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Theorem polid2i 31136

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 12247	. 2 ⊢ 4 ∈ ℂ
2		polid2.1	. . 3 ⊢ 𝐴 ∈ ℋ
3		polid2.2	. . 3 ⊢ 𝐵 ∈ ℋ
4	2, 3	hicli 31060	. 2 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
5		4ne0 12270	. 2 ⊢ 4 ≠ 0
6		2cn 12237	. . . 4 ⊢ 2 ∈ ℂ
7		polid2.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
8		polid2.4	. . . . . 6 ⊢ 𝐷 ∈ ℋ
9	7, 8	hicli 31060	. . . . 5 ⊢ (𝐶 ·_ih 𝐷) ∈ ℂ
10	4, 9	addcli 11156	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) ∈ ℂ
11	4, 9	subcli 11474	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)) ∈ ℂ
12	6, 10, 11	adddii 11162	. . 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 11440	. . . . . . 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 12295	. . . . . 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 7380	. . . 4 ⊢ (2 · (((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐵) − (𝐶 ·_ih 𝐷)))) = (2 · (2 · (𝐴 ·_ih 𝐵)))
18	6, 6, 4	mulassi 11161	. . . 4 ⊢ ((2 · 2) · (𝐴 ·_ih 𝐵)) = (2 · (2 · (𝐴 ·_ih 𝐵)))
19		2t2e4 12321	. . . . 5 ⊢ (2 · 2) = 4
20	19	oveq1i 7379	. . . 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 31060	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
23	7, 3	hicli 31060	. . . . . . 7 ⊢ (𝐶 ·_ih 𝐵) ∈ ℂ
24	22, 23	addcli 11156	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) ∈ ℂ
25	24, 10, 10	pnncani 11493	. . . . 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 31096	. . . . . 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 31097	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐷) + (𝐶 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐶 ·_ih 𝐷)))
28	26, 27	oveq12i 7381	. . . . 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 12295	. . . . 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 11103	. . . . . . . . . 10 ⊢ i ∈ ℂ
32	31, 7	hvmulcli 30993	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐶) ∈ ℋ
33	31, 3	hvmulcli 30993	. . . . . . . . 9 ⊢ (i ·_ℎ 𝐵) ∈ ℋ
34	2, 32, 8, 33	normlem8 31096	. . . . . . . 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 31097	. . . . . . . 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 7381	. . . . . . 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 31060	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
38	22, 37	addcli 11156	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐷) + ((i ·_ℎ 𝐶) ·_ih (i ·_ℎ 𝐵))) ∈ ℂ
39	2, 33	hicli 31060	. . . . . . . . 9 ⊢ (𝐴 ·_ih (i ·_ℎ 𝐵)) ∈ ℂ
40	32, 8	hicli 31060	. . . . . . . . 9 ⊢ ((i ·_ℎ 𝐶) ·_ih 𝐷) ∈ ℂ
41	39, 40	addcli 11156	. . . . . . . 8 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) ∈ ℂ
42	38, 41, 41	pnncani 11493	. . . . . . 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 12295	. . . . . . . 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 31065	. . . . . . . . . . . 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 15101	. . . . . . . . . . . 12 ⊢ (∗‘i) = -i
47	46	oveq1i 7379	. . . . . . . . . . 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 31063	. . . . . . . . . . 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 7381	. . . . . . . . 9 ⊢ ((𝐴 ·_ih (i ·_ℎ 𝐵)) + ((i ·_ℎ 𝐶) ·_ih 𝐷)) = ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))
52	51	oveq2i 7380	. . . . . . . 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 7380	. . . . 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 11398	. . . . . . . 8 ⊢ -i ∈ ℂ
57	56, 4	mulcli 11157	. . . . . . 7 ⊢ (-i · (𝐴 ·_ih 𝐵)) ∈ ℂ
58	31, 9	mulcli 11157	. . . . . . 7 ⊢ (i · (𝐶 ·_ih 𝐷)) ∈ ℂ
59	57, 58	addcli 11156	. . . . . 6 ⊢ ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷))) ∈ ℂ
60	6, 31, 59	mul12i 11345	. . . . 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 11162	. . . . . . 7 ⊢ (i · ((-i · (𝐴 ·_ih 𝐵)) + (i · (𝐶 ·_ih 𝐷)))) = ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷))))
62	31, 31	mulneg2i 11601	. . . . . . . . . . 11 ⊢ (i · -i) = -(i · i)
63		ixi 11783	. . . . . . . . . . . 12 ⊢ (i · i) = -1
64	63	negeqi 11390	. . . . . . . . . . 11 ⊢ -(i · i) = --1
65		negneg1e1 12151	. . . . . . . . . . 11 ⊢ --1 = 1
66	62, 64, 65	3eqtri 2756	. . . . . . . . . 10 ⊢ (i · -i) = 1
67	66	oveq1i 7379	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (1 · (𝐴 ·_ih 𝐵))
68	31, 56, 4	mulassi 11161	. . . . . . . . 9 ⊢ ((i · -i) · (𝐴 ·_ih 𝐵)) = (i · (-i · (𝐴 ·_ih 𝐵)))
69	4	mullidi 11155	. . . . . . . . 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 7379	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (-1 · (𝐶 ·_ih 𝐷))
72	31, 31, 9	mulassi 11161	. . . . . . . . 9 ⊢ ((i · i) · (𝐶 ·_ih 𝐷)) = (i · (i · (𝐶 ·_ih 𝐷)))
73	9	mulm1i 11599	. . . . . . . . 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 7381	. . . . . . 7 ⊢ ((i · (-i · (𝐴 ·_ih 𝐵))) + (i · (i · (𝐶 ·_ih 𝐷)))) = ((𝐴 ·_ih 𝐵) + -(𝐶 ·_ih 𝐷))
76	4, 9	negsubi 11476	. . . . . . 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 7380	. . . . 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 7381	. . 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 11991	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 6499 (class class class)co 7369 ℂcc 11042 1c1 11045 ici 11046 + caddc 11047 · cmul 11049 − cmin 11381 -cneg 11382 / cdiv 11811 2c2 12217 4c4 12219 ∗ccj 15038 ℋchba 30898 +_ℎ cva 30899 ·_ℎ csm 30900 ·_ih csp 30901 −_ℎ cmv 30904

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 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-hfvadd 30979 ax-hfvmul 30984 ax-hfi 31058 ax-his1 31061 ax-his2 31062 ax-his3 31063

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-cj 15041 df-re 15042 df-im 15043 df-hvsub 30950

This theorem is referenced by: polidi 31137 lnopeq0lem1 31984 lnophmlem2 31996

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