opsqrlem6 - Hilbert Space Explorer

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Theorem opsqrlem6 32626

Description: Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
opsqrlem2.1	⊢ 𝑇 ∈ HrmOp
opsqrlem2.2	⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +_op ((1 / 2) ·_op (𝑇 −_op (𝑥 ∘ 𝑥)))))
opsqrlem2.3	⊢ 𝐹 = seq1(𝑆, (ℕ × { 0_hop }))
opsqrlem6.4	⊢ 𝑇 ≤_op I_op

**Assertion**
Ref	Expression
opsqrlem6	⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤_op I_op )

Distinct variable group: 𝑥,𝑦,𝑇 Allowed substitution hints: 𝑆(𝑥, 𝑦) 𝐹(𝑥, 𝑦) 𝑁(𝑥, 𝑦)

Proof of Theorem opsqrlem6 Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6878	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 5113	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘1) ≤_op I_op ))
3		fveq2 6878	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 5113	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘(𝑘 + 1)) ≤_op I_op ))
5		fveq2 6878	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 5113	. 2 ⊢ (𝑗 = 𝑁 → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘𝑁) ≤_op I_op ))
7		opsqrlem2.1	. . . 4 ⊢ 𝑇 ∈ HrmOp
8		opsqrlem2.2	. . . 4 ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +_op ((1 / 2) ·_op (𝑇 −_op (𝑥 ∘ 𝑥)))))
9		opsqrlem2.3	. . . 4 ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0_hop }))
10	7, 8, 9	opsqrlem2 32622	. . 3 ⊢ (𝐹‘1) = 0_hop
11		idleop 32612	. . 3 ⊢ 0_hop ≤_op I_op
12	10, 11	eqbrtri 5126	. 2 ⊢ (𝐹‘1) ≤_op I_op
13		idhmop 32463	. . . . . . . 8 ⊢ I_op ∈ HrmOp
14	7, 8, 9	opsqrlem4 32624	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7076	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 32503	. . . . . . . 8 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 599	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2760	. . . . . . . 8 ⊢ (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘)))
19		hmopco 32504	. . . . . . . 8 ⊢ ((( I_op −_op (𝐹‘𝑘)) ∈ HrmOp ∧ ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp ∧ (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘)))) → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp)
20	18, 19	mp3an3 1479	. . . . . . 7 ⊢ ((( I_op −_op (𝐹‘𝑘)) ∈ HrmOp ∧ ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp) → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp)
21	17, 17, 20	syl2anc 596	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 32610	. . . . . . 7 ⊢ (( I_op −_op (𝐹‘𝑘)) ∈ HrmOp → 0_hop ≤_op (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))))
23	17, 22	syl 18	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))))
24		opsqrlem6.4	. . . . . . . 8 ⊢ 𝑇 ≤_op I_op
25		leop3 32606	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ I_op ∈ HrmOp) → (𝑇 ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op 𝑇)))
26	7, 13, 25	mp2an 705	. . . . . . . 8 ⊢ (𝑇 ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op 𝑇))
27	24, 26	mpbi 233	. . . . . . 7 ⊢ 0_hop ≤_op ( I_op −_op 𝑇)
28		hmopd 32503	. . . . . . . . 9 ⊢ (( I_op ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( I_op −_op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 705	. . . . . . . 8 ⊢ ( I_op −_op 𝑇) ∈ HrmOp
30		leopadd 32613	. . . . . . . 8 ⊢ ((((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp ∧ ( I_op −_op 𝑇) ∈ HrmOp) ∧ ( 0_hop ≤_op (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∧ 0_hop ≤_op ( I_op −_op 𝑇))) → 0_hop ≤_op ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)))
31	29, 30	mpanl2 714	. . . . . . 7 ⊢ (((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp ∧ ( 0_hop ≤_op (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∧ 0_hop ≤_op ( I_op −_op 𝑇))) → 0_hop ≤_op ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)))
32	27, 31	mpanr2 717	. . . . . 6 ⊢ (((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp ∧ 0_hop ≤_op (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘)))) → 0_hop ≤_op ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)))
33	21, 23, 32	syl2anc 596	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)))
34		2cn 12340	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 32355	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 32240	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 599	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 32355	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6727	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 596	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 32254	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 599	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 32355	. . . . . . . . . . . 12 ⊢ ( I_op ∈ HrmOp → I_op : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ I_op : ℋ⟶ ℋ
47		homulcl 32240	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ I_op : ℋ⟶ ℋ) → (2 ·_op I_op ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 705	. . . . . . . . . 10 ⊢ (2 ·_op I_op ): ℋ⟶ ℋ
49		hosubsub4 32299	. . . . . . . . . 10 ⊢ (((2 ·_op I_op ): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·_op I_op ) −_op ((2 ·_op (𝐹‘𝑘)) +_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
50	48, 49	mp3an1 1477	. . . . . . . . 9 ⊢ (((2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·_op I_op ) −_op ((2 ·_op (𝐹‘𝑘)) +_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
51	38, 44, 50	syl2anc 596	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·_op I_op ) −_op ((2 ·_op (𝐹‘𝑘)) +_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
52		hosubcl 32254	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ)
53	42, 38, 52	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 32264	. . . . . . . . . . . . . . 15 ⊢ (( I_op : ℋ⟶ ℋ ∧ (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ ∧ I_op : ℋ⟶ ℋ) → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) = (( I_op +_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
55	46, 46, 54	mp3an13 1481	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) = (( I_op +_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
56	53, 55	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) = (( I_op +_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
57		ho2times 32300	. . . . . . . . . . . . . . 15 ⊢ ( I_op : ℋ⟶ ℋ → (2 ·_op I_op ) = ( I_op +_op I_op ))
58	46, 57	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2 ·_op I_op ) = ( I_op +_op I_op )
59	58	oveq1i 7423	. . . . . . . . . . . . 13 ⊢ ((2 ·_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) = (( I_op +_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))))
60	56, 59	eqtr4di 2813	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) = ((2 ·_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
61		hoaddsubass 32296	. . . . . . . . . . . . . 14 ⊢ (((2 ·_op I_op ): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) = ((2 ·_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
62	48, 61	mp3an1 1477	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) = ((2 ·_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
