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Theorem opsqrlem6 29914
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 6663 . . 3 (𝑗 = 1 → (𝐹𝑗) = (𝐹‘1))
21breq1d 5067 . 2 (𝑗 = 1 → ((𝐹𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3 fveq2 6663 . . 3 (𝑗 = (𝑘 + 1) → (𝐹𝑗) = (𝐹‘(𝑘 + 1)))
43breq1d 5067 . 2 (𝑗 = (𝑘 + 1) → ((𝐹𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5 fveq2 6663 . . 3 (𝑗 = 𝑁 → (𝐹𝑗) = (𝐹𝑁))
65breq1d 5067 . 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 29910 . . 3 (𝐹‘1) = 0hop
11 idleop 29900 . . 3 0hopop Iop
1210, 11eqbrtri 5078 . 2 (𝐹‘1) ≤op Iop
13 idhmop 29751 . . . . . . . 8 Iop ∈ HrmOp
147, 8, 9opsqrlem4 29912 . . . . . . . . 9 𝐹:ℕ⟶HrmOp
1514ffvelrni 6843 . . . . . . . 8 (𝑘 ∈ ℕ → (𝐹𝑘) ∈ HrmOp)
16 hmopd 29791 . . . . . . . 8 (( Iop ∈ HrmOp ∧ (𝐹𝑘) ∈ HrmOp) → ( Iopop (𝐹𝑘)) ∈ HrmOp)
1713, 15, 16sylancr 589 . . . . . . 7 (𝑘 ∈ ℕ → ( Iopop (𝐹𝑘)) ∈ HrmOp)
18 eqid 2819 . . . . . . . 8 (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))
19 hmopco 29792 . . . . . . . 8 ((( Iopop (𝐹𝑘)) ∈ HrmOp ∧ ( Iopop (𝐹𝑘)) ∈ HrmOp ∧ (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
2018, 19mp3an3 1444 . . . . . . 7 ((( Iopop (𝐹𝑘)) ∈ HrmOp ∧ ( Iopop (𝐹𝑘)) ∈ HrmOp) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
2117, 17, 20syl2anc 586 . . . . . 6 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
22 leopsq 29898 . . . . . . 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 29894 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇op Iop ↔ 0hopop ( Iopop 𝑇)))
267, 13, 25mp2an 690 . . . . . . . 8 (𝑇op Iop ↔ 0hopop ( Iopop 𝑇))
2724, 26mpbi 232 . . . . . . 7 0hopop ( Iopop 𝑇)
28 hmopd 29791 . . . . . . . . 9 (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iopop 𝑇) ∈ HrmOp)
2913, 7, 28mp2an 690 . . . . . . . 8 ( Iopop 𝑇) ∈ HrmOp
30 leopadd 29901 . . . . . . . 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 699 . . . . . . 7 (((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ ( 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∧ 0hopop ( Iopop 𝑇))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3227, 31mpanr2 702 . . . . . 6 (((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3321, 23, 32syl2anc 586 . . . . 5 (𝑘 ∈ ℕ → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
34 2cn 11704 . . . . . . . . . 10 2 ∈ ℂ
35 hmopf 29643 . . . . . . . . . . 11 ((𝐹𝑘) ∈ HrmOp → (𝐹𝑘): ℋ⟶ ℋ)
3615, 35syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝐹𝑘): ℋ⟶ ℋ)
37 homulcl 29528 . . . . . . . . . 10 ((2 ∈ ℂ ∧ (𝐹𝑘): ℋ⟶ ℋ) → (2 ·op (𝐹𝑘)): ℋ⟶ ℋ)
3834, 36, 37sylancr 589 . . . . . . . . 9 (𝑘 ∈ ℕ → (2 ·op (𝐹𝑘)): ℋ⟶ ℋ)
39 hmopf 29643 . . . . . . . . . . 11 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
407, 39ax-mp 5 . . . . . . . . . 10 𝑇: ℋ⟶ ℋ
41 fco 6524 . . . . . . . . . . 11 (((𝐹𝑘): ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)
4236, 36, 41syl2anc 586 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)
43 hosubcl 29542 . . . . . . . . . 10 ((𝑇: ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
4440, 42, 43sylancr 589 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
45 hmopf 29643 . . . . . . . . . . . 12 ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
4613, 45ax-mp 5 . . . . . . . . . . 11 Iop : ℋ⟶ ℋ
47 homulcl 29528 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
4834, 46, 47mp2an 690 . . . . . . . . . 10 (2 ·op Iop ): ℋ⟶ ℋ
49 hosubsub4 29587 . . . . . . . . . 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 1442 . . . . . . . . 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 586 . . . . . . . 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 29542 . . . . . . . . . . . . . . 15 ((((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ)
5342, 38, 52syl2anc 586 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ)
54 hoadd32 29552 . . . . . . . . . . . . . . 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 1446 . . . . . . . . . . . . . 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 29588 . . . . . . . . . . . . . . 15 ( Iop : ℋ⟶ ℋ → (2 ·op Iop ) = ( Iop +op Iop ))
5846, 57ax-mp 5 . . . . . . . . . . . . . 14 (2 ·op Iop ) = ( Iop +op Iop )
5958oveq1i 7158 . . . . . . . . . . . . 13 ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))))
6056, 59syl6eqr 2872 . . . . . . . . . . . 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 29584 . . . . . . . . . . . . . 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 1442 . . . . . . . . . . . . 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 586 . . . . . . . . . . . 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 2857 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))))
6564oveq1d 7163 . . . . . . . . . 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 29527 . . . . . . . . . . . 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 589 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ)
