opsqrlem6 - Hilbert Space Explorer

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Theorem opsqrlem6 32216

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 6840	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 5095	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘1) ≤_op I_op ))
3		fveq2 6840	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 5095	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤_op I_op ↔ (𝐹‘(𝑘 + 1)) ≤_op I_op ))
5		fveq2 6840	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 5095	. 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 32212	. . 3 ⊢ (𝐹‘1) = 0_hop
11		idleop 32202	. . 3 ⊢ 0_hop ≤_op I_op
12	10, 11	eqbrtri 5106	. 2 ⊢ (𝐹‘1) ≤_op I_op
13		idhmop 32053	. . . . . . . 8 ⊢ I_op ∈ HrmOp
14	7, 8, 9	opsqrlem4 32214	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7035	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 32093	. . . . . . . 8 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 588	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2736	. . . . . . . 8 ⊢ (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘)))
19		hmopco 32094	. . . . . . . 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 1453	. . . . . . 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 585	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 32200	. . . . . . 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 32196	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ I_op ∈ HrmOp) → (𝑇 ≤_op I_op ↔ 0_hop ≤_op ( I_op −_op 𝑇)))
26	7, 13, 25	mp2an 693	. . . . . . . 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 32093	. . . . . . . . 9 ⊢ (( I_op ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( I_op −_op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 693	. . . . . . . 8 ⊢ ( I_op −_op 𝑇) ∈ HrmOp
30		leopadd 32203	. . . . . . . 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 702	. . . . . . 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 705	. . . . . 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 585	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op ((( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) +_op ( I_op −_op 𝑇)))
34		2cn 12256	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 31945	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 31830	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 588	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·_op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 31945	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6692	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 585	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 31844	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 588	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 31945	. . . . . . . . . . . 12 ⊢ ( I_op ∈ HrmOp → I_op : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ I_op : ℋ⟶ ℋ
47		homulcl 31830	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ I_op : ℋ⟶ ℋ) → (2 ·_op I_op ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 693	. . . . . . . . . 10 ⊢ (2 ·_op I_op ): ℋ⟶ ℋ
49		hosubsub4 31889	. . . . . . . . . 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 1451	. . . . . . . . 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 585	. . . . . . . 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 31844	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 31854	. . . . . . . . . . . . . . 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 1455	. . . . . . . . . . . . . 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 31890	. . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . 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 2789	. . . . . . . . . . . 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 31886	. . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 7382	. . . . . . . . . 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 31829	. . . . . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op +_op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −_op (2 ·_op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 31886	. . . . . . . . . . . 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 1456	. . . . . . . . . . 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 31829	. . . . . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·_op I_op ) +_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 31889	. . . . . . . . . . . 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 1453	. . . . . . . . . . 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 585	. . . . . . . . . 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 2779	. . . . . . . . 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 31885	. . . . . . . . . . . 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 704	. . . . . . . . . . 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 701	. . . . . . . . . 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 585	. . . . . . . . 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 2774	. . . . . . . 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 12391	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 31830	. . . . . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 31874	. . . . . . . . . . . 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 1451	. . . . . . . . . . 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 585	. . . . . . . . . 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 12285	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11886	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 7377	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 31873	. . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . 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 31871	. . . . . . . . . . . . 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 2795	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 7383	. . . . . . . . . 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 2771	. . . . . . . . 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 7383	. . . . . . . 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 2781	. . . . . . 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 31829	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
103		hosubdi 31879	. . . . . . . . 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 1454	. . . . . . . 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 2774	. . . . . 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 31844	. . . . . . . . . 10 ⊢ (( I_op : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 588	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 31856	. . . . . . . . . 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 1451	. . . . . . . . 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 585	. . . . . . . 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 32013	. . . . . . . . . . . . . . 15 ⊢ ( I_op ∈ HrmOp → I_op ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ I_op ∈ LinOp
114		hoddi 32061	. . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . 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 31860	. . . . . . . . . . . . . 14 ⊢ ( I_op ∘ I_op ) = I_op
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ I_op ) = I_op )
119		hoico2 31828	. . . . . . . . . . . . . 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 7385	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
122	116, 121	eqtrd 2771	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
123		hmoplin 32013	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 32061	. . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 31827	. . . . . . . . . . . . . 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 7382	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2771	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 7385	. . . . . . . . . 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 31885	. . . . . . . . . . 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 2771	. . . . . . . . 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 31890	. . . . . . . . . . 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 7383	. . . . . . . . 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 31886	. . . . . . . . . . 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 1451	. . . . . . . . . 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 585	. . . . . . . . 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 2777	. . . . . . . 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 2771	. . . . . . 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 7382	. . . . . 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 32215	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 7383	. . . . . . 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 7383	. . . . . 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 2781	. . . . 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 5111	. . . 4 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
151		peano2nn 12186	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelcdmi 7035	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 32093	. . . . . 6 ⊢ (( I_op ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 588	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( I_op −_op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 12255	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 12284	. . . . . 6 ⊢ 0 < 2
158		leopmul 32205	. . . . . 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 1455	. . . . 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 32196	. . . 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 587	. . 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 12196	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤_op I_op )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {csn 4567 class class class wbr 5085 × cxp 5629 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 / cdiv 11807 ℕcn 12174 2c2 12236 seqcseq 13963 ℋchba 30990 +_op chos 31009 ·_op chot 31010 −_op chod 31011 0_hop ch0o 31014 I_op chio 31015 LinOpclo 31018 HrmOpcho 31021 ≤_op cleo 31029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31070 ax-hfvadd 31071 ax-hvcom 31072 ax-hvass 31073 ax-hv0cl 31074 ax-hvaddid 31075 ax-hfvmul 31076 ax-hvmulid 31077 ax-hvmulass 31078 ax-hvdistr1 31079 ax-hvdistr2 31080 ax-hvmul0 31081 ax-hfi 31150 ax-his1 31153 ax-his2 31154 ax-his3 31155 ax-his4 31156 ax-hcompl 31273

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-lm 23194 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cfil 25222 df-cau 25223 df-cmet 25224 df-grpo 30564 df-gid 30565 df-ginv 30566 df-gdiv 30567 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-vs 30670 df-nmcv 30671 df-ims 30672 df-dip 30772 df-ssp 30793 df-ph 30884 df-cbn 30934 df-hnorm 31039 df-hba 31040 df-hvsub 31042 df-hlim 31043 df-hcau 31044 df-sh 31278 df-ch 31292 df-oc 31323 df-ch0 31324 df-shs 31379 df-pjh 31466 df-hosum 31801 df-homul 31802 df-hodif 31803 df-h0op 31819 df-iop 31820 df-lnop 31912 df-hmop 31915 df-leop 31923

This theorem is referenced by: (None)

W3C validator