Theorem recexsrlem 9924

Description: The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
recexsrlem	|- ( 0R <R A -> E. x e. R. ( A .R x ) = 1R )

Distinct variable group:   x, A

Proof of Theorem recexsrlem

Dummy variables y z w v u f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelsr 9889	. . . 4 |- <R C_ ( R. X. R. )
2	1	brel 5168	. . 3 |- ( 0R <R A -> ( 0R e. R. /\ A e. R. ) )
3	2	simprd 479	. 2 |- ( 0R <R A -> A e. R. )
4		df-nr 9878	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
5		breq2 4657	. . . 4 |- ( [ <. y , z >. ] ~R = A -> ( 0R <R [ <. y , z >. ] ~R <-> 0R <R A ) )
6		oveq1 6657	. . . . . 6 |- ( [ <. y , z >. ] ~R = A -> ( [ <. y , z >. ] ~R .R x ) = ( A .R x ) )
7	6	eqeq1d 2624	. . . . 5 |- ( [ <. y , z >. ] ~R = A -> ( ( [ <. y , z >. ] ~R .R x ) = 1R <-> ( A .R x ) = 1R ) )
8	7	rexbidv 3052	. . . 4 |- ( [ <. y , z >. ] ~R = A -> ( E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R <-> E. x e. R. ( A .R x ) = 1R ) )
9	5, 8	imbi12d 334	. . 3 |- ( [ <. y , z >. ] ~R = A -> ( ( 0R <R [ <. y , z >. ] ~R -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R ) <-> ( 0R <R A -> E. x e. R. ( A .R x ) = 1R ) ) )
10		gt0srpr 9899	. . . . 5 |- ( 0R <R [ <. y , z >. ] ~R <-> z <P y )
11		ltexpri 9865	. . . . 5 |- ( z <P y -> E. w e. P. ( z +P. w ) = y )
12	10, 11	sylbi 207	. . . 4 |- ( 0R <R [ <. y , z >. ] ~R -> E. w e. P. ( z +P. w ) = y )
13		recexpr 9873	. . . . . 6 |- ( w e. P. -> E. v e. P. ( w .P. v ) = 1P )
14		1pr 9837	. . . . . . . . . . . 12 |- 1P e. P.
15		addclpr 9840	. . . . . . . . . . . 12 |- ( ( v e. P. /\ 1P e. P. ) -> ( v +P. 1P ) e. P. )
16	14, 15	mpan2 707	. . . . . . . . . . 11 |- ( v e. P. -> ( v +P. 1P ) e. P. )
17		enrex 9888	. . . . . . . . . . . 12 |- ~R e. _V
18	17, 4	ecopqsi 7804	. . . . . . . . . . 11 |- ( ( ( v +P. 1P ) e. P. /\ 1P e. P. ) -> [ <. ( v +P. 1P ) , 1P >. ] ~R e. R. )
19	16, 14, 18	sylancl 694	. . . . . . . . . 10 |- ( v e. P. -> [ <. ( v +P. 1P ) , 1P >. ] ~R e. R. )
20	19	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) /\ ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) ) -> [ <. ( v +P. 1P ) , 1P >. ] ~R e. R. )
21	16, 14	jctir 561	. . . . . . . . . . . . . 14 |- ( v e. P. -> ( ( v +P. 1P ) e. P. /\ 1P e. P. ) )
22	21	anim2i 593	. . . . . . . . . . . . 13 |- ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) -> ( ( y e. P. /\ z e. P. ) /\ ( ( v +P. 1P ) e. P. /\ 1P e. P. ) ) )
23	22	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) /\ ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) ) -> ( ( y e. P. /\ z e. P. ) /\ ( ( v +P. 1P ) e. P. /\ 1P e. P. ) ) )
24		mulsrpr 9897	. . . . . . . . . . . 12 |- ( ( ( y e. P. /\ z e. P. ) /\ ( ( v +P. 1P ) e. P. /\ 1P e. P. ) ) -> ( [ <. y , z >. ] ~R .R [ <. ( v +P. 1P ) , 1P >. ] ~R ) = [ <. ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) , ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) >. ] ~R )
25	23, 24	syl 17	. . . . . . . . . . 11 |- ( ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) /\ ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) ) -> ( [ <. y , z >. ] ~R .R [ <. ( v +P. 1P ) , 1P >. ] ~R ) = [ <. ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) , ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) >. ] ~R )
26		oveq1 6657	. . . . . . . . . . . . . . . . . . . 20 |- ( ( z +P. w ) = y -> ( ( z +P. w ) .P. v ) = ( y .P. v ) )
27	26	eqcomd 2628	. . . . . . . . . . . . . . . . . . 19 |- ( ( z +P. w ) = y -> ( y .P. v ) = ( ( z +P. w ) .P. v ) )
28		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- z e. _V
29		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- w e. _V
30		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- v e. _V
31		mulcompr 9845	. . . . . . . . . . . . . . . . . . . . 21 |- ( u .P. f ) = ( f .P. u )
