Theorem normpar2i 28013

Description: Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
normpar2.1	|- A e. ~H
normpar2.2	|- B e. ~H
normpar2.3	|- C e. ~H

Assertion

Ref	Expression
normpar2i	|- ( ( normh ` ( A -h B ) ) ^ 2 ) = ( ( ( 2 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) - ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) )

Proof of Theorem normpar2i

Step	Hyp	Ref	Expression
1		normpar2.1	. . . . . . 7 |- A e. ~H
2		normpar2.2	. . . . . . 7 |- B e. ~H
3	1, 2	hvaddcli 27875	. . . . . 6 |- ( A +h B ) e. ~H
4		2cn 11091	. . . . . . 7 |- 2 e. CC
5		normpar2.3	. . . . . . 7 |- C e. ~H
6	4, 5	hvmulcli 27871	. . . . . 6 |- ( 2 .h C ) e. ~H
7	3, 6	hvsubcli 27878	. . . . 5 |- ( ( A +h B ) -h ( 2 .h C ) ) e. ~H
8	7	normcli 27988	. . . 4 |- ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) e. RR
9	8	resqcli 12949	. . 3 |- ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) e. RR
10	9	recni 10052	. 2 |- ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) e. CC
11	1, 2	hvsubcli 27878	. . . . 5 |- ( A -h B ) e. ~H
12	11	normcli 27988	. . . 4 |- ( normh ` ( A -h B ) ) e. RR
13	12	resqcli 12949	. . 3 |- ( ( normh ` ( A -h B ) ) ^ 2 ) e. RR
14	13	recni 10052	. 2 |- ( ( normh ` ( A -h B ) ) ^ 2 ) e. CC
15		4cn 11098	. . . . 5 |- 4 e. CC
16	1, 5	hvsubcli 27878	. . . . . . . 8 |- ( A -h C ) e. ~H
17	16	normcli 27988	. . . . . . 7 |- ( normh ` ( A -h C ) ) e. RR
18	17	resqcli 12949	. . . . . 6 |- ( ( normh ` ( A -h C ) ) ^ 2 ) e. RR
19	18	recni 10052	. . . . 5 |- ( ( normh ` ( A -h C ) ) ^ 2 ) e. CC
20	15, 19	mulcli 10045	. . . 4 |- ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) e. CC
21	2, 5	hvsubcli 27878	. . . . . . . 8 |- ( B -h C ) e. ~H
22	21	normcli 27988	. . . . . . 7 |- ( normh ` ( B -h C ) ) e. RR
23	22	resqcli 12949	. . . . . 6 |- ( ( normh ` ( B -h C ) ) ^ 2 ) e. RR
24	23	recni 10052	. . . . 5 |- ( ( normh ` ( B -h C ) ) ^ 2 ) e. CC
25	15, 24	mulcli 10045	. . . 4 |- ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) e. CC
26		2ne0 11113	. . . 4 |- 2 =/= 0
27	20, 25, 4, 26	divdiri 10782	. . 3 |- ( ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) / 2 ) = ( ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) / 2 ) + ( ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) / 2 ) )
28	20, 25	addcomi 10227	. . . . . . 7 |- ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) = ( ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) )
29		neg1cn 11124	. . . . . . . . . . . . . . . 16 |- -u 1 e. CC
30	29, 6	hvmulcli 27871	. . . . . . . . . . . . . . 15 |- ( -u 1 .h ( 2 .h C ) ) e. ~H
31	29, 11	hvmulcli 27871	. . . . . . . . . . . . . . 15 |- ( -u 1 .h ( A -h B ) ) e. ~H
32	3, 30, 31	hvadd32i 27911	. . . . . . . . . . . . . 14 |- ( ( ( A +h B ) +h ( -u 1 .h ( 2 .h C ) ) ) +h ( -u 1 .h ( A -h B ) ) ) = ( ( ( A +h B ) +h ( -u 1 .h ( A -h B ) ) ) +h ( -u 1 .h ( 2 .h C ) ) )
33	3, 6	hvsubvali 27877	. . . . . . . . . . . . . . 15 |- ( ( A +h B ) -h ( 2 .h C ) ) = ( ( A +h B ) +h ( -u 1 .h ( 2 .h C ) ) )
34	33	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( -u 1 .h ( A -h B ) ) ) = ( ( ( A +h B ) +h ( -u 1 .h ( 2 .h C ) ) ) +h ( -u 1 .h ( A -h B ) ) )
35	4, 2	hvmulcli 27871	. . . . . . . . . . . . . . . 16 |- ( 2 .h B ) e. ~H
