polid2i - Hilbert Space Explorer

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Theorem polid2i 31101

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 12213	. 2 ⊢ 4 ∈ ℂ
2		polid2.1	. . 3 ⊢ 𝐴 ∈ ℋ
3		polid2.2	. . 3 ⊢ 𝐵 ∈ ℋ
4	2, 3	hicli 31025	. 2 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
5		4ne0 12236	. 2 ⊢ 4 ≠ 0
6		2cn 12203	. . . 4 ⊢ 2 ∈ ℂ
7		polid2.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
8		polid2.4	. . . . . 6 ⊢ 𝐷 ∈ ℋ
9	7, 8	hicli 31025	. . . . 5 ⊢ (𝐶 ·_ih 𝐷) ∈ ℂ
10	4, 9	addcli 11121	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) ∈ ℂ
11	4, 9	subcli 11440	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)) ∈ ℂ
12	6, 10, 11	adddii 11127	. . 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 11406	. . . . . . 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 1463	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐵))
15	4	2timesi 12261	. . . . . 6 ⊢ (2 · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐵))
16	14, 15	eqtr4i 2755	. . . . 5 ⊢ (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))) = (2 · (𝐴 ·_ih 𝐵))
17	16	oveq2i 7360	. . . 4 ⊢ (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))) = (2 · (2 · (𝐴 ·_ih 𝐵)))
18	6, 6, 4	mulassi 11126	. . . 4 ⊢ ((2 · 2) · (𝐴 ·_ih 𝐵)) = (2 · (2 · (𝐴 ·_ih 𝐵)))
19		2t2e4 12287	. . . . 5 ⊢ (2 · 2) = 4
20	19	oveq1i 7359	. . . 4 ⊢ ((2 · 2) · (𝐴 ·_ih 𝐵)) = (4 · (𝐴 ·_ih 𝐵))
21	17, 18, 20	3eqtr2ri 2759	. . 3 ⊢ (4 · (𝐴 ·_ih 𝐵)) = (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))))
22	2, 8	hicli 31025	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
23	7, 3	hicli 31025	. . . . . . 7 ⊢ (𝐶 ·_ih 𝐵) ∈ ℂ
24	22, 23	addcli 11121	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) ∈ ℂ
25	24, 10, 10	pnncani 11459	. . . . 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 31061	. . . . . 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 31062	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
28	26, 27	oveq12i 7361	. . . . 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 12261	. . . . 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 2762	. . . 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 11068	. . . . . . . . . 10 ⊢ i ∈ ℂ
32	31, 7	hvmulcli 30958	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐶) ∈ ℋ
33	31, 3	hvmulcli 30958	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐵) ∈ ℋ
34	2, 32, 8, 33	normlem8 31061	. . . . . . . 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 31062	. . . . . . . 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 7361	. . . . . . 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 31025	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
38	22, 37	addcli 11121	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) ∈ ℂ
39	2, 33	hicli 31025	. . . . . . . . 9 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
40	32, 8	hicli 31025	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih 𝐷) ∈ ℂ
41	39, 40	addcli 11121	. . . . . . . 8 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) ∈ ℂ
42	38, 41, 41	pnncani 11459	. . . . . . 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 12261	. . . . . . . 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 31030	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (i ·_ℎ 𝐵)) = ((∗‘i) · (𝐴 ·_ih 𝐵)))
45	31, 2, 3, 44	mp3an 1463	. . . . . . . . . . 11 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) = ((∗‘i) · (𝐴 ·_ih 𝐵))
46		cji 15066	. . . . . . . . . . . 12 ⊢ (∗‘i) = -i
47	46	oveq1i 7359	. . . . . . . . . . 11 ⊢ ((∗‘i) · (𝐴 ·_ih 𝐵)) = (-i · (𝐴 ·_ih 𝐵))
48	45, 47	eqtri 2752	. . . . . . . . . 10 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) = (-i · (𝐴 ·_ih 𝐵))
