Theorem xpsaddlem 16235

Description: Lemma for xpsadd 16236 and xpsmul 16237. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypotheses

Ref	Expression
xpsval.t	|- T = ( R X.s S )
xpsval.x	|- X = ( Base ` R )
xpsval.y	|- Y = ( Base ` S )
xpsval.1	|- ( ph -> R e. V )
xpsval.2	|- ( ph -> S e. W )
xpsadd.3	|- ( ph -> A e. X )
xpsadd.4	|- ( ph -> B e. Y )
xpsadd.5	|- ( ph -> C e. X )
xpsadd.6	|- ( ph -> D e. Y )
xpsadd.7	|- ( ph -> ( A .x. C ) e. X )
xpsadd.8	|- ( ph -> ( B .X. D ) e. Y )
xpsaddlem.m	|- .x. = ( E ` R )
xpsaddlem.n	|- .X. = ( E ` S )
xpsaddlem.p	|- .xb = ( E ` T )
xpsaddlem.f	|- F = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
xpsaddlem.u	|- U = ( (Scalar ` R ) X_s `' ( { R } +c { S } ) )
xpsaddlem.1	|- ( ( ph /\ `' ( { A } +c { B } ) e. ran F /\ `' ( { C } +c { D } ) e. ran F ) -> ( ( `' F ` `' ( { A } +c { B } ) ) .xb ( `' F ` `' ( { C } +c { D } ) ) ) = ( `' F ` ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) ) )
xpsaddlem.2	|- ( ( `' ( { R } +c { S } ) Fn 2o /\ `' ( { A } +c { B } ) e. ( Base ` U ) /\ `' ( { C } +c { D } ) e. ( Base ` U ) ) -> ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) = ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( E ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) )

Assertion

Ref	Expression
xpsaddlem	|- ( ph -> ( <. A , B >. .xb <. C , D >. ) = <. ( A .x. C ) , ( B .X. D ) >. )

Distinct variable groups:   x, k, y, A   B, k, x, y   C, k, x, y   D, k, x, y   S, k   U, k   x, W   ph, k   .x. , k, x, y   .X. , k, x, y   k, X, x, y   R, k, x   k, Y, x, y

Allowed substitution hints:   ph( x, y)   R( y)   S( x, y)   .xb ( x, y, k)   T( x, y, k)   U( x, y)   E( x, y, k)   F( x, y, k)   V( x, y, k)   W( y, k)

