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Theorem polid2i 27398

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 10975	. 2 ⊢ 4 ∈ ℂ
2		polid2.1	. . 3 ⊢ 𝐴 ∈ ℋ
3		polid2.2	. . 3 ⊢ 𝐵 ∈ ℋ
4	2, 3	hicli 27322	. 2 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
5		4ne0 10994	. 2 ⊢ 4 ≠ 0
6		2cn 10968	. . . 4 ⊢ 2 ∈ ℂ
7		polid2.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
8		polid2.4	. . . . . 6 ⊢ 𝐷 ∈ ℋ
9	7, 8	hicli 27322	. . . . 5 ⊢ (𝐶 ·_ih 𝐷) ∈ ℂ
10	4, 9	addcli 9923	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) ∈ ℂ
11	4, 9	subcli 10236	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)) ∈ ℂ
12	6, 10, 11	adddii 9929	. . 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 10202	. . . . . . 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 1416	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐵))
15	4	2timesi 11024	. . . . . 6 ⊢ (2 · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐵))
16	14, 15	eqtr4i 2635	. . . . 5 ⊢ (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))) = (2 · (𝐴 ·_ih 𝐵))
17	16	oveq2i 6560	. . . 4 ⊢ (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))) = (2 · (2 · (𝐴 ·_ih 𝐵)))
18	6, 6, 4	mulassi 9928	. . . 4 ⊢ ((2 · 2) · (𝐴 ·_ih 𝐵)) = (2 · (2 · (𝐴 ·_ih 𝐵)))
19		2t2e4 11054	. . . . 5 ⊢ (2 · 2) = 4
20	19	oveq1i 6559	. . . 4 ⊢ ((2 · 2) · (𝐴 ·_ih 𝐵)) = (4 · (𝐴 ·_ih 𝐵))
21	17, 18, 20	3eqtr2ri 2639	. . 3 ⊢ (4 · (𝐴 ·_ih 𝐵)) = (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))))
22	2, 8	hicli 27322	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
23	7, 3	hicli 27322	. . . . . . 7 ⊢ (𝐶 ·_ih 𝐵) ∈ ℂ
24	22, 23	addcli 9923	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) ∈ ℂ
25	24, 10, 10	pnncani 10255	. . . . 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 27358	. . . . . 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 27359	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
28	26, 27	oveq12i 6561	. . . . 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 11024	. . . . 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 2642	. . . 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 9874	. . . . . . . . . 10 ⊢ i ∈ ℂ
32	31, 7	hvmulcli 27255	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐶) ∈ ℋ
33	31, 3	hvmulcli 27255	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐵) ∈ ℋ
34	2, 32, 8, 33	normlem8 27358	. . . . . . . 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 27359	. . . . . . . 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 6561	. . . . . . 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 27322	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
38	22, 37	addcli 9923	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) ∈ ℂ
39	2, 33	hicli 27322	. . . . . . . . 9 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
40	32, 8	hicli 27322	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih 𝐷) ∈ ℂ
41	39, 40	addcli 9923	. . . . . . . 8 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) ∈ ℂ
42	38, 41, 41	pnncani 10255	. . . . . . 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 11024	. . . . . . . 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 27327	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (i ·_ℎ 𝐵)) = ((∗‘i) · (𝐴 ·_ih 𝐵)))
45	31, 2, 3, 44	mp3an 1416	. . . . . . . . . . 11 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) = ((∗‘i) · (𝐴 ·_ih 𝐵))
46		cji 13747	. . . . . . . . . . . 12 ⊢ (∗‘i) = -i
47	46	oveq1i 6559	. . . . . . . . . . 11 ⊢ ((∗‘i) · (𝐴 ·_ih 𝐵)) = (-i · (𝐴 ·_ih 𝐵))