68 hoaddsubass 29584 . . . . . . . . . . . 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 1447 . . . . . . . . . . 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 29527 . . . . . . . . . . . 12 (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → ((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
7248, 42, 71sylancr 589 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
73 hosubsub4 29587 . . . . . . . . . . . 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 1444 . . . . . . . . . . 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 586 . . . . . . . . . 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 2862 . . . . . . . . 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 29583 . . . . . . . . . . . 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 701 . . . . . . . . . . 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 698 . . . . . . . . . 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 586 . . . . . . . . 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 2857 . . . . . . . 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 11844 . . . . . . . . . . . 12 (1 / 2) ∈ ℂ
83 homulcl 29528 . . . . . . . . . . . 12 (((1 / 2) ∈ ℂ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ)
8482, 44, 83sylancr 589 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ)
85 hoadddi 29572 . . . . . . . . . . . 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 1442 . . . . . . . . . . 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 586 . . . . . . . . . 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 11733 . . . . . . . . . . . . . 14 2 ≠ 0
8934, 88recidi 11363 . . . . . . . . . . . . 13 (2 · (1 / 2)) = 1
9089oveq1i 7158 . . . . . . . . . . . 12 ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (1 ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
91 homulass 29571 . . . . . . . . . . . . . 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 1445 . . . . . . . . . . . . 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 29569 . . . . . . . . . . . . 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 2878 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))) = (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
9796oveq2d 7164 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
9887, 97eqtrd 2854 . . . . . . . . 9 (𝑘 ∈ ℕ → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
9998oveq2d 7164 . . . . . . . 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 2864 . . . . . . 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 29527 . . . . . . . . 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 586 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ)
103 hosubdi 29577 . . . . . . . . 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 1445 . . . . . . . 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 2857 . . . . . 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 29542 . . . . . . . . . 10 (( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ( Iopop (𝐹𝑘)): ℋ⟶ ℋ)
10846, 36, 107sylancr 589 . . . . . . . . 9 (𝑘 ∈ ℕ → ( Iopop (𝐹𝑘)): ℋ⟶ ℋ)
109 hocsubdir 29554 . . . . . . . . . 10 (( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ ∧ ( Iopop (𝐹𝑘)): ℋ⟶ ℋ) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
11046, 109mp3an1 1442 . . . . . . . . 9 (((𝐹𝑘): ℋ⟶ ℋ ∧ ( Iopop (𝐹𝑘)): ℋ⟶ ℋ) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
11136, 108, 110syl2anc 586 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
112 hmoplin 29711 . . . . . . . . . . . . . . 15 ( Iop ∈ HrmOp → Iop ∈ LinOp)
11313, 112ax-mp 5 . . . . . . . . . . . . . 14 Iop ∈ LinOp
114 hoddi 29759 . . . . . . . . . . . . . 14 (( Iop ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ( Iop ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))))
115113, 46, 114mp3an12 1445 . . . . . . . . . . . . 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 29558 . . . . . . . . . . . . . 14 ( Iop ∘ Iop ) = Iop
118117a1i 11 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119 hoico2 29526 . . . . . . . . . . . . . 14 ((𝐹𝑘): ℋ⟶ ℋ → ( Iop ∘ (𝐹𝑘)) = (𝐹𝑘))
12036, 119syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ( Iop ∘ (𝐹𝑘)) = (𝐹𝑘))
121118, 120oveq12d 7166 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))) = ( Iopop (𝐹𝑘)))
122116, 121eqtrd 2854 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop ∘ ( Iopop (𝐹𝑘))) = ( Iopop (𝐹𝑘)))
123 hmoplin 29711 . . . . . . . . . . . . . 14 ((𝐹𝑘) ∈ HrmOp → (𝐹𝑘) ∈ LinOp)
12415, 123syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝐹𝑘) ∈ LinOp)
125 hoddi 29759 . . . . . . . . . . . . . 14 (((𝐹𝑘) ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
12646, 125mp3an2 1443 . . . . . . . . . . . . 13 (((𝐹𝑘) ∈ LinOp ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
127124, 36, 126syl2anc 586 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
128 hoico1 29525 . . . . . . . . . . . . . 14 ((𝐹𝑘): ℋ⟶ ℋ → ((𝐹𝑘) ∘ Iop ) = (𝐹𝑘))
12936, 128syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ Iop ) = (𝐹𝑘))
130129oveq1d 7163 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))) = ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