Proof of Theorem xpsaddlem

Step	Hyp	Ref	Expression
1		df-ov 6653	. . . . 5 |- ( A F B ) = ( F ` <. A , B >. )
2		xpsadd.3	. . . . . 6 |- ( ph -> A e. X )
3		xpsadd.4	. . . . . 6 |- ( ph -> B e. Y )
4		xpsaddlem.f	. . . . . . 7 |- F = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )
5	4	xpsfval 16227	. . . . . 6 |- ( ( A e. X /\ B e. Y ) -> ( A F B ) = `' ( { A } +c { B } ) )
6	2, 3, 5	syl2anc 693	. . . . 5 |- ( ph -> ( A F B ) = `' ( { A } +c { B } ) )
7	1, 6	syl5eqr 2670	. . . 4 |- ( ph -> ( F ` <. A , B >. ) = `' ( { A } +c { B } ) )
8		opelxpi 5148	. . . . . 6 |- ( ( A e. X /\ B e. Y ) -> <. A , B >. e. ( X X. Y ) )
9	2, 3, 8	syl2anc 693	. . . . 5 |- ( ph -> <. A , B >. e. ( X X. Y ) )
10	4	xpsff1o2 16231	. . . . . . 7 |- F : ( X X. Y ) -1-1-onto-> ran F
11		f1of 6137	. . . . . . 7 |- ( F : ( X X. Y ) -1-1-onto-> ran F -> F : ( X X. Y ) --> ran F )
12	10, 11	ax-mp 5	. . . . . 6 |- F : ( X X. Y ) --> ran F
13	12	ffvelrni 6358	. . . . 5 |- ( <. A , B >. e. ( X X. Y ) -> ( F ` <. A , B >. ) e. ran F )
14	9, 13	syl 17	. . . 4 |- ( ph -> ( F ` <. A , B >. ) e. ran F )
15	7, 14	eqeltrrd 2702	. . 3 |- ( ph -> `' ( { A } +c { B } ) e. ran F )
16		df-ov 6653	. . . . 5 |- ( C F D ) = ( F ` <. C , D >. )
17		xpsadd.5	. . . . . 6 |- ( ph -> C e. X )
18		xpsadd.6	. . . . . 6 |- ( ph -> D e. Y )
19	4	xpsfval 16227	. . . . . 6 |- ( ( C e. X /\ D e. Y ) -> ( C F D ) = `' ( { C } +c { D } ) )
20	17, 18, 19	syl2anc 693	. . . . 5 |- ( ph -> ( C F D ) = `' ( { C } +c { D } ) )
21	16, 20	syl5eqr 2670	. . . 4 |- ( ph -> ( F ` <. C , D >. ) = `' ( { C } +c { D } ) )
22		opelxpi 5148	. . . . . 6 |- ( ( C e. X /\ D e. Y ) -> <. C , D >. e. ( X X. Y ) )
23	17, 18, 22	syl2anc 693	. . . . 5 |- ( ph -> <. C , D >. e. ( X X. Y ) )
24	12	ffvelrni 6358	. . . . 5 |- ( <. C , D >. e. ( X X. Y ) -> ( F ` <. C , D >. ) e. ran F )
25	23, 24	syl 17	. . . 4 |- ( ph -> ( F ` <. C , D >. ) e. ran F )
26	21, 25	eqeltrrd 2702	. . 3 |- ( ph -> `' ( { C } +c { D } ) e. ran F )
27		xpsaddlem.1	. . 3 |- ( ( ph /\ `' ( { A } +c { B } ) e. ran F /\ `' ( { C } +c { D } ) e. ran F ) -> ( ( `' F ` `' ( { A } +c { B } ) ) .xb ( `' F ` `' ( { C } +c { D } ) ) ) = ( `' F ` ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) ) )
28	15, 26, 27	mpd3an23 1426	. 2 |- ( ph -> ( ( `' F ` `' ( { A } +c { B } ) ) .xb ( `' F ` `' ( { C } +c { D } ) ) ) = ( `' F ` ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) ) )
29		f1ocnvfv 6534	. . . . 5 |- ( ( F : ( X X. Y ) -1-1-onto-> ran F /\ <. A , B >. e. ( X X. Y ) ) -> ( ( F ` <. A , B >. ) = `' ( { A } +c { B } ) -> ( `' F ` `' ( { A } +c { B } ) ) = <. A , B >. ) )
30	10, 9, 29	sylancr 695	. . . 4 |- ( ph -> ( ( F ` <. A , B >. ) = `' ( { A } +c { B } ) -> ( `' F ` `' ( { A } +c { B } ) ) = <. A , B >. ) )
31	7, 30	mpd 15	. . 3 |- ( ph -> ( `' F ` `' ( { A } +c { B } ) ) = <. A , B >. )
32		f1ocnvfv 6534	. . . . 5 |- ( ( F : ( X X. Y ) -1-1-onto-> ran F /\ <. C , D >. e. ( X X. Y ) ) -> ( ( F ` <. C , D >. ) = `' ( { C } +c { D } ) -> ( `' F ` `' ( { C } +c { D } ) ) = <. C , D >. ) )
33	10, 23, 32	sylancr 695	. . . 4 |- ( ph -> ( ( F ` <. C , D >. ) = `' ( { C } +c { D } ) -> ( `' F ` `' ( { C } +c { D } ) ) = <. C , D >. ) )
34	21, 33	mpd 15	. . 3 |- ( ph -> ( `' F ` `' ( { C } +c { D } ) ) = <. C , D >. )
35	31, 34	oveq12d 6668	. 2 |- ( ph -> ( ( `' F ` `' ( { A } +c { B } ) ) .xb ( `' F ` `' ( { C } +c { D } ) ) ) = ( <. A , B >. .xb <. C , D >. ) )
36		xpsval.1	. . . . . . 7 |- ( ph -> R e. V )
37		xpsval.2	. . . . . . 7 |- ( ph -> S e. W )
38		xpscfn 16219	. . . . . . 7 |- ( ( R e. V /\ S e. W ) -> `' ( { R } +c { S } ) Fn 2o )
39	36, 37, 38	syl2anc 693	. . . . . 6 |- ( ph -> `' ( { R } +c { S } ) Fn 2o )
40		xpsval.t	. . . . . . . 8 |- T = ( R X.s S )
41		xpsval.x	. . . . . . . 8 |- X = ( Base ` R )
42		xpsval.y	. . . . . . . 8 |- Y = ( Base ` S )
43		eqid 2622	. . . . . . . 8 |- (Scalar ` R ) = (Scalar ` R )
44		xpsaddlem.u	. . . . . . . 8 |- U = ( (Scalar ` R ) X_s `' ( { R } +c { S } ) )
45	40, 41, 42, 36, 37, 4, 43, 44	xpslem 16233	. . . . . . 7 |- ( ph -> ran F = ( Base ` U ) )
46	15, 45	eleqtrd 2703	. . . . . 6 |- ( ph -> `' ( { A } +c { B } ) e. ( Base ` U ) )
47	26, 45	eleqtrd 2703	. . . . . 6 |- ( ph -> `' ( { C } +c { D } ) e. ( Base ` U ) )
