Theorem opsqrlem6 29004

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 6191	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 4663	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3		fveq2 6191	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 4663	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5		fveq2 6191	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 4663	. 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 29000	. . 3 ⊢ (𝐹‘1) = 0hop
11		idleop 28990	. . 3 ⊢ 0hop ≤op Iop
12	10, 11	eqbrtri 4674	. 2 ⊢ (𝐹‘1) ≤op Iop
13		idhmop 28841	. . . . . . . 8 ⊢ Iop ∈ HrmOp
14	7, 8, 9	opsqrlem4 29002	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelrni 6358	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 28881	. . . . . . . 8 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 695	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2622	. . . . . . . 8 ⊢ (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))
19		hmopco 28882	. . . . . . . 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 1413	. . . . . . 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 693	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 28988	. . . . . . 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 28984	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇)))
26	7, 13, 25	mp2an 708	. . . . . . . 8 ⊢ (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇))
27	24, 26	mpbi 220	. . . . . . 7 ⊢ 0hop ≤op ( Iop −op 𝑇)
28		hmopd 28881	. . . . . . . . 9 ⊢ (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iop −op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 708	. . . . . . . 8 ⊢ ( Iop −op 𝑇) ∈ HrmOp
30		leopadd 28991	. . . . . . . 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 717	. . . . . . 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 720	. . . . . 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 693	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
34		2cn 11091	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 28733	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 28618	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 695	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 28733	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6058	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 693	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 28632	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 695	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 28733	. . . . . . . . . . . 12 ⊢ ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ Iop : ℋ⟶ ℋ
47		homulcl 28618	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 708	. . . . . . . . . 10 ⊢ (2 ·op Iop ): ℋ⟶ ℋ
49		hosubsub4 28677	. . . . . . . . . 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 1411	. . . . . . . . 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 693	. . . . . . . 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 28632	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 28642	. . . . . . . . . . . . . . 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 1415	. . . . . . . . . . . . . 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 28678	. . . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . 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	syl6eqr 2674	. . . . . . . . . . . 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 28674	. . . . . . . . . . . . . 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 1411	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 2659	. . . . . . . . . . 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 6665	. . . . . . . . . 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 28617	. . . . . . . . . . . 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 695	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 28674	. . . . . . . . . . . 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 1416	. . . . . . . . . . 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 28617	. . . . . . . . . . . 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 695	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 28677	. . . . . . . . . . . 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 1413	. . . . . . . . . . 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 693	. . . . . . . . . 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 2664	. . . . . . . . 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 28673	. . . . . . . . . . . 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 719	. . . . . . . . . . 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 716	. . . . . . . . . 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 693	. . . . . . . . 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 2659	. . . . . . . 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 11247	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 28618	. . . . . . . . . . . 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 695	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 28662	. . . . . . . . . . . 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 1411	. . . . . . . . . . 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 693	. . . . . . . . . 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 11113	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 10756	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 6660	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 28661	. . . . . . . . . . . . . 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 1414	. . . . . . . . . . . . 13 ⊢ ((𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ → ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
93	44, 92	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
94		homulid2 28659	. . . . . . . . . . . . 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 2680	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 6666	. . . . . . . . . 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 2656	. . . . . . . . 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 6666	. . . . . . . 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 2666	. . . . . . 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 28617	. . . . . . . . 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 693	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
103		hosubdi 28667	. . . . . . . . 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 1414	. . . . . . . 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 2659	. . . . . 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 28632	. . . . . . . . . 10 ⊢ (( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 695	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 28644	. . . . . . . . . 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 1411	. . . . . . . . 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 693	. . . . . . . 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 28801	. . . . . . . . . . . . . . 15 ⊢ ( Iop ∈ HrmOp → Iop ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Iop ∈ LinOp
114		hoddi 28849	. . . . . . . . . . . . . 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 1414	. . . . . . . . . . . . 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 28648	. . . . . . . . . . . . . 14 ⊢ ( Iop ∘ Iop ) = Iop
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119		hoico2 28616	. . . . . . . . . . . . . 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 6668	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
122	116, 121	eqtrd 2656	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
123		hmoplin 28801	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 28849	. . . . . . . . . . . . . 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 1412	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 28615	. . . . . . . . . . . . . 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 6665	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2656	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 6668	. . . . . . . . . 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 560	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
134		hosubadd4 28673	. . . . . . . . . . 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 1324	. . . . . . . . . 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 2656	. . . . . . . . 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 28678	. . . . . . . . . . 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 6666	. . . . . . . . 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 28674	. . . . . . . . . . 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 1411	. . . . . . . . . 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 693	. . . . . . . . 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 2662	. . . . . . . 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 2656	. . . . . . 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 6665	. . . . . 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 29003	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 6666	. . . . . . 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 6666	. . . . . 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 2666	. . . . 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 4679	. . . 4 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))))
151		peano2nn 11032	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelrni 6358	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 28881	. . . . . 6 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 695	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 11090	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 11112	. . . . . 6 ⊢ 0 < 2
158		leopmul 28993	. . . . . 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 1415	. . . . 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 247	. . 3 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))))
162		leop3 28984	. . . 4 ⊢ (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
163	153, 13, 162	sylancl 694	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
164	161, 163	mpbird 247	. 2 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤op Iop )
165	2, 4, 6, 12, 164	nn1suc 11041	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {csn 4177   class class class wbr 4653   × cxp 5112   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   / cdiv 10684  ℕcn 11020  2c2 11070  seqcseq 12801   ℋchil 27776   +_op chos 27795   ·_op chot 27796   −_op chod 27797   0_hop ch0o 27800   I_op chio 27801  LinOpclo 27804  HrmOpcho 27807   ≤_op cleo 27815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942  ax-hcompl 28059

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-lm 21033  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cfil 23053  df-cau 23054  df-cmet 23055  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-dip 27556  df-ssp 27577  df-ph 27668  df-cbn 27719  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-hcau 27830  df-sh 28064  df-ch 28078  df-oc 28109  df-ch0 28110  df-shs 28167  df-pjh 28254  df-hosum 28589  df-homul 28590  df-hodif 28591  df-h0op 28607  df-iop 28608  df-lnop 28700  df-hmop 28703  df-leop 28711

This theorem is referenced by: (None)