32		distrpr 9850	. . . . . . . . . . . . . . . . . . . . 21 |- ( u .P. ( f +P. x ) ) = ( ( u .P. f ) +P. ( u .P. x ) )
33	28, 29, 30, 31, 32	caovdir 6868	. . . . . . . . . . . . . . . . . . . 20 |- ( ( z +P. w ) .P. v ) = ( ( z .P. v ) +P. ( w .P. v ) )
34		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 |- ( ( w .P. v ) = 1P -> ( ( z .P. v ) +P. ( w .P. v ) ) = ( ( z .P. v ) +P. 1P ) )
35	33, 34	syl5eq 2668	. . . . . . . . . . . . . . . . . . 19 |- ( ( w .P. v ) = 1P -> ( ( z +P. w ) .P. v ) = ( ( z .P. v ) +P. 1P ) )
36	27, 35	sylan9eqr 2678	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) -> ( y .P. v ) = ( ( z .P. v ) +P. 1P ) )
37	36	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) -> ( ( y .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) = ( ( ( z .P. v ) +P. 1P ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) )
38		ovex 6678	. . . . . . . . . . . . . . . . . 18 |- ( z .P. v ) e. _V
39	14	elexi 3213	. . . . . . . . . . . . . . . . . 18 |- 1P e. _V
40		ovex 6678	. . . . . . . . . . . . . . . . . 18 |- ( ( y .P. 1P ) +P. ( z .P. 1P ) ) e. _V
41		addcompr 9843	. . . . . . . . . . . . . . . . . 18 |- ( u +P. f ) = ( f +P. u )
42		addasspr 9844	. . . . . . . . . . . . . . . . . 18 |- ( ( u +P. f ) +P. x ) = ( u +P. ( f +P. x ) )
43	38, 39, 40, 41, 42	caov32 6861	. . . . . . . . . . . . . . . . 17 |- ( ( ( z .P. v ) +P. 1P ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) = ( ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. 1P )
44	37, 43	syl6eq 2672	. . . . . . . . . . . . . . . 16 |- ( ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) -> ( ( y .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) = ( ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. 1P ) )
45	44	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) -> ( ( ( y .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. 1P ) = ( ( ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. 1P ) +P. 1P ) )
46		addasspr 9844	. . . . . . . . . . . . . . 15 |- ( ( ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. 1P ) +P. 1P ) = ( ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. ( 1P +P. 1P ) )
47	45, 46	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) -> ( ( ( y .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. 1P ) = ( ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. ( 1P +P. 1P ) ) )
48		distrpr 9850	. . . . . . . . . . . . . . . . 17 |- ( y .P. ( v +P. 1P ) ) = ( ( y .P. v ) +P. ( y .P. 1P ) )
49	48	oveq1i 6660	. . . . . . . . . . . . . . . 16 |- ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) = ( ( ( y .P. v ) +P. ( y .P. 1P ) ) +P. ( z .P. 1P ) )
50		addasspr 9844	. . . . . . . . . . . . . . . 16 |- ( ( ( y .P. v ) +P. ( y .P. 1P ) ) +P. ( z .P. 1P ) ) = ( ( y .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) )
51	49, 50	eqtri 2644	. . . . . . . . . . . . . . 15 |- ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) = ( ( y .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) )
52	51	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) +P. 1P ) = ( ( ( y .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. 1P )
53		distrpr 9850	. . . . . . . . . . . . . . . . 17 |- ( z .P. ( v +P. 1P ) ) = ( ( z .P. v ) +P. ( z .P. 1P ) )
54	53	oveq2i 6661	. . . . . . . . . . . . . . . 16 |- ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) = ( ( y .P. 1P ) +P. ( ( z .P. v ) +P. ( z .P. 1P ) ) )
55		ovex 6678	. . . . . . . . . . . . . . . . 17 |- ( y .P. 1P ) e. _V
56		ovex 6678	. . . . . . . . . . . . . . . . 17 |- ( z .P. 1P ) e. _V
57	55, 38, 56, 41, 42	caov12 6862	. . . . . . . . . . . . . . . 16 |- ( ( y .P. 1P ) +P. ( ( z .P. v ) +P. ( z .P. 1P ) ) ) = ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) )
