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Theorem opsqrlem6 28892
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 (𝑇op (𝑥𝑥)))))
opsqrlem2.3 𝐹 = seq1(𝑆, (ℕ × { 0hop }))
opsqrlem6.4 𝑇op Iop
Assertion
Ref Expression
opsqrlem6 (𝑁 ∈ ℕ → (𝐹𝑁) ≤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.
StepHypRef Expression
1 fveq2 6158 . . 3 (𝑗 = 1 → (𝐹𝑗) = (𝐹‘1))
21breq1d 4633 . 2 (𝑗 = 1 → ((𝐹𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3 fveq2 6158 . . 3 (𝑗 = (𝑘 + 1) → (𝐹𝑗) = (𝐹‘(𝑘 + 1)))
43breq1d 4633 . 2 (𝑗 = (𝑘 + 1) → ((𝐹𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5 fveq2 6158 . . 3 (𝑗 = 𝑁 → (𝐹𝑗) = (𝐹𝑁))
65breq1d 4633 . 2 (𝑗 = 𝑁 → ((𝐹𝑗) ≤op Iop ↔ (𝐹𝑁) ≤op Iop ))
7 opsqrlem2.1 . . . 4 𝑇 ∈ HrmOp
8 opsqrlem2.2 . . . 4 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))
9 opsqrlem2.3 . . . 4 𝐹 = seq1(𝑆, (ℕ × { 0hop }))
107, 8, 9opsqrlem2 28888 . . 3 (𝐹‘1) = 0hop
11 idleop 28878 . . 3 0hopop Iop
1210, 11eqbrtri 4644 . 2 (𝐹‘1) ≤op Iop
13 idhmop 28729 . . . . . . . 8 Iop ∈ HrmOp
147, 8, 9opsqrlem4 28890 . . . . . . . . 9 𝐹:ℕ⟶HrmOp
1514ffvelrni 6324 . . . . . . . 8 (𝑘 ∈ ℕ → (𝐹𝑘) ∈ HrmOp)
16 hmopd 28769 . . . . . . . 8 (( Iop ∈ HrmOp ∧ (𝐹𝑘) ∈ HrmOp) → ( Iopop (𝐹𝑘)) ∈ HrmOp)
1713, 15, 16sylancr 694 . . . . . . 7 (𝑘 ∈ ℕ → ( Iopop (𝐹𝑘)) ∈ HrmOp)
18 eqid 2621 . . . . . . . 8 (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))
19 hmopco 28770 . . . . . . . 8 ((( Iopop (𝐹𝑘)) ∈ HrmOp ∧ ( Iopop (𝐹𝑘)) ∈ HrmOp ∧ (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
2018, 19mp3an3 1410 . . . . . . 7 ((( Iopop (𝐹𝑘)) ∈ HrmOp ∧ ( Iopop (𝐹𝑘)) ∈ HrmOp) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
2117, 17, 20syl2anc 692 . . . . . 6 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
22 leopsq 28876 . . . . . . 7 (( Iopop (𝐹𝑘)) ∈ HrmOp → 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))))
2317, 22syl 17 . . . . . 6 (𝑘 ∈ ℕ → 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))))
24 opsqrlem6.4 . . . . . . . 8 𝑇op Iop
25 leop3 28872 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇op Iop ↔ 0hopop ( Iopop 𝑇)))
267, 13, 25mp2an 707 . . . . . . . 8 (𝑇op Iop ↔ 0hopop ( Iopop 𝑇))
2724, 26mpbi 220 . . . . . . 7 0hopop ( Iopop 𝑇)
28 hmopd 28769 . . . . . . . . 9 (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iopop 𝑇) ∈ HrmOp)
2913, 7, 28mp2an 707 . . . . . . . 8 ( Iopop 𝑇) ∈ HrmOp
30 leopadd 28879 . . . . . . . 8 ((((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ ( Iopop 𝑇) ∈ HrmOp) ∧ ( 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∧ 0hopop ( Iopop 𝑇))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3129, 30mpanl2 716 . . . . . . 7 (((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ ( 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∧ 0hopop ( Iopop 𝑇))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3227, 31mpanr2 719 . . . . . 6 (((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3321, 23, 32syl2anc 692 . . . . 5 (𝑘 ∈ ℕ → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
34 2cn 11051 . . . . . . . . . 10 2 ∈ ℂ
35 hmopf 28621 . . . . . . . . . . 11 ((𝐹𝑘) ∈ HrmOp → (𝐹𝑘): ℋ⟶ ℋ)
3615, 35syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝐹𝑘): ℋ⟶ ℋ)
37 homulcl 28506 . . . . . . . . . 10 ((2 ∈ ℂ ∧ (𝐹𝑘): ℋ⟶ ℋ) → (2 ·op (𝐹𝑘)): ℋ⟶ ℋ)
3834, 36, 37sylancr 694 . . . . . . . . 9 (𝑘 ∈ ℕ → (2 ·op (𝐹𝑘)): ℋ⟶ ℋ)
39 hmopf 28621 . . . . . . . . . . 11 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
407, 39ax-mp 5 . . . . . . . . . 10 𝑇: ℋ⟶ ℋ
41 fco 6025 . . . . . . . . . . 11 (((𝐹𝑘): ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)
4236, 36, 41syl2anc 692 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)
43 hosubcl 28520 . . . . . . . . . 10 ((𝑇: ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
4440, 42, 43sylancr 694 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
45 hmopf 28621 . . . . . . . . . . . 12 ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
4613, 45ax-mp 5 . . . . . . . . . . 11 Iop : ℋ⟶ ℋ
47 homulcl 28506 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
4834, 46, 47mp2an 707 . . . . . . . . . 10 (2 ·op Iop ): ℋ⟶ ℋ
49 hosubsub4 28565 . . . . . . . . . 10 (((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
5048, 49mp3an1 1408 . . . . . . . . 9 (((2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
5138, 44, 50syl2anc 692 . . . . . . . 8 (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
52 hosubcl 28520 . . . . . . . . . . . . . . 15 ((((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ)
5342, 38, 52syl2anc 692 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ)