131127, 130eqtrd 2854 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
132122, 131oveq12d 7166 . . . . . . . . . 10 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))))
13336, 46jctil 522 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ))
134 hosubadd4 29583 . . . . . . . . . . 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 834 . . . . . . . . . 10 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
136132, 135eqtrd 2854 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
137 ho2times 29588 . . . . . . . . . . 11 ((𝐹𝑘): ℋ⟶ ℋ → (2 ·op (𝐹𝑘)) = ((𝐹𝑘) +op (𝐹𝑘)))
13836, 137syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → (2 ·op (𝐹𝑘)) = ((𝐹𝑘) +op (𝐹𝑘)))
139138oveq2d 7164 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
140 hoaddsubass 29584 . . . . . . . . . . 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 1442 . . . . . . . . . 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 586 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
143136, 139, 1423eqtr2d 2860 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
144111, 143eqtrd 2854 . . . . . . 7 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
145144oveq1d 7163 . . . . . 6 (𝑘 ∈ ℕ → ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
1467, 8, 9opsqrlem5 29913 . . . . . . . 8 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
147146oveq2d 7164 . . . . . . 7 (𝑘 ∈ ℕ → ( Iopop (𝐹‘(𝑘 + 1))) = ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
148147oveq2d 7164 . . . . . 6 (𝑘 ∈ ℕ → (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))) = (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
149106, 145, 1483eqtr4d 2864 . . . . 5 (𝑘 ∈ ℕ → ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)) = (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))))
15033, 149breqtrd 5083 . . . 4 (𝑘 ∈ ℕ → 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))))
151 peano2nn 11642 . . . . . . 7 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
15214ffvelrni 6843 . . . . . . 7 ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153151, 152syl 17 . . . . . 6 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154 hmopd 29791 . . . . . 6 (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp)
15513, 153, 154sylancr 589 . . . . 5 (𝑘 ∈ ℕ → ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156 2re 11703 . . . . . 6 2 ∈ ℝ
157 2pos 11732 . . . . . 6 0 < 2
158 leopmul 29903 . . . . . 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 1446 . . . . 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 259 . . 3 (𝑘 ∈ ℕ → 0hopop ( Iopop (𝐹‘(𝑘 + 1))))
162 leop3 29894 . . . 4 (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hopop ( Iopop (𝐹‘(𝑘 + 1)))))
163153, 13, 162sylancl 588 . . 3 (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hopop ( Iopop (𝐹‘(𝑘 + 1)))))
164161, 163mpbird 259 . 2 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤op Iop )
1652, 4, 6, 12, 164nn1suc 11651 1 (𝑁 ∈ ℕ → (𝐹𝑁) ≤op Iop )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  {csn 4559   class class class wbr 5057   × cxp 5546  ccom 5552  wf 6344  cfv 6348  (class class class)co 7148  cmpo 7150  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534   < clt 10667   / cdiv 11289  cn 11630  2c2 11684  seqcseq 13361  chba 28688   +op chos 28707   ·op chot 28708  op chod 28709   0hop ch0o 28712   Iop chio 28713  LinOpclo 28716  HrmOpcho 28719  op cleo 28727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cc 9849  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609  ax-hilex 28768  ax-hfvadd 28769  ax-hvcom 28770  ax-hvass 28771  ax-hv0cl 28772  ax-hvaddid 28773  ax-hfvmul 28774  ax-hvmulid 28775  ax-hvmulass 28776  ax-hvdistr1 28777  ax-hvdistr2 28778  ax-hvmul0 28779  ax-hfi 28848  ax-his1 28851  ax-his2 28852  ax-his3 28853  ax-his4 28854  ax-hcompl 28971
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-omul 8099  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-acn 9363  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-fbas 20534  df-fg 20535  df-cnfld 20538  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-ntr 21620  df-cls 21621  df-nei 21698  df-cn 21827  df-cnp 21828  df-lm 21829  df-haus 21915  df-tx 22162  df-hmeo 22355  df-fil 22446  df-fm 22538  df-flim 22539  df-flf 22540  df-xms 22922  df-ms 22923  df-tms 22924  df-cfil 23850  df-cau 23851  df-cmet 23852  df-grpo 28262  df-gid 28263  df-ginv 28264  df-gdiv 28265  df-ablo 28314  df-vc 28328  df-nv 28361  df-va 28364  df-ba 28365  df-sm 28366  df-0v 28367  df-vs 28368  df-nmcv 28369  df-ims 28370  df-dip 28470  df-ssp 28491  df-ph 28582  df-cbn 28632  df-hnorm 28737  df-hba 28738  df-hvsub 28740  df-hlim 28741  df-hcau 28742  df-sh 28976  df-ch 28990  df-oc 29021  df-ch0 29022  df-shs 29077  df-pjh 29164  df-hosum 29499  df-homul 29500  df-hodif 29501  df-h0op 29517  df-iop 29518  df-lnop 29610  df-hmop 29613  df-leop 29621
This theorem is referenced by: (None)
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