opsqrlem6 - Hilbert Space Explorer

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Theorem opsqrlem6 32164

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 6906	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 5153	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘1) ≤_op I_op ))
3		fveq2 6906	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 5153	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘(𝑘 + 1)) ≤_op I_op ))
5		fveq2 6906	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 5153	. 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 32160	. . 3 ⊢ (𝐹‘1) = 0_hop
11		idleop 32150	. . 3 ⊢ 0_hop ≤_op I_op
12	10, 11	eqbrtri 5164	. 2 ⊢ (𝐹‘1) ≤_op I_op
13		idhmop 32001	. . . . . . . 8 ⊢ I_op ∈ HrmOp
14	7, 8, 9	opsqrlem4 32162	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7103	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 32041	. . . . . . . 8 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 587	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2737	. . . . . . . 8 ⊢ (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘)))
19		hmopco 32042	. . . . . . . 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 1452	. . . . . . 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 584	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 32148	. . . . . . 7 ⊢ (( I_op −_op (𝐹‘𝑘)) ∈ HrmOp → 0_hop ≤_op (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))))
23	17, 22	syl 17	. . . . . 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 32144	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ I_op ∈ HrmOp) → (𝑇 ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op 𝑇)))
26	7, 13, 25	mp2an 692	. . . . . . . 8 ⊢ (𝑇 ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op 𝑇))
27	24, 26	mpbi 230	. . . . . . 7 ⊢ 0_hop ≤_op ( I_op −_op 𝑇)
28		hmopd 32041	. . . . . . . . 9 ⊢ (( I_op ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( I_op −_op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 692	. . . . . . . 8 ⊢ ( I_op −_op 𝑇) ∈ HrmOp
30		leopadd 32151	. . . . . . . 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 701	. . . . . . 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 704	. . . . . 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 584	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)))
34		2cn 12341	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 31893	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 31778	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 587	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 31893	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6760	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 31792	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 587	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 31893	. . . . . . . . . . . 12 ⊢ ( I_op ∈ HrmOp → I_op : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ I_op : ℋ⟶ ℋ
47		homulcl 31778	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ I_op : ℋ⟶ ℋ) → (2 ·_op I_op ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 692	. . . . . . . . . 10 ⊢ (2 ·_op I_op ): ℋ⟶ ℋ
49		hosubsub4 31837	. . . . . . . . . 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 1450	. . . . . . . . 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 584	. . . . . . . 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 31792	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 31802	. . . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 31838	. . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . 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 2795	. . . . . . . . . . . 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 31834	. . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 7446	. . . . . . . . . 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 31777	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 31834	. . . . . . . . . . . 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 1455	. . . . . . . . . . 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 17	. . . . . . . . . 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 31777	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 31837	. . . . . . . . . . . 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 1452	. . . . . . . . . . 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 584	. . . . . . . . . 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 2785	. . . . . . . . 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 31833	. . . . . . . . . . . 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 703	. . . . . . . . . . 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 700	. . . . . . . . . 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 584	. . . . . . . . 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 2780	. . . . . . . 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 12481	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 31778	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 31822	. . . . . . . . . . . 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 1450	. . . . . . . . . . 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 584	. . . . . . . . . 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		2ne0 12370	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11998	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 7441	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 31821	. . . . . . . . . . . . . 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))))))
92	34, 82, 91	mp3an12 1453	. . . . . . . . . . . . 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))))))
93	44, 92	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
94		homullid 31819	. . . . . . . . . . . . 13 ⊢ ((𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ → (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
95	44, 94	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
96	90, 93, 95	3eqtr3a 2801	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 7447	. . . . . . . . . 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)))))
98	87, 97	eqtrd 2777	. . . . . . . . 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)))))
99	98	oveq2d 7447	. . . . . . . 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))))))
100	51, 81, 99	3eqtr4d 2787	. . . . . . 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))))))))
101		hoaddcl 31777	. . . . . . . . 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)