54 hoadd32 28530 . . . . . . . . . . . . . . 15 (( Iop : ℋ⟶ ℋ ∧ (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ) → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
5546, 46, 54mp3an13 1412 . . . . . . . . . . . . . 14 ((((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
5653, 55syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
57 ho2times 28566 . . . . . . . . . . . . . . 15 ( Iop : ℋ⟶ ℋ → (2 ·op Iop ) = ( Iop +op Iop ))
5846, 57ax-mp 5 . . . . . . . . . . . . . 14 (2 ·op Iop ) = ( Iop +op Iop )
5958oveq1i 6625 . . . . . . . . . . . . 13 ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))))
6056, 59syl6eqr 2673 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
61 hoaddsubass 28562 . . . . . . . . . . . . . 14 (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
6248, 61mp3an1 1408 . . . . . . . . . . . . 13 ((((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
6342, 38, 62syl2anc 692 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
6460, 63eqtr4d 2658 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))))
6564oveq1d 6630 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇))
66 hoaddcl 28505 . . . . . . . . . . . 12 (( Iop : ℋ⟶ ℋ ∧ (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ) → ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ)
6746, 53, 66sylancr 694 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ)
68 hoaddsubass 28562 . . . . . . . . . . . 12 ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
6946, 40, 68mp3an23 1413 . . . . . . . . . . 11 (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
7067, 69syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
71 hoaddcl 28505 . . . . . . . . . . . 12 (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → ((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
7248, 42, 71sylancr 694 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
73 hosubsub4 28565 . . . . . . . . . . . 12 ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7440, 73mp3an3 1410 . . . . . . . . . . 11 ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7572, 38, 74syl2anc 692 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7665, 70, 753eqtr3d 2663 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
77 hosubadd4 28561 . . . . . . . . . . . 12 ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7840, 77mpanr1 718 . . . . . . . . . . 11 ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7948, 78mpanl1 715 . . . . . . . . . 10 (((2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
8038, 42, 79syl2anc 692 . . . . . . . . 9 (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
8176, 80eqtr4d 2658 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
82 halfcn 11207 . . . . . . . . . . . 12 (1 / 2) ∈ ℂ
83 homulcl 28506 . . . . . . . . . . . 12 (((1 / 2) ∈ ℂ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ)
8482, 44, 83sylancr 694 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ)
85 hoadddi 28550 . . . . . . . . . . . 12 ((2 ∈ ℂ ∧ (𝐹𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ) → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
8634, 85mp3an1 1408 . . . . . . . . . . 11 (((𝐹𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ) → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
8736, 84, 86syl2anc 692 . . . . . . . . . 10 (𝑘 ∈ ℕ → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
88 2ne0 11073 . . . . . . . . . . . . . 14 2 ≠ 0
8934, 88recidi 10716 . . . . . . . . . . . . 13 (2 · (1 / 2)) = 1
9089oveq1i 6625 . . . . . . . . . . . 12 ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (1 ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
91 homulass 28549 . . . . . . . . . . . . . 14 ((2 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
9234, 82, 91mp3an12 1411 . . . . . . . . . . . . 13 ((𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ → ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
9344, 92syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
94 homulid2 28547 . . . . . . . . . . . . 13 ((𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ → (1 ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
9544, 94syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
9690, 93, 953eqtr3a 2679 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))) = (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
9796oveq2d 6631 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
9887, 97eqtrd 2655 . . . . . . . . 9 (𝑘 ∈ ℕ → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
9998oveq2d 6631 . . . . . . . 8 (𝑘 ∈ ℕ → ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
10051, 81, 993eqtr4d 2665 . . . . . . 7 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
