Theorem opsqrlem6 31833

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 6881	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 5148	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3		fveq2 6881	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 5148	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5		fveq2 6881	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 5148	. 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 31829	. . 3 ⊢ (𝐹‘1) = 0hop
11		idleop 31819	. . 3 ⊢ 0hop ≤op Iop
12	10, 11	eqbrtri 5159	. 2 ⊢ (𝐹‘1) ≤op Iop
13		idhmop 31670	. . . . . . . 8 ⊢ Iop ∈ HrmOp
14	7, 8, 9	opsqrlem4 31831	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7075	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 31710	. . . . . . . 8 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 586	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2724	. . . . . . . 8 ⊢ (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))
19		hmopco 31711	. . . . . . . 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 1446	. . . . . . 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 583	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 31817	. . . . . . 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 31813	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇)))
26	7, 13, 25	mp2an 689	. . . . . . . 8 ⊢ (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇))
27	24, 26	mpbi 229	. . . . . . 7 ⊢ 0hop ≤op ( Iop −op 𝑇)
28		hmopd 31710	. . . . . . . . 9 ⊢ (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iop −op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 689	. . . . . . . 8 ⊢ ( Iop −op 𝑇) ∈ HrmOp
30		leopadd 31820	. . . . . . . 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 698	. . . . . . 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 701	. . . . . 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 583	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
34		2cn 12283	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 31562	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 31447	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 586	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 31562	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6731	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 583	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 31461	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 586	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 31562	. . . . . . . . . . . 12 ⊢ ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ Iop : ℋ⟶ ℋ
47		homulcl 31447	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 689	. . . . . . . . . 10 ⊢ (2 ·op Iop ): ℋ⟶ ℋ
49		hosubsub4 31506	. . . . . . . . . 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 1444	. . . . . . . . 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 583	. . . . . . . 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 31461	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 31471	. . . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . . 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 31507	. . . . . . . . . . . . . . 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 7411	. . . . . . . . . . . . 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 2782	. . . . . . . . . . . 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 31503	. . . . . . . . . . . . . 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 1444	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 2767	. . . . . . . . . . 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 7416	. . . . . . . . . 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 31446	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 31503	. . . . . . . . . . . 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 1449	. . . . . . . . . . 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 31446	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 31506	. . . . . . . . . . . 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 1446	. . . . . . . . . . 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 583	. . . . . . . . . 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 2772	. . . . . . . . 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 31502	. . . . . . . . . . . 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 700	. . . . . . . . . . 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 697	. . . . . . . . . 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 583	. . . . . . . . 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 2767	. . . . . . . 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 12423	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 31447	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 31491	. . . . . . . . . . . 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 1444	. . . . . . . . . . 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 583	. . . . . . . . . 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 12312	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11941	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 7411	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 31490	. . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . 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 31488	. . . . . . . . . . . . 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 7417	. . . . . . . . . 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 7417	. . . . . . . 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 2774	. . . . . . 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 31446	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
103		hosubdi 31496	. . . . . . . . 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 1447	. . . . . . . 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 2767	. . . . . 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 31461	. . . . . . . . . 10 ⊢ (( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 586	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 31473	. . . . . . . . . 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 1444	. . . . . . . . 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 583	. . . . . . . 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 31630	. . . . . . . . . . . . . . 15 ⊢ ( Iop ∈ HrmOp → Iop ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Iop ∈ LinOp
114		hoddi 31678	. . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . 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 31477	. . . . . . . . . . . . . 14 ⊢ ( Iop ∘ Iop ) = Iop
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119		hoico2 31445	. . . . . . . . . . . . . 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 7419	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
122	116, 121	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
123		hmoplin 31630	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 31678	. . . . . . . . . . . . . 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 1445	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 31444	. . . . . . . . . . . . . 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 7416	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 7419	. . . . . . . . . 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 519	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
134		hosubadd4 31502	. . . . . . . . . . 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 834	. . . . . . . . . 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 2764	. . . . . . . . 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 31507	. . . . . . . . . . 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 7417	. . . . . . . . 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 31503	. . . . . . . . . . 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 1444	. . . . . . . . . 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 583	. . . . . . . . 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 2770	. . . . . . . 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 2764	. . . . . . 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 7416	. . . . . 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 31832	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 7417	. . . . . . 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 7417	. . . . . 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 2774	. . . . 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 5164	. . . 4 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))))
151		peano2nn 12220	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelcdmi 7075	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 31710	. . . . . 6 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 586	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 12282	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 12311	. . . . . 6 ⊢ 0 < 2
158		leopmul 31822	. . . . . 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 1448	. . . . 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 31813	. . . 4 ⊢ (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
163	153, 13, 162	sylancl 585	. . 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 12230	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {csn 4620   class class class wbr 5138   × cxp 5664   ∘ ccom 5670  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401   ∈ cmpo 7403  ℂcc 11103  ℝcr 11104  0cc0 11105  1c1 11106   + caddc 11108   · cmul 11110   < clt 11244   / cdiv 11867  ℕcn 12208  2c2 12263  seqcseq 13962   ℋchba 30607   +_op chos 30626   ·_op chot 30627   −_op chod 30628   0_hop ch0o 30631   I_op chio 30632  LinOpclo 30635  HrmOpcho 30638   ≤_op cleo 30646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-cc 10425  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183  ax-addf 11184  ax-mulf 11185  ax-hilex 30687  ax-hfvadd 30688  ax-hvcom 30689  ax-hvass 30690  ax-hv0cl 30691  ax-hvaddid 30692  ax-hfvmul 30693  ax-hvmulid 30694  ax-hvmulass 30695  ax-hvdistr1 30696  ax-hvdistr2 30697  ax-hvmul0 30698  ax-hfi 30767  ax-his1 30770  ax-his2 30771  ax-his3 30772  ax-his4 30773  ax-hcompl 30890