36	35, 6	hvsubvali 27877	. . . . . . . . . . . . . . 15 |- ( ( 2 .h B ) -h ( 2 .h C ) ) = ( ( 2 .h B ) +h ( -u 1 .h ( 2 .h C ) ) )
37	1, 2	hvcomi 27876	. . . . . . . . . . . . . . . . . 18 |- ( A +h B ) = ( B +h A )
38	1, 2	hvnegdii 27919	. . . . . . . . . . . . . . . . . 18 |- ( -u 1 .h ( A -h B ) ) = ( B -h A )
39	37, 38	oveq12i 6662	. . . . . . . . . . . . . . . . 17 |- ( ( A +h B ) +h ( -u 1 .h ( A -h B ) ) ) = ( ( B +h A ) +h ( B -h A ) )
40	2, 1	hvsubcan2i 27921	. . . . . . . . . . . . . . . . 17 |- ( ( B +h A ) +h ( B -h A ) ) = ( 2 .h B )
41	39, 40	eqtri 2644	. . . . . . . . . . . . . . . 16 |- ( ( A +h B ) +h ( -u 1 .h ( A -h B ) ) ) = ( 2 .h B )
42	41	oveq1i 6660	. . . . . . . . . . . . . . 15 |- ( ( ( A +h B ) +h ( -u 1 .h ( A -h B ) ) ) +h ( -u 1 .h ( 2 .h C ) ) ) = ( ( 2 .h B ) +h ( -u 1 .h ( 2 .h C ) ) )
43	36, 42	eqtr4i 2647	. . . . . . . . . . . . . 14 |- ( ( 2 .h B ) -h ( 2 .h C ) ) = ( ( ( A +h B ) +h ( -u 1 .h ( A -h B ) ) ) +h ( -u 1 .h ( 2 .h C ) ) )
44	32, 34, 43	3eqtr4i 2654	. . . . . . . . . . . . 13 |- ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( -u 1 .h ( A -h B ) ) ) = ( ( 2 .h B ) -h ( 2 .h C ) )
45	7, 11	hvsubvali 27877	. . . . . . . . . . . . 13 |- ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) = ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( -u 1 .h ( A -h B ) ) )
46	4, 2, 5	hvsubdistr1i 27909	. . . . . . . . . . . . 13 |- ( 2 .h ( B -h C ) ) = ( ( 2 .h B ) -h ( 2 .h C ) )
47	44, 45, 46	3eqtr4i 2654	. . . . . . . . . . . 12 |- ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) = ( 2 .h ( B -h C ) )
48	47	fveq2i 6194	. . . . . . . . . . 11 |- ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) ) = ( normh ` ( 2 .h ( B -h C ) ) )
49	4, 21	norm-iii-i 27996	. . . . . . . . . . 11 |- ( normh ` ( 2 .h ( B -h C ) ) ) = ( ( abs ` 2 ) x. ( normh ` ( B -h C ) ) )
50		0le2 11111	. . . . . . . . . . . . 13 |- 0 <_ 2
51		2re 11090	. . . . . . . . . . . . . 14 |- 2 e. RR
52	51	absidi 14117	. . . . . . . . . . . . 13 |- ( 0 <_ 2 -> ( abs ` 2 ) = 2 )
53	50, 52	ax-mp 5	. . . . . . . . . . . 12 |- ( abs ` 2 ) = 2
54	53	oveq1i 6660	. . . . . . . . . . 11 |- ( ( abs ` 2 ) x. ( normh ` ( B -h C ) ) ) = ( 2 x. ( normh ` ( B -h C ) ) )
55	48, 49, 54	3eqtri 2648	. . . . . . . . . 10 |- ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) ) = ( 2 x. ( normh ` ( B -h C ) ) )
56	55	oveq1i 6660	. . . . . . . . 9 |- ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) ) ^ 2 ) = ( ( 2 x. ( normh ` ( B -h C ) ) ) ^ 2 )
57	22	recni 10052	. . . . . . . . . 10 |- ( normh ` ( B -h C ) ) e. CC
58	4, 57	sqmuli 12947	. . . . . . . . 9 |- ( ( 2 x. ( normh ` ( B -h C ) ) ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( ( normh ` ( B -h C ) ) ^ 2 ) )
59		sq2 12960	. . . . . . . . . 10 |- ( 2 ^ 2 ) = 4
60	59	oveq1i 6660	. . . . . . . . 9 |- ( ( 2 ^ 2 ) x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) = ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) )
61	56, 58, 60	3eqtri 2648	. . . . . . . 8 |- ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) ) ^ 2 ) = ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) )
62	1, 2	hvsubcan2i 27921	. . . . . . . . . . . . . . 15 |- ( ( A +h B ) +h ( A -h B ) ) = ( 2 .h A )
63	62	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( ( A +h B ) +h ( A -h B ) ) +h ( -u 1 .h ( 2 .h C ) ) ) = ( ( 2 .h A ) +h ( -u 1 .h ( 2 .h C ) ) )