101 hoaddcl 28505 . . . . . . . . 9 (((𝐹𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ) → ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ)
10236, 84, 101syl2anc 692 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ)
103 hosubdi 28555 . . . . . . . . 9 ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ ∧ ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ) → (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
10434, 46, 103mp3an12 1411 . . . . . . . 8 (((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ → (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
105102, 104syl 17 . . . . . . 7 (𝑘 ∈ ℕ → (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
106100, 105eqtr4d 2658 . . . . . 6 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
107 hosubcl 28520 . . . . . . . . . 10 (( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ( Iopop (𝐹𝑘)): ℋ⟶ ℋ)
10846, 36, 107sylancr 694 . . . . . . . . 9 (𝑘 ∈ ℕ → ( Iopop (𝐹𝑘)): ℋ⟶ ℋ)
109 hocsubdir 28532 . . . . . . . . . 10 (( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ ∧ ( Iopop (𝐹𝑘)): ℋ⟶ ℋ) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
11046, 109mp3an1 1408 . . . . . . . . 9 (((𝐹𝑘): ℋ⟶ ℋ ∧ ( Iopop (𝐹𝑘)): ℋ⟶ ℋ) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
11136, 108, 110syl2anc 692 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
112 hmoplin 28689 . . . . . . . . . . . . . . 15 ( Iop ∈ HrmOp → Iop ∈ LinOp)
11313, 112ax-mp 5 . . . . . . . . . . . . . 14 Iop ∈ LinOp
114 hoddi 28737 . . . . . . . . . . . . . 14 (( Iop ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ( Iop ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))))
115113, 46, 114mp3an12 1411 . . . . . . . . . . . . 13 ((𝐹𝑘): ℋ⟶ ℋ → ( Iop ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))))
11636, 115syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ( Iop ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))))
11746hoid1i 28536 . . . . . . . . . . . . . 14 ( Iop ∘ Iop ) = Iop
118117a1i 11 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119 hoico2 28504 . . . . . . . . . . . . . 14 ((𝐹𝑘): ℋ⟶ ℋ → ( Iop ∘ (𝐹𝑘)) = (𝐹𝑘))
12036, 119syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ( Iop ∘ (𝐹𝑘)) = (𝐹𝑘))
121118, 120oveq12d 6633 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))) = ( Iopop (𝐹𝑘)))
122116, 121eqtrd 2655 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop ∘ ( Iopop (𝐹𝑘))) = ( Iopop (𝐹𝑘)))
123 hmoplin 28689 . . . . . . . . . . . . . 14 ((𝐹𝑘) ∈ HrmOp → (𝐹𝑘) ∈ LinOp)
12415, 123syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝐹𝑘) ∈ LinOp)
125 hoddi 28737 . . . . . . . . . . . . . 14 (((𝐹𝑘) ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
12646, 125mp3an2 1409 . . . . . . . . . . . . 13 (((𝐹𝑘) ∈ LinOp ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
127124, 36, 126syl2anc 692 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
128 hoico1 28503 . . . . . . . . . . . . . 14 ((𝐹𝑘): ℋ⟶ ℋ → ((𝐹𝑘) ∘ Iop ) = (𝐹𝑘))
12936, 128syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ Iop ) = (𝐹𝑘))
130129oveq1d 6630 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))) = ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
131127, 130eqtrd 2655 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
132122, 131oveq12d 6633 . . . . . . . . . 10 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))))
13336, 46jctil 559 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ))
134 hosubadd4 28561 . . . . . . . . . . 11 ((( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) ∧ ((𝐹𝑘): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)) → (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
135133, 36, 42, 134syl12anc 1321 . . . . . . . . . 10 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
136132, 135eqtrd 2655 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
137 ho2times 28566 . . . . . . . . . . 11 ((𝐹𝑘): ℋ⟶ ℋ → (2 ·op (𝐹𝑘)) = ((𝐹𝑘) +op (𝐹𝑘)))
13836, 137syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → (2 ·op (𝐹𝑘)) = ((𝐹𝑘) +op (𝐹𝑘)))
139138oveq2d 6631 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
140 hoaddsubass 28562 . . . . . . . . . . 11 (( Iop : ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
14146, 140mp3an1 1408 . . . . . . . . . 10 ((((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
14242, 38, 141syl2anc 692 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
143136, 139, 1423eqtr2d 2661 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
144111, 143eqtrd 2655 . . . . . . 7 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
