opsqrlem6 - Hilbert Space Explorer

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Theorem opsqrlem6 32047

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 5112	. 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 5112	. 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 5112	. 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 32043	. . 3 ⊢ (𝐹‘1) = 0_hop
11		idleop 32033	. . 3 ⊢ 0_hop ≤_op I_op
12	10, 11	eqbrtri 5123	. 2 ⊢ (𝐹‘1) ≤_op I_op
13		idhmop 31884	. . . . . . . 8 ⊢ I_op ∈ HrmOp
14	7, 8, 9	opsqrlem4 32045	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7037	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 31924	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘))) = (( I_op −_op (𝐹‘𝑘)) ∘ ( I_op −_op (𝐹‘𝑘)))
19		hmopco 31925	. . . . . . . 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 32031	. . . . . . 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 32027	. . . . . . . . 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 31924	. . . . . . . . 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 32034	. . . . . . . 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 12237	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 31776	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 31661	. . . . . . . . . 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 31776	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6694	. . . . . . . . . . 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 31675	. . . . . . . . . 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 31776	. . . . . . . . . . . 12 ⊢ ( I_op ∈ HrmOp → I_op : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ I_op : ℋ⟶ ℋ
47		homulcl 31661	. . . . . . . . . . 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 31720	. . . . . . . . . 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 31675	. . . . . . . . . . . . . . 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 31685	. . . . . . . . . . . . . . 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 31721	. . . . . . . . . . . . . . 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 7379	. . . . . . . . . . . . 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 2782	. . . . . . . . . . . 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 31717	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . 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 7384	. . . . . . . . . 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 31660	. . . . . . . . . . . 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 31717	. . . . . . . . . . . 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 31660	. . . . . . . . . . . 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 31720	. . . . . . . . . . . 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 2772	. . . . . . . . 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 31716	. . . . . . . . . . . 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 2767	. . . . . . . 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 12372	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 31661	. . . . . . . . . . . 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 31705	. . . . . . . . . . . 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 12266	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11889	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 7379	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 31704	. . . . . . . . . . . . . 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 31702	. . . . . . . . . . . . 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 2788	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 7385	. . . . . . . . . 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 2764	. . . . . . . . 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 7385	. . . . . . . 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 2774	. . . . . . 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 31660	. . . . . . . . 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 31710	. . . . . . . . 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 2767	. . . . . 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 31675	. . . . . . . . . 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 31687	. . . . . . . . . 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 31844	. . . . . . . . . . . . . . 15 ⊢ ( I_op ∈ HrmOp → I_op ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ I_op ∈ LinOp
114		hoddi 31892	. . . . . . . . . . . . . 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 31691	. . . . . . . . . . . . . 14 ⊢ ( I_op ∘ I_op ) = I_op
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ I_op ) = I_op )
119		hoico2 31659	. . . . . . . . . . . . . 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 7387	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( I_op ∘ I_op ) −_op ( I_op ∘ (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
122	116, 121	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( I_op ∘ ( I_op −_op (𝐹‘𝑘))) = ( I_op −_op (𝐹‘𝑘)))
123		hmoplin 31844	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 31892	. . . . . . . . . . . . . 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 31658	. . . . . . . . . . . . . 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 7384	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ I_op ) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( I_op −_op (𝐹‘𝑘))) = ((𝐹‘𝑘) −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 7387	. . . . . . . . . 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 31716	. . . . . . . . . . 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 836	. . . . . . . . . 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 2764	. . . . . . . . 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 31721	. . . . . . . . . . 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 7385	. . . . . . . . 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 31717	. . . . . . . . . . 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 2770	. . . . . . . 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 2764	. . . . . . 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 7384	. . . . . 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 32046	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +_op ((1 / 2) ·_op (𝑇 −_op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 7385	. . . . . . 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 7385	. . . . . 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 2774	. . . . 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 5128	. . . 4 ⊢ (𝑘 ∈ ℕ → 0_hop ≤_op (2 ·_op ( I_op −_op (𝐹‘(𝑘 + 1)))))
151		peano2nn 12174	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelcdmi 7037	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 31924	. . . . . 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 12236	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 12265	. . . . . 6 ⊢ 0 < 2
158		leopmul 32036	. . . . . 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 32027	. . . 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 12184	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 2109 {csn 4585 class class class wbr 5102 × cxp 5629 ∘ ccom 5635 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 / cdiv 11811 ℕcn 12162 2c2 12217 seqcseq 13942 ℋchba 30821 +_op chos 30840 ·_op chot 30841 −_op chod 30842 0_hop ch0o 30845 I_op chio 30846 LinOpclo 30849 HrmOpcho 30852 ≤_op cleo 30860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124 ax-hilex 30901 ax-hfvadd 30902 ax-hvcom 30903 ax-hvass 30904 ax-hv0cl 30905 ax-hvaddid 30906 ax-hfvmul 30907 ax-hvmulid 30908 ax-hvmulass 30909 ax-hvdistr1 30910 ax-hvdistr2 30911 ax-hvmul0 30912 ax-hfi 30981 ax-his1 30984 ax-his2 30985 ax-his3 30986 ax-his4 30987 ax-hcompl 31104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-cn 23090 df-cnp 23091 df-lm 23092 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cfil 25131 df-cau 25132 df-cmet 25133 df-grpo 30395 df-gid 30396 df-ginv 30397 df-gdiv 30398 df-ablo 30447 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-vs 30501 df-nmcv 30502 df-ims 30503 df-dip 30603 df-ssp 30624 df-ph 30715 df-cbn 30765 df-hnorm 30870 df-hba 30871 df-hvsub 30873 df-hlim 30874 df-hcau 30875 df-sh 31109 df-ch 31123 df-oc 31154 df-ch0 31155 df-shs 31210 df-pjh 31297 df-hosum 31632 df-homul 31633 df-hodif 31634 df-h0op 31650 df-iop 31651 df-lnop 31743 df-hmop 31746 df-leop 31754

This theorem is referenced by: (None)

W3C validator