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Theorem polid2i 29420

Description: Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
polid2.1	⊢ 𝐴 ∈ ℋ
polid2.2	⊢ 𝐵 ∈ ℋ
polid2.3	⊢ 𝐶 ∈ ℋ
polid2.4	⊢ 𝐷 ∈ ℋ

**Assertion**
Ref	Expression
polid2i	⊢ (𝐴 ·_ih 𝐵) = (((((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) − ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵))) + (i · (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵)))))) / 4)

**Proof of Theorem polid2i**
Step	Hyp	Ref	Expression
1		4cn 11988	. 2 ⊢ 4 ∈ ℂ
2		polid2.1	. . 3 ⊢ 𝐴 ∈ ℋ
3		polid2.2	. . 3 ⊢ 𝐵 ∈ ℋ
4	2, 3	hicli 29344	. 2 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
5		4ne0 12011	. 2 ⊢ 4 ≠ 0
6		2cn 11978	. . . 4 ⊢ 2 ∈ ℂ
7		polid2.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
8		polid2.4	. . . . . 6 ⊢ 𝐷 ∈ ℋ
9	7, 8	hicli 29344	. . . . 5 ⊢ (𝐶 ·_ih 𝐷) ∈ ℂ
10	4, 9	addcli 10912	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) ∈ ℂ
11	4, 9	subcli 11227	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)) ∈ ℂ
12	6, 10, 11	adddii 10918	. . 3 ⊢ (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))) = ((2 · ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷))) + (2 · ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))))
13		ppncan 11193	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) ∈ ℂ ∧ (𝐶 ·_ih 𝐷) ∈ ℂ ∧ (𝐴 ·_ih 𝐵) ∈ ℂ) → (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐵)))
14	4, 9, 4, 13	mp3an 1459	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐵))
15	4	2timesi 12041	. . . . . 6 ⊢ (2 · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐵))
16	14, 15	eqtr4i 2769	. . . . 5 ⊢ (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))) = (2 · (𝐴 ·_ih 𝐵))
17	16	oveq2i 7266	. . . 4 ⊢ (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))) = (2 · (2 · (𝐴 ·_ih 𝐵)))
18	6, 6, 4	mulassi 10917	. . . 4 ⊢ ((2 · 2) · (𝐴 ·_ih 𝐵)) = (2 · (2 · (𝐴 ·_ih 𝐵)))
19		2t2e4 12067	. . . . 5 ⊢ (2 · 2) = 4
20	19	oveq1i 7265	. . . 4 ⊢ ((2 · 2) · (𝐴 ·_ih 𝐵)) = (4 · (𝐴 ·_ih 𝐵))
21	17, 18, 20	3eqtr2ri 2773	. . 3 ⊢ (4 · (𝐴 ·_ih 𝐵)) = (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))))
22	2, 8	hicli 29344	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
23	7, 3	hicli 29344	. . . . . . 7 ⊢ (𝐶 ·_ih 𝐵) ∈ ℂ
24	22, 23	addcli 10912	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) ∈ ℂ
25	24, 10, 10	pnncani 11246	. . . . 5 ⊢ ((((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷))) − (((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))) = (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
26	2, 7, 8, 3	normlem8 29380	. . . . . 6 ⊢ ((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
27	2, 7, 8, 3	normlem9 29381	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
28	26, 27	oveq12i 7267	. . . . 5 ⊢ (((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) − ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵))) = ((((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷))) − (((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷))))
29	10	2timesi 12041	. . . . 5 ⊢ (2 · ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷))) = (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
30	25, 28, 29	3eqtr4i 2776	. . . 4 ⊢ (((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) − ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵))) = (2 · ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
31		ax-icn 10861	. . . . . . . . . 10 ⊢ i ∈ ℂ
32	31, 7	hvmulcli 29277	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐶) ∈ ℋ
33	31, 3	hvmulcli 29277	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐵) ∈ ℋ
34	2, 32, 8, 33	normlem8 29380	. . . . . . . 8 ⊢ ((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) = (((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) + ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)))
35	2, 32, 8, 33	normlem9 29381	. . . . . . . 8 ⊢ ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵))) = (((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) − ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)))
36	34, 35	oveq12i 7267	. . . . . . 7 ⊢ (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵)))) = ((((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) + ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷))) − (((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) − ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷))))
37	32, 33	hicli 29344	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
38	22, 37	addcli 10912	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) ∈ ℂ
39	2, 33	hicli 29344	. . . . . . . . 9 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
40	32, 8	hicli 29344	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih 𝐷) ∈ ℂ
41	39, 40	addcli 10912	. . . . . . . 8 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) ∈ ℂ
42	38, 41, 41	pnncani 11246	. . . . . . 7 ⊢ ((((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) + ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷))) − (((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) − ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)))) = (((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) + ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)))
43	41	2timesi 12041	. . . . . . . 8 ⊢ (2 · ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷))) = (((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) + ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)))