63	42, 38, 62	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) = ((2 ·_op I_op ) +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
64	60, 63	eqtr4d 2798	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))))
65	64	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) −_op 𝑇) = ((((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) −_op 𝑇))
66		hoaddcl 32239	. . . . . . . . . . . 12 ⊢ (( I_op : ℋ⟶ ℋ ∧ (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ) → ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))): ℋ⟶ ℋ)
67	46, 53, 66	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 32296	. . . . . . . . . . . 12 ⊢ ((( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))): ℋ⟶ ℋ ∧ I_op : ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) −_op 𝑇) = (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)))
69	46, 40, 68	mp3an23 1482	. . . . . . . . . . 11 ⊢ (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))): ℋ⟶ ℋ → ((( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) −_op 𝑇) = (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)))
70	67, 69	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op I_op ) −_op 𝑇) = (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)))
71		hoaddcl 32239	. . . . . . . . . . . 12 ⊢ (((2 ·_op I_op ): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → ((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
72	48, 42, 71	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 32299	. . . . . . . . . . . 12 ⊢ ((((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) −_op 𝑇) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
74	40, 73	mp3an3 1479	. . . . . . . . . . 11 ⊢ ((((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) → ((((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) −_op 𝑇) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
75	72, 38, 74	syl2anc 596	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) −_op 𝑇) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
76	65, 70, 75	3eqtr3d 2803	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
77		hosubadd4 32295	. . . . . . . . . . . 12 ⊢ ((((2 ·_op I_op ): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)) → (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
78	40, 77	mpanr1 716	. . . . . . . . . . 11 ⊢ ((((2 ·_op I_op ): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
79	48, 78	mpanl1 713	. . . . . . . . . 10 ⊢ (((2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
80	38, 42, 79	syl2anc 596	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((2 ·_op (𝐹‘𝑘)) +_op 𝑇)))
81	76, 80	eqtr4d 2798	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)) = (((2 ·_op I_op ) −_op (2 ·_op (𝐹‘𝑘))) −_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
82		halfcn 12482	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 32240	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℂ ∧ (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
84	82, 44, 83	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 32284	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ) → (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·_op (𝐹‘𝑘)) +_op (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
86	34, 85	mp3an1 1477	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ) → (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·_op (𝐹‘𝑘)) +_op (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
87	36, 84, 86	syl2anc 596	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·_op (𝐹‘𝑘)) +_op (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
88		2thalfe1 12372	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
89	88	oveq1i 7423	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
90		homulass 32283	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
91	34, 82, 90	mp3an12 1480	. . . . . . . . . . . . 13 ⊢ ((𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ → ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
92	44, 91	syl 18	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
93		homullid 32281	. . . . . . . . . . . . 13 ⊢ ((𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ → (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
94	44, 93	syl 18	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
95	89, 92, 94	3eqtr3a 2819	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
96	95	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((2 ·_op (𝐹‘𝑘)) +_op (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·_op (𝐹‘𝑘)) +_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
97	87, 96	eqtrd 2795	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·_op (𝐹‘𝑘)) +_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
98	97	oveq2d 7429	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((2 ·_op I_op ) −_op (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·_op I_op ) −_op ((2 ·_op (𝐹‘𝑘)) +_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
99	51, 81, 98	3eqtr4d 2805	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)) = ((2 ·_op I_op ) −_op (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
100		hoaddcl 32239	. . . . . . . . 9 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ) → ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
101	36, 84, 100	syl2anc 596	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
102		hosubdi 32289	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ I_op : ℋ⟶ ℋ ∧ ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ) → (2 ·_op ( I_op −_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·_op I_op ) −_op (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
103	34, 46, 102	mp3an12 1480	. . . . . . . 8 ⊢ (((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ → (2 ·_op ( I_op −_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·_op I_op ) −_op (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
104	101, 103	syl 18	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (2 ·_op ( I_op −_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·_op I_op ) −_op (2 ·_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
105	99, 104	eqtr4d 2798	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)) = (2 ·_op ( I_op −_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
106		hosubcl 32254	. . . . . . . . . 10 ⊢ (( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
107	46, 36, 106	sylancr 599	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
108		hocsubdir 32266	. . . . . . . . . 10 ⊢ (( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ ∧ ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ ( I_op −_op (𝐹‘𝑘))) −_op ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘)))))
109	46, 108	mp3an1 1477	. . . . . . . . 9 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ ( I_op −_op (𝐹‘𝑘))) −_op ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘)))))