64	3, 30, 11	hvadd32i 27911	. . . . . . . . . . . . . 14 |- ( ( ( A +h B ) +h ( -u 1 .h ( 2 .h C ) ) ) +h ( A -h B ) ) = ( ( ( A +h B ) +h ( A -h B ) ) +h ( -u 1 .h ( 2 .h C ) ) )
65	4, 1	hvmulcli 27871	. . . . . . . . . . . . . . 15 |- ( 2 .h A ) e. ~H
66	65, 6	hvsubvali 27877	. . . . . . . . . . . . . 14 |- ( ( 2 .h A ) -h ( 2 .h C ) ) = ( ( 2 .h A ) +h ( -u 1 .h ( 2 .h C ) ) )
67	63, 64, 66	3eqtr4i 2654	. . . . . . . . . . . . 13 |- ( ( ( A +h B ) +h ( -u 1 .h ( 2 .h C ) ) ) +h ( A -h B ) ) = ( ( 2 .h A ) -h ( 2 .h C ) )
68	33	oveq1i 6660	. . . . . . . . . . . . 13 |- ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) = ( ( ( A +h B ) +h ( -u 1 .h ( 2 .h C ) ) ) +h ( A -h B ) )
69	4, 1, 5	hvsubdistr1i 27909	. . . . . . . . . . . . 13 |- ( 2 .h ( A -h C ) ) = ( ( 2 .h A ) -h ( 2 .h C ) )
70	67, 68, 69	3eqtr4i 2654	. . . . . . . . . . . 12 |- ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) = ( 2 .h ( A -h C ) )
71	70	fveq2i 6194	. . . . . . . . . . 11 |- ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) ) = ( normh ` ( 2 .h ( A -h C ) ) )
72	4, 16	norm-iii-i 27996	. . . . . . . . . . 11 |- ( normh ` ( 2 .h ( A -h C ) ) ) = ( ( abs ` 2 ) x. ( normh ` ( A -h C ) ) )
73	53	oveq1i 6660	. . . . . . . . . . 11 |- ( ( abs ` 2 ) x. ( normh ` ( A -h C ) ) ) = ( 2 x. ( normh ` ( A -h C ) ) )
74	71, 72, 73	3eqtri 2648	. . . . . . . . . 10 |- ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) ) = ( 2 x. ( normh ` ( A -h C ) ) )
75	74	oveq1i 6660	. . . . . . . . 9 |- ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) ) ^ 2 ) = ( ( 2 x. ( normh ` ( A -h C ) ) ) ^ 2 )
76	17	recni 10052	. . . . . . . . . 10 |- ( normh ` ( A -h C ) ) e. CC
77	4, 76	sqmuli 12947	. . . . . . . . 9 |- ( ( 2 x. ( normh ` ( A -h C ) ) ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( ( normh ` ( A -h C ) ) ^ 2 ) )
78	59	oveq1i 6660	. . . . . . . . 9 |- ( ( 2 ^ 2 ) x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) = ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) )
79	75, 77, 78	3eqtri 2648	. . . . . . . 8 |- ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) ) ^ 2 ) = ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) )
80	61, 79	oveq12i 6662	. . . . . . 7 |- ( ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) ) ^ 2 ) + ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) ) ^ 2 ) ) = ( ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) )
81	28, 80	eqtr4i 2647	. . . . . 6 |- ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) = ( ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) ) ^ 2 ) + ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) ) ^ 2 ) )
82	7, 11	normpari 28011	. . . . . 6 |- ( ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) -h ( A -h B ) ) ) ^ 2 ) + ( ( normh ` ( ( ( A +h B ) -h ( 2 .h C ) ) +h ( A -h B ) ) ) ^ 2 ) ) = ( ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) )
83	81, 82	eqtri 2644	. . . . 5 |- ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) = ( ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) )
84	83	oveq1i 6660	. . . 4 |- ( ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) / 2 ) = ( ( ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) ) / 2 )
