Theorem opsqrlem6 31386

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 × { 0hop }))
opsqrlem6.4	⊢ 𝑇 ≤op Iop

Assertion

Ref	Expression
opsqrlem6	⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )

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 6889	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 5158	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3		fveq2 6889	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 5158	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5		fveq2 6889	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 5158	. 2 ⊢ (𝑗 = 𝑁 → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘𝑁) ≤op Iop ))
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 × { 0hop }))
10	7, 8, 9	opsqrlem2 31382	. . 3 ⊢ (𝐹‘1) = 0hop
11		idleop 31372	. . 3 ⊢ 0hop ≤op Iop
12	10, 11	eqbrtri 5169	. 2 ⊢ (𝐹‘1) ≤op Iop
13		idhmop 31223	. . . . . . . 8 ⊢ Iop ∈ HrmOp
14	7, 8, 9	opsqrlem4 31384	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7083	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 31263	. . . . . . . 8 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 588	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2733	. . . . . . . 8 ⊢ (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))
19		hmopco 31264	. . . . . . . 8 ⊢ ((( Iop −op (𝐹‘𝑘)) ∈ HrmOp ∧ ( Iop −op (𝐹‘𝑘)) ∈ HrmOp ∧ (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
20	18, 19	mp3an3 1451	. . . . . . 7 ⊢ ((( Iop −op (𝐹‘𝑘)) ∈ HrmOp ∧ ( Iop −op (𝐹‘𝑘)) ∈ HrmOp) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
21	17, 17, 20	syl2anc 585	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 31370	. . . . . . 7 ⊢ (( Iop −op (𝐹‘𝑘)) ∈ HrmOp → 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))))
23	17, 22	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))))
24		opsqrlem6.4	. . . . . . . 8 ⊢ 𝑇 ≤op Iop
25		leop3 31366	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇)))
26	7, 13, 25	mp2an 691	. . . . . . . 8 ⊢ (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇))
27	24, 26	mpbi 229	. . . . . . 7 ⊢ 0hop ≤op ( Iop −op 𝑇)
28		hmopd 31263	. . . . . . . . 9 ⊢ (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iop −op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 691	. . . . . . . 8 ⊢ ( Iop −op 𝑇) ∈ HrmOp
30		leopadd 31373	. . . . . . . 8 ⊢ ((((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp ∧ ( Iop −op 𝑇) ∈ HrmOp) ∧ ( 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∧ 0hop ≤op ( Iop −op 𝑇))) → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
31	29, 30	mpanl2 700	. . . . . . 7 ⊢ (((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp ∧ ( 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∧ 0hop ≤op ( Iop −op 𝑇))) → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
32	27, 31	mpanr2 703	. . . . . 6 ⊢ (((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp ∧ 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))) → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
33	21, 23, 32	syl2anc 585	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
34		2cn 12284	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 31115	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 31000	. . . . . . . . . 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 31115	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6739	. . . . . . . . . . 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 31014	. . . . . . . . . 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 31115	. . . . . . . . . . . 12 ⊢ ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ Iop : ℋ⟶ ℋ
47		homulcl 31000	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 691	. . . . . . . . . 10 ⊢ (2 ·op Iop ): ℋ⟶ ℋ
49		hosubsub4 31059	. . . . . . . . . 10 ⊢ (((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
50	48, 49	mp3an1 1449	. . . . . . . . 9 ⊢ (((2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
51	38, 44, 50	syl2anc 585	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
52		hosubcl 31014	. . . . . . . . . . . . . . 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 31024	. . . . . . . . . . . . . . 15 ⊢ (( Iop : ℋ⟶ ℋ ∧ (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ) → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
55	46, 46, 54	mp3an13 1453	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
56	53, 55	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
57		ho2times 31060	. . . . . . . . . . . . . . 15 ⊢ ( Iop : ℋ⟶ ℋ → (2 ·op Iop ) = ( Iop +op Iop ))
58	46, 57	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2 ·op Iop ) = ( Iop +op Iop )
59	58	oveq1i 7416	. . . . . . . . . . . . 13 ⊢ ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))))
60	56, 59	eqtr4di 2791	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
61		hoaddsubass 31056	. . . . . . . . . . . . . 14 ⊢ (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
62	48, 61	mp3an1 1449	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
63	42, 38, 62	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
64	60, 63	eqtr4d 2776	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))))
65	64	oveq1d 7421	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇))
66		hoaddcl 30999	. . . . . . . . . . . 12 ⊢ (( Iop : ℋ⟶ ℋ ∧ (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ) → ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ)
67	46, 53, 66	sylancr 588	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 31056	. . . . . . . . . . . 12 ⊢ ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
69	46, 40, 68	mp3an23 1454	. . . . . . . . . . 11 ⊢ (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
70	67, 69	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
71		hoaddcl 30999	. . . . . . . . . . . 12 ⊢ (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → ((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
72	48, 42, 71	sylancr 588	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 31059	. . . . . . . . . . . 12 ⊢ ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
74	40, 73	mp3an3 1451	. . . . . . . . . . 11 ⊢ ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
75	72, 38, 74	syl2anc 585	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
76	65, 70, 75	3eqtr3d 2781	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
77		hosubadd4 31055	. . . . . . . . . . . 12 ⊢ ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
78	40, 77	mpanr1 702	. . . . . . . . . . 11 ⊢ ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
79	48, 78	mpanl1 699	. . . . . . . . . 10 ⊢ (((2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
80	38, 42, 79	syl2anc 585	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
81	76, 80	eqtr4d 2776	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
82		halfcn 12424	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 31000	. . . . . . . . . . . 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 31044	. . . . . . . . . . . 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 1449	. . . . . . . . . . 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 12313	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11942	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 7416	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 31043	. . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . 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 31041	. . . . . . . . . . . . 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 2797	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 7422	. . . . . . . . . 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 2773	. . . . . . . . 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 7422	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