48	45, 47	eqtri 2632	. . . . . . . . . 10 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) = (-i · (𝐴 ·_ih 𝐵))
49		ax-his3 27325	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((i ·_ℎ 𝐶) ·_ih 𝐷) = (i · (𝐶 ·_ih 𝐷)))
50	31, 7, 8, 49	mp3an 1416	. . . . . . . . . 10 ⊢ ((i ·_ℎ 𝐶) ·_ih 𝐷) = (i · (𝐶 ·_ih 𝐷))
51	48, 50	oveq12i 6561	. . . . . . . . 9 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) = ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))
52	51	oveq2i 6560	. . . . . . . 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 2634	. . . . . . 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 2636	. . . . . 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 6560	. . . . 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 10161	. . . . . . . 8 ⊢ -i ∈ ℂ
57	56, 4	mulcli 9924	. . . . . . 7 ⊢ (-i · (𝐴 ·_ih 𝐵)) ∈ ℂ
58	31, 9	mulcli 9924	. . . . . . 7 ⊢ (i · (𝐶 ·_ih 𝐷)) ∈ ℂ
59	57, 58	addcli 9923	. . . . . 6 ⊢ ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))) ∈ ℂ
60	6, 31, 59	mul12i 10110	. . . . 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 9929	. . . . . . 7 ⊢ (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))) = ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷))))
62	31, 31	mulneg2i 10356	. . . . . . . . . . 11 ⊢ (i · -i) = -(i · i)
63		ixi 10535	. . . . . . . . . . . 12 ⊢ (i · i) = -1
64	63	negeqi 10153	. . . . . . . . . . 11 ⊢ -(i · i) = --1
65		negneg1e1 11005	. . . . . . . . . . 11 ⊢ --1 = 1
66	62, 64, 65	3eqtri 2636	. . . . . . . . . 10 ⊢ (i · -i) = 1
67	66	oveq1i 6559	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (1 · (𝐴 ·_ih 𝐵))
68	31, 56, 4	mulassi 9928	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (i · (-i · (𝐴 ·_ih 𝐵)))
69	4	mulid2i 9922	. . . . . . . . 9 ⊢ (1 · (𝐴 ·_ih 𝐵)) = (𝐴 ·_ih 𝐵)
70	67, 68, 69	3eqtr3i 2640	. . . . . . . 8 ⊢ (i · (-i · (𝐴 ·_ih 𝐵))) = (𝐴 ·_ih 𝐵)
71	63	oveq1i 6559	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (-1 · (𝐶 ·_ih 𝐷))
72	31, 31, 9	mulassi 9928	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (i · (i · (𝐶 ·_ih 𝐷)))
73	9	mulm1i 10354	. . . . . . . . 9 ⊢ (-1 · (𝐶 ·_ih 𝐷)) = -(𝐶 ·_ih 𝐷)
74	71, 72, 73	3eqtr3i 2640	. . . . . . . 8 ⊢ (i · (i · (𝐶 ·_ih 𝐷))) = -(𝐶 ·_ih 𝐷)
75	70, 74	oveq12i 6561	. . . . . . 7 ⊢ ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷)))) = ((𝐴 ·_ih 𝐵) + -(𝐶 ·_ih 𝐷))
76	4, 9	negsubi 10238	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + -(𝐶 ·_ih 𝐷)) = ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))
77	61, 75, 76	3eqtri 2636	. . . . . 6 ⊢ (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))) = ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷))
78	77	oveq2i 6560	. . . . 5 ⊢ (2 · (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))))) = (2 · ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))
79	55, 60, 78	3eqtr2i 2638	. . . 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 6561	. . 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 2642	. 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 10737	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 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 4c4 10949 ∗ccj 13684 ℋchil 27160 +_ℎ cva 27161 ·_ℎ csm 27162 ·_ih csp 27163 −_ℎ cmv 27166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hfvadd 27241 ax-hfvmul 27246 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-3 10957 df-4 10958 df-cj 13687 df-re 13688 df-im 13689 df-hvsub 27212

This theorem is referenced by: polidi 27399 lnopeq0lem1 28248 lnophmlem2 28260

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