85	4, 10	mulcli 10045	. . . . 5 |- ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) e. CC
86	4, 14	mulcli 10045	. . . . 5 |- ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) e. CC
87	85, 86, 4, 26	divdiri 10782	. . . 4 |- ( ( ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) ) / 2 ) = ( ( ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) / 2 ) + ( ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) / 2 ) )
88	10, 4, 26	divcan3i 10771	. . . . 5 |- ( ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) / 2 ) = ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 )
89	14, 4, 26	divcan3i 10771	. . . . 5 |- ( ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) / 2 ) = ( ( normh ` ( A -h B ) ) ^ 2 )
90	88, 89	oveq12i 6662	. . . 4 |- ( ( ( 2 x. ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) ) / 2 ) + ( ( 2 x. ( ( normh ` ( A -h B ) ) ^ 2 ) ) / 2 ) ) = ( ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) + ( ( normh ` ( A -h B ) ) ^ 2 ) )
91	84, 87, 90	3eqtri 2648	. . 3 |- ( ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) / 2 ) = ( ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) + ( ( normh ` ( A -h B ) ) ^ 2 ) )
92	15, 19, 4, 26	div23i 10783	. . . . 5 |- ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) / 2 ) = ( ( 4 / 2 ) x. ( ( normh ` ( A -h C ) ) ^ 2 ) )
93		4d2e2 11184	. . . . . 6 |- ( 4 / 2 ) = 2
94	93	oveq1i 6660	. . . . 5 |- ( ( 4 / 2 ) x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) = ( 2 x. ( ( normh ` ( A -h C ) ) ^ 2 ) )
95	92, 94	eqtri 2644	. . . 4 |- ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) / 2 ) = ( 2 x. ( ( normh ` ( A -h C ) ) ^ 2 ) )
96	15, 24, 4, 26	div23i 10783	. . . . 5 |- ( ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) / 2 ) = ( ( 4 / 2 ) x. ( ( normh ` ( B -h C ) ) ^ 2 ) )
97	93	oveq1i 6660	. . . . 5 |- ( ( 4 / 2 ) x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) = ( 2 x. ( ( normh ` ( B -h C ) ) ^ 2 ) )
98	96, 97	eqtri 2644	. . . 4 |- ( ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) / 2 ) = ( 2 x. ( ( normh ` ( B -h C ) ) ^ 2 ) )
99	95, 98	oveq12i 6662	. . 3 |- ( ( ( 4 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) / 2 ) + ( ( 4 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) / 2 ) ) = ( ( 2 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) )
100	27, 91, 99	3eqtr3i 2652	. 2 |- ( ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) + ( ( normh ` ( A -h B ) ) ^ 2 ) ) = ( ( 2 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) )
101	10, 14, 100	mvlladdi 10299	1 |- ( ( normh ` ( A -h B ) ) ^ 2 ) = ( ( ( 2 x. ( ( normh ` ( A -h C ) ) ^ 2 ) ) + ( 2 x. ( ( normh ` ( B -h C ) ) ^ 2 ) ) ) - ( ( normh ` ( ( A +h B ) -h ( 2 .h C ) ) ) ^ 2 ) )

Syntax hints:   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   4c4 11072   ^cexp 12860   abscabs 13974   ~Hchil 27776   +h cva 27777   .h csm 27778   normhcno 27780   -h cmv 27782

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  ax-pre-sup 10014  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

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-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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-hnorm 27825  df-hvsub 27828

This theorem is referenced by: (None)