110	36, 107, 109	syl2anc 596	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ ( I_op −_op (𝐹‘𝑘))) −_op ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘)))))
111		hmoplin 32423	. . . . . . . . . . . . . . 15 ⊢ ( I_op ∈ HrmOp → I_op ∈ LinOp)
112	13, 111	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ I_op ∈ LinOp
113		hoddi 32471	. . . . . . . . . . . . . 14 ⊢ (( I_op ∈ LinOp ∧ I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))))
114	112, 46, 113	mp3an12 1480	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))))
115	36, 114	syl 18	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))))
116	46	hoid1i 32270	. . . . . . . . . . . . . 14 ⊢ ( I_op ∘ I_op ) = I_op
117	116	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ I_op ) = I_op )
118		hoico2 32238	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( I_op ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
119	36, 118	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
120	117, 119	oveq12d 7431	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
121	115, 120	eqtrd 2795	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
122		hmoplin 32423	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
123	15, 122	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
124		hoddi 32471	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ LinOp ∧ I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
125	46, 124	mp3an2 1478	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ LinOp ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
126	123, 36, 125	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
127		hoico1 32237	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ((𝐹‘𝑘) ∘ I_op ) = (𝐹‘𝑘))
128	36, 127	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ I_op ) = (𝐹‘𝑘))
129	128	oveq1d 7428	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
130	126, 129	eqtrd 2795	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	121, 130	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ ( I_op −_op (𝐹‘𝑘))) −_op ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘)))) = (( I_op −_op (𝐹‘𝑘)) −_op ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
132	36, 46	jctil 529	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
133		hosubadd4 32295	. . . . . . . . . . 11 ⊢ ((( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) ∧ ((𝐹‘𝑘): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)) → (( I_op −_op (𝐹‘𝑘)) −_op ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((𝐹‘𝑘) +_op (𝐹‘𝑘))))
134	132, 36, 42, 133	syl12anc 850	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (( I_op −_op (𝐹‘𝑘)) −_op ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((𝐹‘𝑘) +_op (𝐹‘𝑘))))
135	131, 134	eqtrd 2795	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ ( I_op −_op (𝐹‘𝑘))) −_op ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘)))) = (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((𝐹‘𝑘) +_op (𝐹‘𝑘))))
136		ho2times 32300	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → (2 ·_op (𝐹‘𝑘)) = ((𝐹‘𝑘) +_op (𝐹‘𝑘)))
137	36, 136	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2 ·_op (𝐹‘𝑘)) = ((𝐹‘𝑘) +_op (𝐹‘𝑘)))
138	137	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) = (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op ((𝐹‘𝑘) +_op (𝐹‘𝑘))))
139		hoaddsubass 32296	. . . . . . . . . . 11 ⊢ (( I_op : ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) = ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
140	46, 139	mp3an1 1477	. . . . . . . . . 10 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) = ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
141	42, 38, 140	syl2anc 596	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( I_op +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −_op (2 ·_op (𝐹‘𝑘))) = ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
142	135, 138, 141	3eqtr2d 2801	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ ( I_op −_op (𝐹‘𝑘))) −_op ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘)))) = ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
143	110, 142	eqtrd 2795	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))))
144	143	oveq1d 7428	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)) = (( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))) +_op ( I_op −_op 𝑇)))
145	7, 8, 9	opsqrlem5 32625	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
146	145	oveq2d 7429	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘(𝑘 + 1))) = ( I_op −_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
147	146	oveq2d 7429	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1)))) = (2 ·_op ( I_op −_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
148	105, 144, 147	3eqtr4d 2805	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)) = (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
149	33, 148	breqtrd 5131	. . . 4 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
150		peano2nn 12269	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
151	14	ffvelcdmi 7076	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
152	150, 151	syl 18	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153		hmopd 32503	. . . . . 6 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
154	13, 152, 153	sylancr 599	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155		2re 12339	. . . . . 6 ⊢ 2 ∈ ℝ
156		2pos 12369	. . . . . 6 ⊢ 0 < 2
157		leopmul 32615	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp ∧ 0 < 2) → ( 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1))) ↔ 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1))))))
158	155, 156, 157	mp3an13 1481	. . . . 5 ⊢ (( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp → ( 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1))) ↔ 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1))))))
159	154, 158	syl 18	. . . 4 ⊢ (𝑘 ∈ ℕ → ( 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1))) ↔ 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1))))))
160	149, 159	mpbird 260	. . 3 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1))))
161		leop3 32606	. . . 4 ⊢ (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ I_op ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
162	152, 13, 161	sylancl 598	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
163	160, 162	mpbird 260	. 2 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤_op I_op )
164	2, 4, 6, 12, 163	nn1suc 12279	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤_op I_op )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {csn 4584 class class class wbr 5103 × cxp 5653 ∘ ccom 5659 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ∈ cmpo 7415 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 < clt 11267 / cdiv 11895 ℕcn 12257 2c2 12319 seqcseq 14065 ℋchba 31400 +_op chos 31419 ·_op chot 31420 −_op chod 31421 0_hop ch0o 31424 I_op chio 31425 LinOpclo 31428 HrmOpcho 31431 ≤_op cleo 31439