58	54, 57	eqtri 2644	. . . . . . . . . . . . . . 15 |- ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) = ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) )
59	58	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) +P. ( 1P +P. 1P ) ) = ( ( ( z .P. v ) +P. ( ( y .P. 1P ) +P. ( z .P. 1P ) ) ) +P. ( 1P +P. 1P ) )
60	47, 52, 59	3eqtr4g 2681	. . . . . . . . . . . . 13 |- ( ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) -> ( ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) +P. 1P ) = ( ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) +P. ( 1P +P. 1P ) ) )
61		mulclpr 9842	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. P. /\ ( v +P. 1P ) e. P. ) -> ( y .P. ( v +P. 1P ) ) e. P. )
62	16, 61	sylan2 491	. . . . . . . . . . . . . . . . 17 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. ( v +P. 1P ) ) e. P. )
63		mulclpr 9842	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. P. /\ 1P e. P. ) -> ( z .P. 1P ) e. P. )
64	14, 63	mpan2 707	. . . . . . . . . . . . . . . . 17 |- ( z e. P. -> ( z .P. 1P ) e. P. )
65		addclpr 9840	. . . . . . . . . . . . . . . . 17 |- ( ( ( y .P. ( v +P. 1P ) ) e. P. /\ ( z .P. 1P ) e. P. ) -> ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) e. P. )
66	62, 64, 65	syl2an 494	. . . . . . . . . . . . . . . 16 |- ( ( ( y e. P. /\ v e. P. ) /\ z e. P. ) -> ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) e. P. )
67	66	an32s 846	. . . . . . . . . . . . . . 15 |- ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) -> ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) e. P. )
68		mulclpr 9842	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. P. /\ 1P e. P. ) -> ( y .P. 1P ) e. P. )
69	14, 68	mpan2 707	. . . . . . . . . . . . . . . . 17 |- ( y e. P. -> ( y .P. 1P ) e. P. )
70		mulclpr 9842	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. P. /\ ( v +P. 1P ) e. P. ) -> ( z .P. ( v +P. 1P ) ) e. P. )
71	16, 70	sylan2 491	. . . . . . . . . . . . . . . . 17 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. ( v +P. 1P ) ) e. P. )
72		addclpr 9840	. . . . . . . . . . . . . . . . 17 |- ( ( ( y .P. 1P ) e. P. /\ ( z .P. ( v +P. 1P ) ) e. P. ) -> ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) e. P. )
73	69, 71, 72	syl2an 494	. . . . . . . . . . . . . . . 16 |- ( ( y e. P. /\ ( z e. P. /\ v e. P. ) ) -> ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) e. P. )
74	73	anassrs 680	. . . . . . . . . . . . . . 15 |- ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) -> ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) e. P. )
75	67, 74	jca 554	. . . . . . . . . . . . . 14 |- ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) -> ( ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) e. P. /\ ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) e. P. ) )
76		addclpr 9840	. . . . . . . . . . . . . . . 16 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
77	14, 14, 76	mp2an 708	. . . . . . . . . . . . . . 15 |- ( 1P +P. 1P ) e. P.
78	77, 14	pm3.2i 471	. . . . . . . . . . . . . 14 |- ( ( 1P +P. 1P ) e. P. /\ 1P e. P. )
79		enreceq 9887	. . . . . . . . . . . . . 14 |- ( ( ( ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) e. P. /\ ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) e. P. ) /\ ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) ) -> ( [ <. ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) , ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) >. ] ~R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R <-> ( ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) +P. 1P ) = ( ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) +P. ( 1P +P. 1P ) ) ) )