100	51, 81, 99	3eqtr4d 2783	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
101		hoaddcl 30999	. . . . . . . . 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 31049	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ ∧ ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ) → (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
104	34, 46, 103	mp3an12 1452	. . . . . . . 8 ⊢ (((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ → (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −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 ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
106	100, 105	eqtr4d 2776	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
107		hosubcl 31014	. . . . . . . . . 10 ⊢ (( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 588	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 31026	. . . . . . . . . 10 ⊢ (( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ ∧ ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))))
110	46, 109	mp3an1 1449	. . . . . . . . 9 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))))
111	36, 108, 110	syl2anc 585	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))))
112		hmoplin 31183	. . . . . . . . . . . . . . 15 ⊢ ( Iop ∈ HrmOp → Iop ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Iop ∈ LinOp
114		hoddi 31231	. . . . . . . . . . . . . 14 ⊢ (( Iop ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
115	113, 46, 114	mp3an12 1452	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
116	36, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
117	46	hoid1i 31030	. . . . . . . . . . . . . 14 ⊢ ( Iop ∘ Iop ) = Iop
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119		hoico2 30998	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( Iop ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
120	36, 119	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
121	118, 120	oveq12d 7424	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
122	116, 121	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
123		hmoplin 31183	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 31231	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
126	46, 125	mp3an2 1450	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ LinOp ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
127	124, 36, 126	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 30997	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ((𝐹‘𝑘) ∘ Iop ) = (𝐹‘𝑘))
129	36, 128	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ Iop ) = (𝐹‘𝑘))
130	129	oveq1d 7421	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 7424	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))) = (( Iop −op (𝐹‘𝑘)) −op ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
133	36, 46	jctil 521	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
134		hosubadd4 31055	. . . . . . . . . . 11 ⊢ ((( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) ∧ ((𝐹‘𝑘): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)) → (( Iop −op (𝐹‘𝑘)) −op ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
135	133, 36, 42, 134	syl12anc 836	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) −op ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
136	132, 135	eqtrd 2773	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
137		ho2times 31060	. . . . . . . . . . 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 7422	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
140		hoaddsubass 31056	. . . . . . . . . . 11 ⊢ (( Iop : ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
141	46, 140	mp3an1 1449	. . . . . . . . . 10 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
142	42, 38, 141	syl2anc 585	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
143	136, 139, 142	3eqtr2d 2779	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
144	111, 143	eqtrd 2773	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
145	144	oveq1d 7421	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
146	7, 8, 9	opsqrlem5 31385	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 7422	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘(𝑘 + 1))) = ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
148	147	oveq2d 7422	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))) = (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
149	106, 145, 148	3eqtr4d 2783	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)) = (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))))
150	33, 149	breqtrd 5174	. . . 4 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))))
151		peano2nn 12221	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelcdmi 7083	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 31263	. . . . . 6 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 588	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 12283	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 12312	. . . . . 6 ⊢ 0 < 2
158		leopmul 31375	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp ∧ 0 < 2) → ( 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))) ↔ 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1))))))
159	156, 157, 158	mp3an13 1453	. . . . 5 ⊢ (( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp → ( 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))) ↔ 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1))))))
160	155, 159	syl 17	. . . 4 ⊢ (𝑘 ∈ ℕ → ( 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))) ↔ 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1))))))
161	150, 160	mpbird 257	. . 3 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))))
162		leop3 31366	. . . 4 ⊢ (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
163	153, 13, 162	sylancl 587	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
164	161, 163	mpbird 257	. 2 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤op Iop )
165	2, 4, 6, 12, 164	nn1suc 12231	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {csn 4628   class class class wbr 5148   × cxp 5674   ∘ ccom 5680  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406   ∈ cmpo 7408  ℂcc 11105  ℝcr 11106  0cc0 11107  1c1 11108   + caddc 11110   · cmul 11112   < clt 11245   / cdiv 11868  ℕcn 12209  2c2 12264  seqcseq 13963   ℋchba 30160   +_op chos 30179   ·_op chot 30180   −_op chod 30181   0_hop ch0o 30184   I_op chio 30185  LinOpclo 30188  HrmOpcho 30191   ≤_op cleo 30199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cc 10427  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185  ax-addf 11186  ax-mulf 11187  ax-hilex 30240  ax-hfvadd 30241  ax-hvcom 30242  ax-hvass 30243  ax-hv0cl 30244  ax-hvaddid 30245  ax-hfvmul 30246  ax-hvmulid 30247  ax-hvmulass 30248  ax-hvdistr1 30249  ax-hvdistr2 30250  ax-hvmul0 30251  ax-hfi 30320  ax-his1 30323  ax-his2 30324  ax-his3 30325  ax-his4 30326  ax-hcompl 30443