44		his5 29349	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (i ·_ℎ 𝐵)) = ((∗‘i) · (𝐴 ·_ih 𝐵)))
45	31, 2, 3, 44	mp3an 1459	. . . . . . . . . . 11 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) = ((∗‘i) · (𝐴 ·_ih 𝐵))
46		cji 14798	. . . . . . . . . . . 12 ⊢ (∗‘i) = -i
47	46	oveq1i 7265	. . . . . . . . . . 11 ⊢ ((∗‘i) · (𝐴 ·_ih 𝐵)) = (-i · (𝐴 ·_ih 𝐵))
48	45, 47	eqtri 2766	. . . . . . . . . 10 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) = (-i · (𝐴 ·_ih 𝐵))
49		ax-his3 29347	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((i ·_ℎ 𝐶) ·_ih 𝐷) = (i · (𝐶 ·_ih 𝐷)))
50	31, 7, 8, 49	mp3an 1459	. . . . . . . . . 10 ⊢ ((i ·_ℎ 𝐶) ·_ih 𝐷) = (i · (𝐶 ·_ih 𝐷))
51	48, 50	oveq12i 7267	. . . . . . . . 9 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) = ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))
52	51	oveq2i 7266	. . . . . . . 8 ⊢ (2 · ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷))) = (2 · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))))
53	43, 52	eqtr3i 2768	. . . . . . 7 ⊢ (((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) + ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷))) = (2 · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))))
54	36, 42, 53	3eqtri 2770	. . . . . 6 ⊢ (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵)))) = (2 · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))))
55	54	oveq2i 7266	. . . . 5 ⊢ (i · (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵))))) = (i · (2 · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))))
56		negicn 11152	. . . . . . . 8 ⊢ -i ∈ ℂ
57	56, 4	mulcli 10913	. . . . . . 7 ⊢ (-i · (𝐴 ·_ih 𝐵)) ∈ ℂ
58	31, 9	mulcli 10913	. . . . . . 7 ⊢ (i · (𝐶 ·_ih 𝐷)) ∈ ℂ
59	57, 58	addcli 10912	. . . . . 6 ⊢ ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))) ∈ ℂ
60	6, 31, 59	mul12i 11100	. . . . 5 ⊢ (2 · (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))))) = (i · (2 · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))))
61	31, 57, 58	adddii 10918	. . . . . . 7 ⊢ (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))) = ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷))))
62	31, 31	mulneg2i 11352	. . . . . . . . . . 11 ⊢ (i · -i) = -(i · i)
63		ixi 11534	. . . . . . . . . . . 12 ⊢ (i · i) = -1
64	63	negeqi 11144	. . . . . . . . . . 11 ⊢ -(i · i) = --1
65		negneg1e1 12021	. . . . . . . . . . 11 ⊢ --1 = 1
66	62, 64, 65	3eqtri 2770	. . . . . . . . . 10 ⊢ (i · -i) = 1
67	66	oveq1i 7265	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (1 · (𝐴 ·_ih 𝐵))
68	31, 56, 4	mulassi 10917	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (i · (-i · (𝐴 ·_ih 𝐵)))
69	4	mulid2i 10911	. . . . . . . . 9 ⊢ (1 · (𝐴 ·_ih 𝐵)) = (𝐴 ·_ih 𝐵)
70	67, 68, 69	3eqtr3i 2774	. . . . . . . 8 ⊢ (i · (-i · (𝐴 ·_ih 𝐵))) = (𝐴 ·_ih 𝐵)
71	63	oveq1i 7265	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (-1 · (𝐶 ·_ih 𝐷))
72	31, 31, 9	mulassi 10917	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (i · (i · (𝐶 ·_ih 𝐷)))
73	9	mulm1i 11350	. . . . . . . . 9 ⊢ (-1 · (𝐶 ·_ih 𝐷)) = -(𝐶 ·_ih 𝐷)
74	71, 72, 73	3eqtr3i 2774	. . . . . . . 8 ⊢ (i · (i · (𝐶 ·_ih 𝐷))) = -(𝐶 ·_ih 𝐷)
75	70, 74	oveq12i 7267	. . . . . . 7 ⊢ ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷)))) = ((𝐴 ·_ih 𝐵) + -(𝐶 ·_ih 𝐷))
76	4, 9	negsubi 11229	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + -(𝐶 ·_ih 𝐷)) = ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))
77	61, 75, 76	3eqtri 2770	. . . . . 6 ⊢ (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))) = ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))
78	77	oveq2i 7266	. . . . 5 ⊢ (2 · (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))))) = (2 · ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))
79	55, 60, 78	3eqtr2i 2772	. . . 4 ⊢ (i · (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵))))) = (2 · ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))
80	30, 79	oveq12i 7267	. . 3 ⊢ ((((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) − ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵))) + (i · (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵)))))) = ((2 · ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷))) + (2 · ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))))
81	12, 21, 80	3eqtr4i 2776	. 2 ⊢ (4 · (𝐴 ·_ih 𝐵)) = ((((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) − ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵))) + (i · (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵))))))
82	1, 4, 5, 81	mvllmuli 11738	1 ⊢ (𝐴 ·_ih 𝐵) = (((((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) − ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵))) + (i · (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵)))))) / 4)

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 1c1 10803 ici 10804 + caddc 10805 · cmul 10807 − cmin 11135 -cneg 11136 / cdiv 11562 2c2 11958 4c4 11960 ∗ccj 14735 ℋchba 29182 +_ℎ cva 29183 ·_ℎ csm 29184 ·_ih csp 29185 −_ℎ cmv 29188

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-hfvadd 29263 ax-hfvmul 29268 ax-hfi 29342 ax-his1 29345 ax-his2 29346 ax-his3 29347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-2 11966 df-3 11967 df-4 11968 df-cj 14738 df-re 14739 df-im 14740 df-hvsub 29234

This theorem is referenced by: polidi 29421 lnopeq0lem1 30268 lnophmlem2 30280

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