Theorem opsqrlem6 32434

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 6879	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 5120	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3		fveq2 6879	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 5120	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5		fveq2 6879	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 5120	. 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 32430	. . 3 ⊢ (𝐹‘1) = 0hop
11		idleop 32420	. . 3 ⊢ 0hop ≤op Iop
12	10, 11	eqbrtri 5133	. 2 ⊢ (𝐹‘1) ≤op Iop
13		idhmop 32271	. . . . . . . 8 ⊢ Iop ∈ HrmOp
14	7, 8, 9	opsqrlem4 32432	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7076	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 32311	. . . . . . . 8 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 598	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2769	. . . . . . . 8 ⊢ (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))
19		hmopco 32312	. . . . . . . 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 1476	. . . . . . 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 595	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 32418	. . . . . . 7 ⊢ (( Iop −op (𝐹‘𝑘)) ∈ HrmOp → 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))))
23	17, 22	syl 18	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))))
24		opsqrlem6.4	. . . . . . . 8 ⊢ 𝑇 ≤op Iop
25		leop3 32414	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇)))
26	7, 13, 25	mp2an 704	. . . . . . . 8 ⊢ (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇))
27	24, 26	mpbi 233	. . . . . . 7 ⊢ 0hop ≤op ( Iop −op 𝑇)
28		hmopd 32311	. . . . . . . . 9 ⊢ (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iop −op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 704	. . . . . . . 8 ⊢ ( Iop −op 𝑇) ∈ HrmOp
30		leopadd 32421	. . . . . . . 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 713	. . . . . . 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 716	. . . . . 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 595	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
34		2cn 12312	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 32163	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 32048	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 598	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 32163	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6728	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 595	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 32062	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 598	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 32163	. . . . . . . . . . . 12 ⊢ ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ Iop : ℋ⟶ ℋ
47		homulcl 32048	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 704	. . . . . . . . . 10 ⊢ (2 ·op Iop ): ℋ⟶ ℋ
49		hosubsub4 32107	. . . . . . . . . 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 1474	. . . . . . . . 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 595	. . . . . . . 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 32062	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 32072	. . . . . . . . . . . . . . 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 1478	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 32108	. . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . 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 2822	. . . . . . . . . . . 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 32104	. . . . . . . . . . . . . 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 1474	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 2807	. . . . . . . . . . 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 7423	. . . . . . . . . 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 32047	. . . . . . . . . . . 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 598	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 32104	. . . . . . . . . . . 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 1479	. . . . . . . . . . 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 18	. . . . . . . . . 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 32047	. . . . . . . . . . . 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 598	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 32107	. . . . . . . . . . . 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 1476	. . . . . . . . . . 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 595	. . . . . . . . . 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 2812	. . . . . . . . 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 32103	. . . . . . . . . . . 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 715	. . . . . . . . . . 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 712	. . . . . . . . . 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 595	. . . . . . . . 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 2807	. . . . . . . 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 12454	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 32048	. . . . . . . . . . . 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 598	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 32092	. . . . . . . . . . . 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 1474	. . . . . . . . . . 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 595	. . . . . . . . . 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 12343	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11942	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 7418	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 32091	. . . . . . . . . . . . . 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 1477	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 32089	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
96	90, 93, 95	3eqtr3a 2828	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 7424	. . . . . . . . . 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 2804	. . . . . . . . 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 7424	. . . . . . . 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 2814	. . . . . . 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 32047	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
103		hosubdi 32097	. . . . . . . . 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 1477	. . . . . . . 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 18	. . . . . . 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 2807	. . . . . 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 32062	. . . . . . . . . 10 ⊢ (( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 598	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 32074	. . . . . . . . . 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 1474	. . . . . . . . 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 595	. . . . . . . 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 32231	. . . . . . . . . . . . . . 15 ⊢ ( Iop ∈ HrmOp → Iop ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Iop ∈ LinOp
114		hoddi 32279	. . . . . . . . . . . . . 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 1477	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
116	36, 115	syl 18	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
117	46	hoid1i 32078	. . . . . . . . . . . . . 14 ⊢ ( Iop ∘ Iop ) = Iop
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119		hoico2 32046	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( Iop ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
120	36, 119	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
121	118, 120	oveq12d 7426	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
122	116, 121	eqtrd 2804	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
123		hmoplin 32231	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 32279	. . . . . . . . . . . . . 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 1475	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 32045	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ((𝐹‘𝑘) ∘ Iop ) = (𝐹‘𝑘))
129	36, 128	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ Iop ) = (𝐹‘𝑘))
130	129	oveq1d 7423	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2804	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 7426	. . . . . . . . . 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 528	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
134		hosubadd4 32103	. . . . . . . . . . 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 849	. . . . . . . . . 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 2804	. . . . . . . . 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 32108	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → (2 ·op (𝐹‘𝑘)) = ((𝐹‘𝑘) +op (𝐹‘𝑘)))
138	36, 137	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2 ·op (𝐹‘𝑘)) = ((𝐹‘𝑘) +op (𝐹‘𝑘)))
139	138	oveq2d 7424	. . . . . . . . 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 32104	. . . . . . . . . . 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 1474	. . . . . . . . . 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 595	. . . . . . . . 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 2810	. . . . . . . 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 2804	. . . . . . 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 7423	. . . . . 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 32433	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 7424	. . . . . . 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 7424	. . . . . 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 2814	. . . . 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 5138	. . . 4 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))))
151		peano2nn 12241	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelcdmi 7076	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 18	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 32311	. . . . . 6 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 598	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 12311	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 12341	. . . . . 6 ⊢ 0 < 2
158		leopmul 32423	. . . . . 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 1478	. . . . 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 18	. . . 4 ⊢ (𝑘 ∈ ℕ → ( 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))) ↔ 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1))))))
161	150, 160	mpbird 260	. . 3 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))))
162		leop3 32414	. . . 4 ⊢ (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
163	153, 13, 162	sylancl 597	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
164	161, 163	mpbird 260	. 2 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤op Iop )
165	2, 4, 6, 12, 164	nn1suc 12251	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {csn 4591   class class class wbr 5110   × cxp 5657   ∘ ccom 5663  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  ℂcc 11094  ℝcr 11095  0cc0 11096  1c1 11097   + caddc 11099   · cmul 11101   < clt 11239   / cdiv 11867  ℕcn 12229  2c2 12291  seqcseq 14033   ℋchba 31208   +_op chos 31227   ·_op chot 31228   −_op chod 31229   0_hop ch0o 31232   I_op chio 31233  LinOpclo 31236  HrmOpcho 31239   ≤_op cleo 31247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606  ax-cc 10415  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174  ax-addf 11175  ax-mulf 11176  ax-hilex 31288  ax-hfvadd 31289  ax-hvcom 31290  ax-hvass 31291  ax-hv0cl 31292  ax-hvaddid 31293  ax-hfvmul 31294  ax-hvmulid 31295  ax-hvmulass 31296  ax-hvdistr1 31297  ax-hvdistr2 31298  ax-hvmul0 31299  ax-hfi 31368  ax-his1 31371  ax-his2 31372  ax-his3 31373  ax-his4 31374  ax-hcompl 31491