49		ax-his3 31028	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((i ·_ℎ 𝐶) ·_ih 𝐷) = (i · (𝐶 ·_ih 𝐷)))
50	31, 7, 8, 49	mp3an 1463	. . . . . . . . . 10 ⊢ ((i ·_ℎ 𝐶) ·_ih 𝐷) = (i · (𝐶 ·_ih 𝐷))
51	48, 50	oveq12i 7361	. . . . . . . . 9 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) = ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))
52	51	oveq2i 7360	. . . . . . . 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 2754	. . . . . . 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 2756	. . . . . 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 7360	. . . . 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 11364	. . . . . . . 8 ⊢ -i ∈ ℂ
57	56, 4	mulcli 11122	. . . . . . 7 ⊢ (-i · (𝐴 ·_ih 𝐵)) ∈ ℂ
58	31, 9	mulcli 11122	. . . . . . 7 ⊢ (i · (𝐶 ·_ih 𝐷)) ∈ ℂ
59	57, 58	addcli 11121	. . . . . 6 ⊢ ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))) ∈ ℂ
60	6, 31, 59	mul12i 11311	. . . . 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 11127	. . . . . . 7 ⊢ (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))) = ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷))))
62	31, 31	mulneg2i 11567	. . . . . . . . . . 11 ⊢ (i · -i) = -(i · i)
63		ixi 11749	. . . . . . . . . . . 12 ⊢ (i · i) = -1
64	63	negeqi 11356	. . . . . . . . . . 11 ⊢ -(i · i) = --1
65		negneg1e1 12117	. . . . . . . . . . 11 ⊢ --1 = 1
66	62, 64, 65	3eqtri 2756	. . . . . . . . . 10 ⊢ (i · -i) = 1
67	66	oveq1i 7359	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (1 · (𝐴 ·_ih 𝐵))
68	31, 56, 4	mulassi 11126	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (i · (-i · (𝐴 ·_ih 𝐵)))
69	4	mullidi 11120	. . . . . . . . 9 ⊢ (1 · (𝐴 ·_ih 𝐵)) = (𝐴 ·_ih 𝐵)
70	67, 68, 69	3eqtr3i 2760	. . . . . . . 8 ⊢ (i · (-i · (𝐴 ·_ih 𝐵))) = (𝐴 ·_ih 𝐵)
71	63	oveq1i 7359	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (-1 · (𝐶 ·_ih 𝐷))
72	31, 31, 9	mulassi 11126	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (i · (i · (𝐶 ·_ih 𝐷)))
73	9	mulm1i 11565	. . . . . . . . 9 ⊢ (-1 · (𝐶 ·_ih 𝐷)) = -(𝐶 ·_ih 𝐷)
74	71, 72, 73	3eqtr3i 2760	. . . . . . . 8 ⊢ (i · (i · (𝐶 ·_ih 𝐷))) = -(𝐶 ·_ih 𝐷)
75	70, 74	oveq12i 7361	. . . . . . 7 ⊢ ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷)))) = ((𝐴 ·_ih 𝐵) + -(𝐶 ·_ih 𝐷))
76	4, 9	negsubi 11442	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + -(𝐶 ·_ih 𝐷)) = ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))
77	61, 75, 76	3eqtri 2756	. . . . . 6 ⊢ (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))) = ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))
78	77	oveq2i 7360	. . . . 5 ⊢ (2 · (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))))) = (2 · ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))
79	55, 60, 78	3eqtr2i 2758	. . . 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 7361	. . 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 2762	. 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 11957	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 1540 ∈ wcel 2109 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 1c1 11010 ici 11011 + caddc 11012 · cmul 11014 − cmin 11347 -cneg 11348 / cdiv 11777 2c2 12183 4c4 12185 ∗ccj 15003 ℋchba 30863 +_ℎ cva 30864 ·_ℎ csm 30865 ·_ih csp 30866 −_ℎ cmv 30869

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-hfvadd 30944 ax-hfvmul 30949 ax-hfi 31023 ax-his1 31026 ax-his2 31027 ax-his3 31028

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-cj 15006 df-re 15007 df-im 15008 df-hvsub 30915

This theorem is referenced by: polidi 31102 lnopeq0lem1 31949 lnophmlem2 31961

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