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-supp 8144  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-oadd 8467  df-omul 8468  df-er 8700  df-map 8819  df-pm 8820  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9502  df-card 9931  df-acn 9934  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-rlim 15430  df-sum 15630  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-mulr 17208  df-starv 17209  df-sca 17210  df-vsca 17211  df-ip 17212  df-tset 17213  df-ple 17214  df-ds 17216  df-unif 17217  df-hom 17218  df-cco 17219  df-rest 17365  df-topn 17366  df-0g 17384  df-gsum 17385  df-topgen 17386  df-pt 17387  df-prds 17390  df-xrs 17445  df-qtop 17450  df-imas 17451  df-xps 17453  df-mre 17527  df-mrc 17528  df-acs 17530  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-submnd 18669  df-mulg 18946  df-cntz 19176  df-cmn 19645  df-psmet 20929  df-xmet 20930  df-met 20931  df-bl 20932  df-mopn 20933  df-fbas 20934  df-fg 20935  df-cnfld 20938  df-top 22388  df-topon 22405  df-topsp 22427  df-bases 22441  df-cld 22515  df-ntr 22516  df-cls 22517  df-nei 22594  df-cn 22723  df-cnp 22724  df-lm 22725  df-haus 22811  df-tx 23058  df-hmeo 23251  df-fil 23342  df-fm 23434  df-flim 23435  df-flf 23436  df-xms 23818  df-ms 23819  df-tms 23820  df-cfil 24764  df-cau 24765  df-cmet 24766  df-grpo 29734  df-gid 29735  df-ginv 29736  df-gdiv 29737  df-ablo 29786  df-vc 29800  df-nv 29833  df-va 29836  df-ba 29837  df-sm 29838  df-0v 29839  df-vs 29840  df-nmcv 29841  df-ims 29842  df-dip 29942  df-ssp 29963  df-ph 30054  df-cbn 30104  df-hnorm 30209  df-hba 30210  df-hvsub 30212  df-hlim 30213  df-hcau 30214  df-sh 30448  df-ch 30462  df-oc 30493  df-ch0 30494  df-shs 30549  df-pjh 30636  df-hosum 30971  df-homul 30972  df-hodif 30973  df-h0op 30989  df-iop 30990  df-lnop 31082  df-hmop 31085  df-leop 31093

This theorem is referenced by: (None)