opsqrlem6 - Hilbert Space Explorer

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Theorem opsqrlem6 29576

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 6446	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 4896	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘1) ≤_op I_op ))
3		fveq2 6446	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 4896	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘(𝑘 + 1)) ≤_op I_op ))
5		fveq2 6446	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 4896	. 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 29572	. . 3 ⊢ (𝐹‘1) = 0_hop
11		idleop 29562	. . 3 ⊢ 0_hop ≤_op I_op
12	10, 11	eqbrtri 4907	. 2 ⊢ (𝐹‘1) ≤_op I_op
13		idhmop 29413	. . . . . . . 8 ⊢ I_op ∈ HrmOp
14	7, 8, 9	opsqrlem4 29574	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelrni 6622	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 29453	. . . . . . . 8 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 581	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2778	. . . . . . . 8 ⊢ (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘)))
19		hmopco 29454	. . . . . . . 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 1523	. . . . . . 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 579	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 29560	. . . . . . 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 29556	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ I_op ∈ HrmOp) → (𝑇 ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op 𝑇)))
26	7, 13, 25	mp2an 682	. . . . . . . 8 ⊢ (𝑇 ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op 𝑇))
27	24, 26	mpbi 222	. . . . . . 7 ⊢ 0_hop ≤_op ( I_op −_op 𝑇)
28		hmopd 29453	. . . . . . . . 9 ⊢ (( I_op ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( I_op −_op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 682	. . . . . . . 8 ⊢ ( I_op −_op 𝑇) ∈ HrmOp
30		leopadd 29563	. . . . . . . 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 691	. . . . . . 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 694	. . . . . 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 579	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)))
34		2cn 11450	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 29305	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 29190	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 581	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 29305	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6308	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 579	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 29204	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 581	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 29305	. . . . . . . . . . . 12 ⊢ ( I_op ∈ HrmOp → I_op : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ I_op : ℋ⟶ ℋ
47		homulcl 29190	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ I_op : ℋ⟶ ℋ) → (2 ·_op I_op ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 682	. . . . . . . . . 10 ⊢ (2 ·_op I_op ): ℋ⟶ ℋ
49		hosubsub4 29249	. . . . . . . . . 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 1521	. . . . . . . . 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 579	. . . . . . . 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 29204	. . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 29214	. . . . . . . . . . . . . . 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 1525	. . . . . . . . . . . . . 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 29250	. . . . . . . . . . . . . . 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 6932	. . . . . . . . . . . . 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	syl6eqr 2832	. . . . . . . . . . . 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 29246	. . . . . . . . . . . . . 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 1521	. . . . . . . . . . . . 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 579	. . . . . . . . . . . 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 2817	. . . . . . . . . . 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 6937	. . . . . . . . . 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 29189	. . . . . . . . . . . 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 581	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 29246	. . . . . . . . . . . 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 1526	. . . . . . . . . . 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 29189	. . . . . . . . . . . 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 581	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 29249	. . . . . . . . . . . 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 1523	. . . . . . . . . . 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 579	. . . . . . . . . 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 2822	. . . . . . . . 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 29245	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 690	. . . . . . . . . 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 579	. . . . . . . . 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 2817	. . . . . . . 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 11597	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 29190	. . . . . . . . . . . 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 581	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 29234	. . . . . . . . . . . 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 1521	. . . . . . . . . . 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 579	. . . . . . . . . 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 11486	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11106	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 6932	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 29233	. . . . . . . . . . . . . 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 1524	. . . . . . . . . . . . 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		homulid2 29231	. . . . . . . . . . . . 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 2838	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 6938	. . . . . . . . . 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 2814	. . . . . . . . 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 6938	. . . . . . . 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 2824	. . . . . . 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 29189	. . . . . . . . 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 579	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
103		hosubdi 29239	. . . . . . . . 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 1524	. . . . . . . 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 2817	. . . . . 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 29204	. . . . . . . . . 10 ⊢ (( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 581	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 29216	. . . . . . . . . 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 1521	. . . . . . . . 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 579	. . . . . . . 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 29373	. . . . . . . . . . . . . . 15 ⊢ ( I_op ∈ HrmOp → I_op ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ I_op ∈ LinOp
114		hoddi 29421	. . . . . . . . . . . . . 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 1524	. . . . . . . . . . . . 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 29220	. . . . . . . . . . . . . 14 ⊢ ( I_op ∘ I_op ) = I_op
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ I_op ) = I_op )
119		hoico2 29188	. . . . . . . . . . . . . 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 6940	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
122	116, 121	eqtrd 2814	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
123		hmoplin 29373	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 29421	. . . . . . . . . . . . . 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 1522	. . . . . . . . . . . . 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 579	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 29187	. . . . . . . . . . . . . 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 6937	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2814	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 6940	. . . . . . . . . 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 515	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
134		hosubadd4 29245	. . . . . . . . . . 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 827	. . . . . . . . . 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 2814	. . . . . . . . 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 29250	. . . . . . . . . . 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 6938	. . . . . . . . 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 29246	. . . . . . . . . . 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 1521	. . . . . . . . . 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 579	. . . . . . . . 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 2820	. . . . . . . 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 2814	. . . . . . 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 6937	. . . . . 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 29575	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 6938	. . . . . . 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 6938	. . . . . 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 2824	. . . . 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 4912	. . . 4 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
151		peano2nn 11388	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelrni 6622	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 29453	. . . . . 6 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 581	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 11449	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 11485	. . . . . 6 ⊢ 0 < 2
158		leopmul 29565	. . . . . 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 1525	. . . . 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 249	. . 3 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1))))
162		leop3 29556	. . . 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 580	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
164	161, 163	mpbird 249	. 2 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤_op I_op )
165	2, 4, 6, 12, 164	nn1suc 11397	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤_op I_op )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {csn 4398 class class class wbr 4886 × cxp 5353 ∘ ccom 5359 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 / cdiv 11032 ℕcn 11374 2c2 11430 seqcseq 13119 ℋchba 28348 +_op chos 28367 ·_op chot 28368 −_op chod 28369 0_hop ch0o 28372 I_op chio 28373 LinOpclo 28376 HrmOpcho 28379 ≤_op cleo 28387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 ax-hilex 28428 ax-hfvadd 28429 ax-hvcom 28430 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvmulass 28436 ax-hvdistr1 28437 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his1 28511 ax-his2 28512 ax-his3 28513 ax-his4 28514 ax-hcompl 28631

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-rlim 14628 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-cn 21439 df-cnp 21440 df-lm 21441 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cfil 23461 df-cau 23462 df-cmet 23463 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ims 28028 df-dip 28128 df-ssp 28149 df-ph 28240 df-cbn 28291 df-hnorm 28397 df-hba 28398 df-hvsub 28400 df-hlim 28401 df-hcau 28402 df-sh 28636 df-ch 28650 df-oc 28681 df-ch0 28682 df-shs 28739 df-pjh 28826 df-hosum 29161 df-homul 29162 df-hodif 29163 df-h0op 29179 df-iop 29180 df-lnop 29272 df-hmop 29275 df-leop 29283

This theorem is referenced by: (None)

W3C validator