48		xpsaddlem.2	. . . . . 6 |- ( ( `' ( { R } +c { S } ) Fn 2o /\ `' ( { A } +c { B } ) e. ( Base ` U ) /\ `' ( { C } +c { D } ) e. ( Base ` U ) ) -> ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) = ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( E ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) )
49	39, 46, 47, 48	syl3anc 1326	. . . . 5 |- ( ph -> ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) = ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( E ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) )
50		xpsadd.7	. . . . . . . 8 |- ( ph -> ( A .x. C ) e. X )
51		xpsadd.8	. . . . . . . 8 |- ( ph -> ( B .X. D ) e. Y )
52		xpscfn 16219	. . . . . . . 8 |- ( ( ( A .x. C ) e. X /\ ( B .X. D ) e. Y ) -> `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) Fn 2o )
53	50, 51, 52	syl2anc 693	. . . . . . 7 |- ( ph -> `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) Fn 2o )
54		dffn5 6241	. . . . . . 7 |- ( `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) Fn 2o <-> `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) = ( k e. 2o |-> ( `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ` k ) ) )
55	53, 54	sylib 208	. . . . . 6 |- ( ph -> `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) = ( k e. 2o |-> ( `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ` k ) ) )
56		iftrue 4092	. . . . . . . . . . . . 13 |- ( k = (/) -> if ( k = (/) , R , S ) = R )
57	56	fveq2d 6195	. . . . . . . . . . . 12 |- ( k = (/) -> ( E ` if ( k = (/) , R , S ) ) = ( E ` R ) )
58		xpsaddlem.m	. . . . . . . . . . . 12 |- .x. = ( E ` R )
59	57, 58	syl6eqr 2674	. . . . . . . . . . 11 |- ( k = (/) -> ( E ` if ( k = (/) , R , S ) ) = .x. )
60		iftrue 4092	. . . . . . . . . . 11 |- ( k = (/) -> if ( k = (/) , A , B ) = A )
61		iftrue 4092	. . . . . . . . . . 11 |- ( k = (/) -> if ( k = (/) , C , D ) = C )
62	59, 60, 61	oveq123d 6671	. . . . . . . . . 10 |- ( k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = ( A .x. C ) )
63		iftrue 4092	. . . . . . . . . 10 |- ( k = (/) -> if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) = ( A .x. C ) )
64	62, 63	eqtr4d 2659	. . . . . . . . 9 |- ( k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) )
65		iffalse 4095	. . . . . . . . . . . . 13 |- ( -. k = (/) -> if ( k = (/) , R , S ) = S )
66	65	fveq2d 6195	. . . . . . . . . . . 12 |- ( -. k = (/) -> ( E ` if ( k = (/) , R , S ) ) = ( E ` S ) )
67		xpsaddlem.n	. . . . . . . . . . . 12 |- .X. = ( E ` S )
68	66, 67	syl6eqr 2674	. . . . . . . . . . 11 |- ( -. k = (/) -> ( E ` if ( k = (/) , R , S ) ) = .X. )
69		iffalse 4095	. . . . . . . . . . 11 |- ( -. k = (/) -> if ( k = (/) , A , B ) = B )
70		iffalse 4095	. . . . . . . . . . 11 |- ( -. k = (/) -> if ( k = (/) , C , D ) = D )
71	68, 69, 70	oveq123d 6671	. . . . . . . . . 10 |- ( -. k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = ( B .X. D ) )
72		iffalse 4095	. . . . . . . . . 10 |- ( -. k = (/) -> if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) = ( B .X. D ) )
73	71, 72	eqtr4d 2659	. . . . . . . . 9 |- ( -. k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) )
74	64, 73	pm2.61i 176	. . . . . . . 8 |- ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) )
75	36	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ k e. 2o ) -> R e. V )
76	37	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ k e. 2o ) -> S e. W )
77		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ k e. 2o ) -> k e. 2o )
78		xpscfv 16222	. . . . . . . . . . 11 |- ( ( R e. V /\ S e. W /\ k e. 2o ) -> ( `' ( { R } +c { S } ) ` k ) = if ( k = (/) , R , S ) )
79	75, 76, 77, 78	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ k e. 2o ) -> ( `' ( { R } +c { S } ) ` k ) = if ( k = (/) , R , S ) )
80	79	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ k e. 2o ) -> ( E ` ( `' ( { R } +c { S } ) ` k ) ) = ( E ` if ( k = (/) , R , S ) ) )
81	2	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. 2o ) -> A e. X )
82	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. 2o ) -> B e. Y )
83		xpscfv 16222	. . . . . . . . . 10 |- ( ( A e. X /\ B e. Y /\ k e. 2o ) -> ( `' ( { A } +c { B } ) ` k ) = if ( k = (/) , A , B ) )
84	81, 82, 77, 83	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ k e. 2o ) -> ( `' ( { A } +c { B } ) ` k ) = if ( k = (/) , A , B ) )
85	17	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. 2o ) -> C e. X )