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cc 10437 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 ax-mulf 11204 ax-hilex 31480 ax-hfvadd 31481 ax-hvcom 31482 ax-hvass 31483 ax-hv0cl 31484 ax-hvaddid 31485 ax-hfvmul 31486 ax-hvmulid 31487 ax-hvmulass 31488 ax-hvdistr1 31489 ax-hvdistr2 31490 ax-hvmul0 31491 ax-hfi 31560 ax-his1 31563 ax-his2 31564 ax-his3 31565 ax-his4 31566 ax-hcompl 31683

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-rlim 15576 df-sum 15774 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-cn 23452 df-cnp 23453 df-lm 23454 df-haus 23540 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cfil 25483 df-cau 25484 df-cmet 25485 df-grpo 30974 df-gid 30975 df-ginv 30976 df-gdiv 30977 df-ablo 31026 df-vc 31040 df-nv 31073 df-va 31076 df-ba 31077 df-sm 31078 df-0v 31079 df-vs 31080 df-nmcv 31081 df-ims 31082 df-dip 31182 df-ssp 31203 df-ph 31294 df-cbn 31344 df-hnorm 31449 df-hba 31450 df-hvsub 31452 df-hlim 31453 df-hcau 31454 df-sh 31688 df-ch 31702 df-oc 31733 df-ch0 31734 df-shs 31789 df-pjh 31876 df-hosum 32211 df-homul 32212 df-hodif 32213 df-h0op 32229 df-iop 32230 df-lnop 32322 df-hmop 32325 df-leop 32333

This theorem is used by: (None)

W3C validator