 0 [,) +oo ) e. _V -> ( 0 [,] +oo ) = ( Base ` W ) )
53	4, 52	ax-mp 5	. . 3 |- ( 0 [,] +oo ) = ( Base ` W )
54		xrge0plusg 29687	. . . . . 6 |- +e = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
55	1	fveq2i 6194	. . . . . 6 |- ( +g ` G ) = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
56	54, 55	eqtr4i 2647	. . . . 5 |- +e = ( +g ` G )
57	5, 56	resvplusg 29833	. . . 4 |- ( ( 0 [,) +oo ) e. _V -> +e = ( +g ` W ) )
58	4, 57	ax-mp 5	. . 3 |- +e = ( +g ` W )
59		ovex 6678	. . . . . 6 |- ( 0 [,] +oo ) e. _V
60		ax-xrsvsca 29674	. . . . . . 7 |- xe = ( .s ` RR*s )
61	1, 60	ressvsca 16032	. . . . . 6 |- ( ( 0 [,] +oo ) e. _V -> xe = ( .s ` G ) )
62	59, 61	ax-mp 5	. . . . 5 |- xe = ( .s ` G )
63	5, 62	resvvsca 29834	. . . 4 |- ( ( 0 [,) +oo ) e. _V -> xe = ( .s ` W ) )
64	4, 63	ax-mp 5	. . 3 |- xe = ( .s ` W )
65		xrge00 29686	. . . . . 6 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
66	1	fveq2i 6194	. . . . . 6 |- ( 0g ` G ) = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
67	65, 66	eqtr4i 2647	. . . . 5 |- 0 = ( 0g ` G )
68	5, 67	resv0g 29836	. . . 4 |- ( ( 0 [,) +oo ) e. _V -> 0 = ( 0g ` W ) )
69	4, 68	ax-mp 5	. . 3 |- 0 = ( 0g ` W )
70		df-refld 19951	. . . . . 6 |- RRfld = (ℂfld ↾s RR )
71	70	oveq1i 6660	. . . . 5 |- (RRfld ↾s ( 0 [,) +oo ) ) = ( (ℂfld ↾s RR ) ↾s ( 0 [,) +oo ) )
72		reex 10027	. . . . . 6 |- RR e. _V
73		ressress 15938	. . . . . 6 |- ( ( RR e. _V /\ ( 0 [,) +oo ) e. _V ) -> ( (ℂfld ↾s RR ) ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( RR i^i ( 0 [,) +oo ) ) ) )
74	72, 4, 73	mp2an 708	. . . . 5 |- ( (ℂfld ↾s RR ) ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( RR i^i ( 0 [,) +oo ) ) )
75	71, 74	eqtri 2644	. . . 4 |- (RRfld ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( RR i^i ( 0 [,) +oo ) ) )
76		ax-xrssca 29673	. . . . . . . 8 |- RRfld = (Scalar ` RR*s )
77	1, 76	resssca 16031	. . . . . . 7 |- ( ( 0 [,] +oo ) e. _V -> RRfld = (Scalar ` G ) )
78	59, 77	ax-mp 5	. . . . . 6 |- RRfld = (Scalar ` G )
79		rebase 19952	. . . . . 6 |- RR = ( Base ` RRfld )
80	5, 78, 79	resvsca 29830	. . . . 5 |- ( ( 0 [,) +oo ) e. _V -> (RRfld ↾s ( 0 [,) +oo ) ) = (Scalar ` W ) )
81	4, 80	ax-mp 5	. . . 4 |- (RRfld ↾s ( 0 [,) +oo ) ) = (Scalar ` W )
82		incom 3805	. . . . . 6 |- ( ( 0 [,) +oo ) i^i RR ) = ( RR i^i ( 0 [,) +oo ) )
83		df-ss 3588	. . . . . . 7 |- ( ( 0 [,) +oo ) C_ RR <-> ( ( 0 [,) +oo ) i^i RR ) = ( 0 [,) +oo ) )
84	19, 83	mpbi 220	. . . . . 6 |- ( ( 0 [,) +oo ) i^i RR ) = ( 0 [,) +oo )
85	82, 84	eqtr3i 2646	. . . . 5 |- ( RR i^i ( 0 [,) +oo ) ) = ( 0 [,) +oo )
86	85	oveq2i 6661	. . . 4 |- (ℂfld ↾s ( RR i^i ( 0 [,) +oo ) ) ) = (ℂfld ↾s ( 0 [,) +oo ) )
87	75, 81, 86	3eqtr3ri 2653	. . 3 |- (ℂfld ↾s ( 0 [,) +oo ) ) = (Scalar ` W )
88		ax-resscn 9993	. . . . 5 |- RR C_ CC
89	19, 88	sstri 3612	. . . 4 |- ( 0 [,) +oo ) C_ CC
90		eqid 2622	. . . . 5 |- (ℂfld ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( 0 [,) +oo ) )
91		cnfldbas 19750	. . . . 5 |- CC = ( Base ` ℂfld )
92	90, 91	ressbas2 15931	. . . 4 |- ( ( 0 [,) +oo ) C_ CC -> ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
93	89, 92	ax-mp 5	. . 3 |- ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
94		cnfldadd 19751	. . . . 5 |- + = ( +g ` ℂfld )
95	90, 94	ressplusg 15993	. . . 4 |- ( ( 0 [,) +oo ) e. _V -> + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