102	36, 84, 101	syl2anc 584	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
103		hosubdi 31827	. . . . . . . . 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))))))))
104	34, 46, 103	mp3an12 1453	. . . . . . . 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))))))))
105	102, 104	syl 17	. . . . . . 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))))))))
106	100, 105	eqtr4d 2780	. . . . . 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))))))))
107		hosubcl 31792	. . . . . . . . . 10 ⊢ (( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 587	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 31804	. . . . . . . . . 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)))))
110	46, 109	mp3an1 1450	. . . . . . . . 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)))))
111	36, 108, 110	syl2anc 584	. . . . . . . 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)))))
112		hmoplin 31961	. . . . . . . . . . . . . . 15 ⊢ ( I_op ∈ HrmOp → I_op ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ I_op ∈ LinOp
114		hoddi 32009	. . . . . . . . . . . . . 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))))
115	113, 46, 114	mp3an12 1453	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))))
116	36, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))))
117	46	hoid1i 31808	. . . . . . . . . . . . . 14 ⊢ ( I_op ∘ I_op ) = I_op
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ I_op ) = I_op )
119		hoico2 31776	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( I_op ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
120	36, 119	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
121	118, 120	oveq12d 7449	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
122	116, 121	eqtrd 2777	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
123		hmoplin 31961	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 32009	. . . . . . . . . . . . . 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))))
126	46, 125	mp3an2 1451	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ LinOp ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
127	124, 36, 126	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 31775	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ((𝐹‘𝑘) ∘ I_op ) = (𝐹‘𝑘))
129	36, 128	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ I_op ) = (𝐹‘𝑘))
130	129	oveq1d 7446	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2777	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 7449	. . . . . . . . . 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)))))
133	36, 46	jctil 519	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
134		hosubadd4 31833	. . . . . . . . . . 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))))
135	133, 36, 42, 134	syl12anc 837	. . . . . . . . . 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))))
136	132, 135	eqtrd 2777	. . . . . . . . 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))))
137		ho2times 31838	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → (2 ·_op (𝐹‘𝑘)) = ((𝐹‘𝑘) +_op (𝐹‘𝑘)))
138	36, 137	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2 ·_op (𝐹‘𝑘)) = ((𝐹‘𝑘) +_op (𝐹‘𝑘)))
139	138	oveq2d 7447	. . . . . . . . 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))))
140		hoaddsubass 31834	. . . . . . . . . . 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)))))
141	46, 140	mp3an1 1450	. . . . . . . . . 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)))))
142	42, 38, 141	syl2anc 584	. . . . . . . . 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)))))
143	136, 139, 142	3eqtr2d 2783	. . . . . . . 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)))))
144	111, 143	eqtrd 2777	. . . . . . 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)))))
145	144	oveq1d 7446	. . . . . 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)))
146	7, 8, 9	opsqrlem5 32163	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 7447	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘(𝑘 + 1))) = ( I_op −_op ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
148	147	oveq2d 7447	. . . . . 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))))))))
149	106, 145, 148	3eqtr4d 2787	. . . . 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)))))
150	33, 149	breqtrd 5169	. . . 4 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
151		peano2nn 12278	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelcdmi 7103	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 32041	. . . . . 6 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 587	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 12340	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 12369	. . . . . 6 ⊢ 0 < 2
158		leopmul 32153	. . . . . 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))))))
159	156, 157, 158	mp3an13 1454	. . . . 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))))))
160	155, 159	syl 17	. . . 4 ⊢ (𝑘 ∈ ℕ → ( 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1))) ↔ 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1))))))
161	150, 160	mpbird 257	. . 3 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1))))
162		leop3 32144	. . . 4 ⊢ (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ I_op ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
163	153, 13, 162	sylancl 586	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
164	161, 163	mpbird 257	. 2 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤_op I_op )
165	2, 4, 6, 12, 164	nn1suc 12288	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤_op I_op )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 class class class wbr 5143 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 / cdiv 11920 ℕcn 12266 2c2 12321 seqcseq 14042 ℋchba 30938 +_op chos 30957 ·_op chot 30958 −_op chod 30959 0_hop ch0o 30962 I_op chio 30963 LinOpclo 30966 HrmOpcho 30969 ≤_op cleo 30977

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104 ax-hcompl 31221

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-lm 23237 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cfil 25289 df-cau 25290 df-cmet 25291 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-dip 30720 df-ssp 30741 df-ph 30832 df-cbn 30882 df-hnorm 30987 df-hba 30988 df-hvsub 30990 df-hlim 30991 df-hcau 30992 df-sh 31226 df-ch 31240 df-oc 31271 df-ch0 31272 df-shs 31327 df-pjh 31414 df-hosum 31749 df-homul 31750 df-hodif 31751 df-h0op 31767 df-iop 31768 df-lnop 31860 df-hmop 31863 df-leop 31871

This theorem is referenced by: (None)

W3C validator