145144oveq1d 6630 . . . . . 6 (𝑘 ∈ ℕ → ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
1467, 8, 9opsqrlem5 28891 . . . . . . . 8 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
147146oveq2d 6631 . . . . . . 7 (𝑘 ∈ ℕ → ( Iopop (𝐹‘(𝑘 + 1))) = ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
148147oveq2d 6631 . . . . . 6 (𝑘 ∈ ℕ → (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))) = (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
149106, 145, 1483eqtr4d 2665 . . . . 5 (𝑘 ∈ ℕ → ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)) = (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))))
15033, 149breqtrd 4649 . . . 4 (𝑘 ∈ ℕ → 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))))
151 peano2nn 10992 . . . . . . 7 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
15214ffvelrni 6324 . . . . . . 7 ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153151, 152syl 17 . . . . . 6 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154 hmopd 28769 . . . . . 6 (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp)
15513, 153, 154sylancr 694 . . . . 5 (𝑘 ∈ ℕ → ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156 2re 11050 . . . . . 6 2 ∈ ℝ
157 2pos 11072 . . . . . 6 0 < 2
158 leopmul 28881 . . . . . 6 ((2 ∈ ℝ ∧ ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp ∧ 0 < 2) → ( 0hopop ( Iopop (𝐹‘(𝑘 + 1))) ↔ 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1))))))
159156, 157, 158mp3an13 1412 . . . . 5 (( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp → ( 0hopop ( Iopop (𝐹‘(𝑘 + 1))) ↔ 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1))))))
160155, 159syl 17 . . . 4 (𝑘 ∈ ℕ → ( 0hopop ( Iopop (𝐹‘(𝑘 + 1))) ↔ 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1))))))
161150, 160mpbird 247 . . 3 (𝑘 ∈ ℕ → 0hopop ( Iopop (𝐹‘(𝑘 + 1))))
162 leop3 28872 . . . 4 (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hopop ( Iopop (𝐹‘(𝑘 + 1)))))
163153, 13, 162sylancl 693 . . 3 (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hopop ( Iopop (𝐹‘(𝑘 + 1)))))
164161, 163mpbird 247 . 2 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤op Iop )
1652, 4, 6, 12, 164nn1suc 11001 1 (𝑁 ∈ ℕ → (𝐹𝑁) ≤op Iop )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {csn 4155   class class class wbr 4623   × cxp 5082  ccom 5088  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617  cc 9894  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   · cmul 9901   < clt 10034   / cdiv 10644  cn 10980  2c2 11030  seqcseq 12757  chil 27664   +op chos 27683   ·op chot 27684  op chod 27685   0hop ch0o 27688   Iop chio 27689  LinOpclo 27692  HrmOpcho 27695  op cleo 27703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cc 9217  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975  ax-mulf 9976  ax-hilex 27744  ax-hfvadd 27745  ax-hvcom 27746  ax-hvass 27747  ax-hv0cl 27748  ax-hvaddid 27749  ax-hfvmul 27750  ax-hvmulid 27751  ax-hvmulass 27752  ax-hvdistr1 27753  ax-hvdistr2 27754  ax-hvmul0 27755  ax-hfi 27824  ax-his1 27827  ax-his2 27828  ax-his3 27829  ax-his4 27830  ax-hcompl 27947
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-omul 7525  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-acn 8728  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-starv 15896  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-unif 15905  df-hom 15906  df-cco 15907  df-rest 16023  df-topn 16024  df-0g 16042  df-gsum 16043  df-topgen 16044  df-pt 16045  df-prds 16048  df-xrs 16102  df-qtop 16107  df-imas 16108  df-xps 16110  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-mulg 17481  df-cntz 17690  df-cmn 18135  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-fbas 19683  df-fg 19684  df-cnfld 19687  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cld 20763  df-ntr 20764  df-cls 20765  df-nei 20842  df-cn 20971  df-cnp 20972  df-lm 20973  df-haus 21059  df-tx 21305  df-hmeo 21498  df-fil 21590  df-fm 21682  df-flim 21683  df-flf 21684  df-xms 22065  df-ms 22066  df-tms 22067  df-cfil 22993  df-cau 22994  df-cmet 22995  df-grpo 27235  df-gid 27236  df-ginv 27237  df-gdiv 27238  df-ablo 27287  df-vc 27302  df-nv 27335  df-va 27338  df-ba 27339  df-sm 27340  df-0v 27341  df-vs 27342  df-nmcv 27343  df-ims 27344  df-dip 27444  df-ssp 27465  df-ph 27556  df-cbn 27607  df-hnorm 27713  df-hba 27714  df-hvsub 27716  df-hlim 27717  df-hcau 27718  df-sh 27952  df-ch 27966  df-oc 27997  df-ch0 27998  df-shs 28055  df-pjh 28142  df-hosum 28477  df-homul 28478  df-hodif 28479  df-h0op 28495  df-iop 28496  df-lnop 28588  df-hmop 28591  df-leop 28599
This theorem is referenced by: (None)
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