96	4, 95	ax-mp 5	. . 3 |- + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
97		cnfldmul 19752	. . . . 5 |- x. = ( .r ` ℂfld )
98	90, 97	ressmulr 16006	. . . 4 |- ( ( 0 [,) +oo ) e. _V -> x. = ( .r ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
99	4, 98	ax-mp 5	. . 3 |- x. = ( .r ` (ℂfld ↾s ( 0 [,) +oo ) ) )
100		cndrng 19775	. . . . 5 |- ℂfld e. DivRing
101		drngring 18754	. . . . 5 |- (ℂfld e. DivRing -> ℂfld e. Ring )
102	100, 101	ax-mp 5	. . . 4 |- ℂfld e. Ring
103		1re 10039	. . . . . 6 |- 1 e. RR
104		0le1 10551	. . . . . 6 |- 0 <_ 1
105		ltpnf 11954	. . . . . . 7 |- ( 1 e. RR -> 1 < +oo )
106	103, 105	ax-mp 5	. . . . . 6 |- 1 < +oo
107	103, 104, 106	3pm3.2i 1239	. . . . 5 |- ( 1 e. RR /\ 0 <_ 1 /\ 1 < +oo )
108		0re 10040	. . . . . 6 |- 0 e. RR
109		pnfxr 10092	. . . . . 6 |- +oo e. RR*
110		elico2 12237	. . . . . 6 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( 1 e. ( 0 [,) +oo ) <-> ( 1 e. RR /\ 0 <_ 1 /\ 1 < +oo ) ) )
111	108, 109, 110	mp2an 708	. . . . 5 |- ( 1 e. ( 0 [,) +oo ) <-> ( 1 e. RR /\ 0 <_ 1 /\ 1 < +oo ) )
112	107, 111	mpbir 221	. . . 4 |- 1 e. ( 0 [,) +oo )
113		cnfld1 19771	. . . . 5 |- 1 = ( 1r ` ℂfld )
114	90, 91, 113	ress1r 29789	. . . 4 |- ( (ℂfld e. Ring /\ 1 e. ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ CC ) -> 1 = ( 1r ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
115	102, 112, 89, 114	mp3an 1424	. . 3 |- 1 = ( 1r ` (ℂfld ↾s ( 0 [,) +oo ) ) )
116		ringmnd 18556	. . . . 5 |- (ℂfld e. Ring -> ℂfld e. Mnd )
117	100, 101, 116	mp2b 10	. . . 4 |- ℂfld e. Mnd
118		0e0icopnf 12282	. . . 4 |- 0 e. ( 0 [,) +oo )
119		cnfld0 19770	. . . . 5 |- 0 = ( 0g ` ℂfld )
120	90, 91, 119	ress0g 17319	. . . 4 |- ( (ℂfld e. Mnd /\ 0 e. ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ CC ) -> 0 = ( 0g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
121	117, 118, 89, 120	mp3an 1424	. . 3 |- 0 = ( 0g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
122	53, 58, 64, 69, 87, 93, 96, 99, 115, 121	isslmd 29755	. 2 |- ( W e. SLMod <-> ( W e. CMnd /\ (ℂfld ↾s ( 0 [,) +oo ) ) e. SRing /\ A. q e. ( 0 [,) +oo ) A. r e. ( 0 [,) +oo ) A. x e. ( 0 [,] +oo ) A. w e. ( 0 [,] +oo ) ( ( ( r xe w ) e. ( 0 [,] +oo ) /\ ( r xe ( w +e x ) ) = ( ( r xe w ) +e ( r xe x ) ) /\ ( ( q + r ) xe w ) = ( ( q xe w ) +e ( r xe w ) ) ) /\ ( ( ( q x. r ) xe w ) = ( q xe ( r xe w ) ) /\ ( 1 xe w ) = w /\ ( 0 xe w ) = 0 ) ) ) )
123	8, 9, 48, 122	mpbir3an 1244	1 |- W e. SLMod

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   +ecxad 11944   xecxmu 11945   [,)cico 12177   [,]cicc 12178   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   RR*scxrs 16160   Mndcmnd 17294  CMndccmn 18193   1rcur 18501  SRingcsrg 18505   Ringcrg 18547   DivRingcdr 18747  ℂ_fldccnfld 19746  RRfldcrefld 19950  SLModcslmd 29753   ↾_v cresv 29824

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-rep 4771  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  ax-addf 10015  ax-mulf 10016  ax-xrssca 29673  ax-xrsvsca 29674

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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-xrs 16162  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-cnfld 19747  df-refld 19951  df-slmd 29754  df-resv 29825

This theorem is referenced by: (None)