80	75, 78, 79	sylancl 694	. . . . . . . . . . . . 13 |- ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) -> ( [ <. ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) , ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) >. ] ~R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R <-> ( ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) +P. 1P ) = ( ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) +P. ( 1P +P. 1P ) ) ) )
81	60, 80	syl5ibr 236	. . . . . . . . . . . 12 |- ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) -> ( ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) -> [ <. ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) , ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) >. ] ~R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) )
82	81	imp 445	. . . . . . . . . . 11 |- ( ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) /\ ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) ) -> [ <. ( ( y .P. ( v +P. 1P ) ) +P. ( z .P. 1P ) ) , ( ( y .P. 1P ) +P. ( z .P. ( v +P. 1P ) ) ) >. ] ~R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
83	25, 82	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) /\ ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) ) -> ( [ <. y , z >. ] ~R .R [ <. ( v +P. 1P ) , 1P >. ] ~R ) = [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
84		df-1r 9883	. . . . . . . . . 10 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
85	83, 84	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) /\ ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) ) -> ( [ <. y , z >. ] ~R .R [ <. ( v +P. 1P ) , 1P >. ] ~R ) = 1R )
86		oveq2 6658	. . . . . . . . . . 11 |- ( x = [ <. ( v +P. 1P ) , 1P >. ] ~R -> ( [ <. y , z >. ] ~R .R x ) = ( [ <. y , z >. ] ~R .R [ <. ( v +P. 1P ) , 1P >. ] ~R ) )
87	86	eqeq1d 2624	. . . . . . . . . 10 |- ( x = [ <. ( v +P. 1P ) , 1P >. ] ~R -> ( ( [ <. y , z >. ] ~R .R x ) = 1R <-> ( [ <. y , z >. ] ~R .R [ <. ( v +P. 1P ) , 1P >. ] ~R ) = 1R ) )
88	87	rspcev 3309	. . . . . . . . 9 |- ( ( [ <. ( v +P. 1P ) , 1P >. ] ~R e. R. /\ ( [ <. y , z >. ] ~R .R [ <. ( v +P. 1P ) , 1P >. ] ~R ) = 1R ) -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R )
89	20, 85, 88	syl2anc 693	. . . . . . . 8 |- ( ( ( ( y e. P. /\ z e. P. ) /\ v e. P. ) /\ ( ( w .P. v ) = 1P /\ ( z +P. w ) = y ) ) -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R )
90	89	exp43 640	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( v e. P. -> ( ( w .P. v ) = 1P -> ( ( z +P. w ) = y -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R ) ) ) )
91	90	rexlimdv 3030	. . . . . 6 |- ( ( y e. P. /\ z e. P. ) -> ( E. v e. P. ( w .P. v ) = 1P -> ( ( z +P. w ) = y -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R ) ) )
92	13, 91	syl5 34	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( w e. P. -> ( ( z +P. w ) = y -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R ) ) )
93	92	rexlimdv 3030	. . . 4 |- ( ( y e. P. /\ z e. P. ) -> ( E. w e. P. ( z +P. w ) = y -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R ) )
94	12, 93	syl5 34	. . 3 |- ( ( y e. P. /\ z e. P. ) -> ( 0R <R [ <. y , z >. ] ~R -> E. x e. R. ( [ <. y , z >. ] ~R .R x ) = 1R ) )
95	4, 9, 94	ecoptocl 7837	. 2 |- ( A e. R. -> ( 0R <R A -> E. x e. R. ( A .R x ) = 1R ) )
96	3, 95	mpcom 38	1 |- ( 0R <R A -> E. x e. R. ( A .R x ) = 1R )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653  (class class class)co 6650   [cec 7740   P.cnp 9681   1Pc1p 9682   +P. cpp 9683   .P. cmp 9684   <P cltp 9685   ~R cer 9686   R.cnr 9687   0Rc0r 9688   1Rc1r 9689   .R cmr 9692   <R cltr 9693

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-inf2 8538

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-mr 9880  df-ltr 9881  df-0r 9882  df-1r 9883

This theorem is referenced by:  recexsr  9928