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  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 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-omul 8454  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-fi 9367  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-acn 9924  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-q 12969  df-rp 13013  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ioo 13372  df-ico 13374  df-icc 13375  df-fz 13532  df-fzo 13679  df-fl 13821  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-starv 17321  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-unif 17329  df-hom 17330  df-cco 17331  df-rest 17471  df-topn 17472  df-0g 17490  df-gsum 17491  df-topgen 17492  df-pt 17493  df-prds 17496  df-xrs 17552  df-qtop 17557  df-imas 17558  df-xps 17560  df-mre 17634  df-mrc 17635  df-acs 17637  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-submnd 18838  df-mulg 19130  df-cntz 19383  df-cmn 19848  df-psmet 21479  df-xmet 21480  df-met 21481  df-bl 21482  df-mopn 21483  df-fbas 21484  df-fg 21485  df-cnfld 21488  df-top 23016  df-topon 23033  df-topsp 23055  df-bases 23068  df-cld 23141  df-ntr 23142  df-cls 23143  df-nei 23220  df-cn 23349  df-cnp 23350  df-lm 23351  df-haus 23437  df-tx 23684  df-hmeo 23877  df-fil 23968  df-fm 24060  df-flim 24061  df-flf 24062  df-xms 24442  df-ms 24443  df-tms 24444  df-cfil 25379  df-cau 25380  df-cmet 25381  df-grpo 30782  df-gid 30783  df-ginv 30784  df-gdiv 30785  df-ablo 30834  df-vc 30848  df-nv 30881  df-va 30884  df-ba 30885  df-sm 30886  df-0v 30887  df-vs 30888  df-nmcv 30889  df-ims 30890  df-dip 30990  df-ssp 31011  df-ph 31102  df-cbn 31152  df-hnorm 31257  df-hba 31258  df-hvsub 31260  df-hlim 31261  df-hcau 31262  df-sh 31496  df-ch 31510  df-oc 31541  df-ch0 31542  df-shs 31597  df-pjh 31684  df-hosum 32019  df-homul 32020  df-hodif 32021  df-h0op 32037  df-iop 32038  df-lnop 32130  df-hmop 32133  df-leop 32141

This theorem is referenced by: (None)