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-er 8698  df-map 8817  df-pm 8818  df-ixp 8887  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fsupp 9357  df-fi 9401  df-sup 9432  df-inf 9433  df-oi 9500  df-card 9929  df-acn 9932  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-struct 17078  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17143  df-ress 17172  df-plusg 17208  df-mulr 17209  df-starv 17210  df-sca 17211  df-vsca 17212  df-ip 17213  df-tset 17214  df-ple 17215  df-ds 17217  df-unif 17218  df-hom 17219  df-cco 17220  df-rest 17366  df-topn 17367  df-0g 17385  df-gsum 17386  df-topgen 17387  df-pt 17388  df-prds 17391  df-xrs 17446  df-qtop 17451  df-imas 17452  df-xps 17454  df-mre 17528  df-mrc 17529  df-acs 17531  df-mgm 18562  df-sgrp 18641  df-mnd 18657  df-submnd 18703  df-mulg 18985  df-cntz 19222  df-cmn 19691  df-psmet 21219  df-xmet 21220  df-met 21221  df-bl 21222  df-mopn 21223  df-fbas 21224  df-fg 21225  df-cnfld 21228  df-top 22717  df-topon 22734  df-topsp 22756  df-bases 22770  df-cld 22844  df-ntr 22845  df-cls 22846  df-nei 22923  df-cn 23052  df-cnp 23053  df-lm 23054  df-haus 23140  df-tx 23387  df-hmeo 23580  df-fil 23671  df-fm 23763  df-flim 23764  df-flf 23765  df-xms 24147  df-ms 24148  df-tms 24149  df-cfil 25104  df-cau 25105  df-cmet 25106  df-grpo 30181  df-gid 30182  df-ginv 30183  df-gdiv 30184  df-ablo 30233  df-vc 30247  df-nv 30280  df-va 30283  df-ba 30284  df-sm 30285  df-0v 30286  df-vs 30287  df-nmcv 30288  df-ims 30289  df-dip 30389  df-ssp 30410  df-ph 30501  df-cbn 30551  df-hnorm 30656  df-hba 30657  df-hvsub 30659  df-hlim 30660  df-hcau 30661  df-sh 30895  df-ch 30909  df-oc 30940  df-ch0 30941  df-shs 30996  df-pjh 31083  df-hosum 31418  df-homul 31419  df-hodif 31420  df-h0op 31436  df-iop 31437  df-lnop 31529  df-hmop 31532  df-leop 31540

This theorem is referenced by: (None)