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Theorem iserodd 16165
 Description: Collect the odd terms in a sequence. (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 10-Jul-2022.)
Hypotheses
Ref Expression
iserodd.f ((𝜑𝑘 ∈ ℕ0) → 𝐶 ∈ ℂ)
iserodd.h (𝑛 = ((2 · 𝑘) + 1) → 𝐵 = 𝐶)
Assertion
Ref Expression
iserodd (𝜑 → (seq0( + , (𝑘 ∈ ℕ0𝐶)) ⇝ 𝐴 ↔ seq1( + , (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))) ⇝ 𝐴))
Distinct variable groups:   𝐵,𝑘   𝐶,𝑛   𝑘,𝑛,𝜑
Allowed substitution hints:   𝐴(𝑘,𝑛)   𝐵(𝑛)   𝐶(𝑘)

Proof of Theorem iserodd
Dummy variables 𝑖 𝑗 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 12272 . 2 0 = (ℤ‘0)
2 nnuz 12273 . 2 ℕ = (ℤ‘1)
3 0zd 11985 . 2 (𝜑 → 0 ∈ ℤ)
4 1zzd 12005 . 2 (𝜑 → 1 ∈ ℤ)
5 2nn0 11906 . . . . . 6 2 ∈ ℕ0
65a1i 11 . . . . 5 (𝜑 → 2 ∈ ℕ0)
7 nn0mulcl 11925 . . . . 5 ((2 ∈ ℕ0𝑚 ∈ ℕ0) → (2 · 𝑚) ∈ ℕ0)
86, 7sylan 583 . . . 4 ((𝜑𝑚 ∈ ℕ0) → (2 · 𝑚) ∈ ℕ0)
9 nn0p1nn 11928 . . . 4 ((2 · 𝑚) ∈ ℕ0 → ((2 · 𝑚) + 1) ∈ ℕ)
108, 9syl 17 . . 3 ((𝜑𝑚 ∈ ℕ0) → ((2 · 𝑚) + 1) ∈ ℕ)
1110fmpttd 6860 . 2 (𝜑 → (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)):ℕ0⟶ℕ)
12 nn0mulcl 11925 . . . . . 6 ((2 ∈ ℕ0𝑖 ∈ ℕ0) → (2 · 𝑖) ∈ ℕ0)
136, 12sylan 583 . . . . 5 ((𝜑𝑖 ∈ ℕ0) → (2 · 𝑖) ∈ ℕ0)
1413nn0red 11948 . . . 4 ((𝜑𝑖 ∈ ℕ0) → (2 · 𝑖) ∈ ℝ)
15 peano2nn0 11929 . . . . . 6 (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
16 nn0mulcl 11925 . . . . . 6 ((2 ∈ ℕ0 ∧ (𝑖 + 1) ∈ ℕ0) → (2 · (𝑖 + 1)) ∈ ℕ0)
176, 15, 16syl2an 598 . . . . 5 ((𝜑𝑖 ∈ ℕ0) → (2 · (𝑖 + 1)) ∈ ℕ0)
1817nn0red 11948 . . . 4 ((𝜑𝑖 ∈ ℕ0) → (2 · (𝑖 + 1)) ∈ ℝ)
19 1red 10635 . . . 4 ((𝜑𝑖 ∈ ℕ0) → 1 ∈ ℝ)
20 nn0re 11898 . . . . . . 7 (𝑖 ∈ ℕ0𝑖 ∈ ℝ)
2120adantl 485 . . . . . 6 ((𝜑𝑖 ∈ ℕ0) → 𝑖 ∈ ℝ)
2221ltp1d 11563 . . . . 5 ((𝜑𝑖 ∈ ℕ0) → 𝑖 < (𝑖 + 1))
23 1red 10635 . . . . . . . 8 (𝑖 ∈ ℕ0 → 1 ∈ ℝ)
2420, 23readdcld 10663 . . . . . . 7 (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℝ)
25 2rp 12386 . . . . . . . 8 2 ∈ ℝ+
2625a1i 11 . . . . . . 7 (𝑖 ∈ ℕ0 → 2 ∈ ℝ+)
2720, 24, 26ltmul2d 12465 . . . . . 6 (𝑖 ∈ ℕ0 → (𝑖 < (𝑖 + 1) ↔ (2 · 𝑖) < (2 · (𝑖 + 1))))
2827adantl 485 . . . . 5 ((𝜑𝑖 ∈ ℕ0) → (𝑖 < (𝑖 + 1) ↔ (2 · 𝑖) < (2 · (𝑖 + 1))))
2922, 28mpbid 235 . . . 4 ((𝜑𝑖 ∈ ℕ0) → (2 · 𝑖) < (2 · (𝑖 + 1)))
3014, 18, 19, 29ltadd1dd 11244 . . 3 ((𝜑𝑖 ∈ ℕ0) → ((2 · 𝑖) + 1) < ((2 · (𝑖 + 1)) + 1))
31 oveq2 7147 . . . . . 6 (𝑚 = 𝑖 → (2 · 𝑚) = (2 · 𝑖))
3231oveq1d 7154 . . . . 5 (𝑚 = 𝑖 → ((2 · 𝑚) + 1) = ((2 · 𝑖) + 1))
33 eqid 2801 . . . . 5 (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)) = (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))
34 ovex 7172 . . . . 5 ((2 · 𝑖) + 1) ∈ V
3532, 33, 34fvmpt 6749 . . . 4 (𝑖 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖) = ((2 · 𝑖) + 1))