86	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. 2o ) -> D e. Y )
87		xpscfv 16222	. . . . . . . . . 10 |- ( ( C e. X /\ D e. Y /\ k e. 2o ) -> ( `' ( { C } +c { D } ) ` k ) = if ( k = (/) , C , D ) )
88	85, 86, 77, 87	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ k e. 2o ) -> ( `' ( { C } +c { D } ) ` k ) = if ( k = (/) , C , D ) )
89	80, 84, 88	oveq123d 6671	. . . . . . . 8 |- ( ( ph /\ k e. 2o ) -> ( ( `' ( { A } +c { B } ) ` k ) ( E ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) = ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) )
90	50	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. 2o ) -> ( A .x. C ) e. X )
91	51	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. 2o ) -> ( B .X. D ) e. Y )
92		xpscfv 16222	. . . . . . . . 9 |- ( ( ( A .x. C ) e. X /\ ( B .X. D ) e. Y /\ k e. 2o ) -> ( `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ` k ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) )
93	90, 91, 77, 92	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ k e. 2o ) -> ( `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ` k ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) )
94	74, 89, 93	3eqtr4a 2682	. . . . . . 7 |- ( ( ph /\ k e. 2o ) -> ( ( `' ( { A } +c { B } ) ` k ) ( E ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) = ( `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ` k ) )
95	94	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( E ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) = ( k e. 2o |-> ( `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ` k ) ) )
96	55, 95	eqtr4d 2659	. . . . 5 |- ( ph -> `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) = ( k e. 2o |-> ( ( `' ( { A } +c { B } ) ` k ) ( E ` ( `' ( { R } +c { S } ) ` k ) ) ( `' ( { C } +c { D } ) ` k ) ) ) )
97	49, 96	eqtr4d 2659	. . . 4 |- ( ph -> ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) = `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) )
98	97	fveq2d 6195	. . 3 |- ( ph -> ( `' F ` ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) ) = ( `' F ` `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ) )
99		df-ov 6653	. . . . 5 |- ( ( A .x. C ) F ( B .X. D ) ) = ( F ` <. ( A .x. C ) , ( B .X. D ) >. )
100	4	xpsfval 16227	. . . . . 6 |- ( ( ( A .x. C ) e. X /\ ( B .X. D ) e. Y ) -> ( ( A .x. C ) F ( B .X. D ) ) = `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) )
101	50, 51, 100	syl2anc 693	. . . . 5 |- ( ph -> ( ( A .x. C ) F ( B .X. D ) ) = `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) )
102	99, 101	syl5eqr 2670	. . . 4 |- ( ph -> ( F ` <. ( A .x. C ) , ( B .X. D ) >. ) = `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) )
103		opelxpi 5148	. . . . . 6 |- ( ( ( A .x. C ) e. X /\ ( B .X. D ) e. Y ) -> <. ( A .x. C ) , ( B .X. D ) >. e. ( X X. Y ) )
104	50, 51, 103	syl2anc 693	. . . . 5 |- ( ph -> <. ( A .x. C ) , ( B .X. D ) >. e. ( X X. Y ) )
105		f1ocnvfv 6534	. . . . 5 |- ( ( F : ( X X. Y ) -1-1-onto-> ran F /\ <. ( A .x. C ) , ( B .X. D ) >. e. ( X X. Y ) ) -> ( ( F ` <. ( A .x. C ) , ( B .X. D ) >. ) = `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) -> ( `' F ` `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ) = <. ( A .x. C ) , ( B .X. D ) >. ) )
106	10, 104, 105	sylancr 695	. . . 4 |- ( ph -> ( ( F ` <. ( A .x. C ) , ( B .X. D ) >. ) = `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) -> ( `' F ` `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ) = <. ( A .x. C ) , ( B .X. D ) >. ) )
107	102, 106	mpd 15	. . 3 |- ( ph -> ( `' F ` `' ( { ( A .x. C ) } +c { ( B .X. D ) } ) ) = <. ( A .x. C ) , ( B .X. D ) >. )
108	98, 107	eqtrd 2656	. 2 |- ( ph -> ( `' F ` ( `' ( { A } +c { B } ) ( E ` U ) `' ( { C } +c { D } ) ) ) = <. ( A .x. C ) , ( B .X. D ) >. )
109	28, 35, 108	3eqtr3d 2664	1 |- ( ph -> ( <. A , B >. .xb <. C , D >. ) = <. ( A .x. C ) , ( B .X. D ) >. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2oc2o 7554   +c ccda 8989   Basecbs 15857  Scalarcsca 15944   X__scprds 16106   X._s cxps 16166

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

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  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-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108

This theorem is referenced by:  xpsadd  16236  xpsmul  16237