3635adantl 485 . . 3 ((𝜑𝑖 ∈ ℕ0) → ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖) = ((2 · 𝑖) + 1))
3715adantl 485 . . . 4 ((𝜑𝑖 ∈ ℕ0) → (𝑖 + 1) ∈ ℕ0)
38 oveq2 7147 . . . . . 6 (𝑚 = (𝑖 + 1) → (2 · 𝑚) = (2 · (𝑖 + 1)))
3938oveq1d 7154 . . . . 5 (𝑚 = (𝑖 + 1) → ((2 · 𝑚) + 1) = ((2 · (𝑖 + 1)) + 1))
40 ovex 7172 . . . . 5 ((2 · (𝑖 + 1)) + 1) ∈ V
4139, 33, 40fvmpt 6749 . . . 4 ((𝑖 + 1) ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘(𝑖 + 1)) = ((2 · (𝑖 + 1)) + 1))
4237, 41syl 17 . . 3 ((𝜑𝑖 ∈ ℕ0) → ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘(𝑖 + 1)) = ((2 · (𝑖 + 1)) + 1))
4330, 36, 423brtr4d 5065 . 2 ((𝜑𝑖 ∈ ℕ0) → ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖) < ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘(𝑖 + 1)))
44 eldifi 4057 . . . . . . 7 (𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))) → 𝑛 ∈ ℕ)
45 simpr 488 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
46 0cnd 10627 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 0 ∈ ℂ)
47 nnz 11996 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
4847adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
49 odd2np1 15685 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (¬ 2 ∥ 𝑛 ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = 𝑛))
5048, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛 ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = 𝑛))
51 simprl 770 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 𝑘 ∈ ℤ)
52 nnm1nn0 11930 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
5352ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (𝑛 − 1) ∈ ℕ0)
5453nn0red 11948 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (𝑛 − 1) ∈ ℝ)
5525a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 2 ∈ ℝ+)
5653nn0ge0d 11950 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 0 ≤ (𝑛 − 1))
5754, 55, 56divge0d 12463 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 0 ≤ ((𝑛 − 1) / 2))
58 simprr 772 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((2 · 𝑘) + 1) = 𝑛)
5958oveq1d 7154 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (((2 · 𝑘) + 1) − 1) = (𝑛 − 1))
60 2cn 11704 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℂ
61 zcn 11978 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
6261ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 𝑘 ∈ ℂ)
63 mulcl 10614 . . . . . . . . . . . . . . . . . . . . . 22 ((2 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (2 · 𝑘) ∈ ℂ)
6460, 62, 63sylancr 590 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (2 · 𝑘) ∈ ℂ)
65 ax-1cn 10588 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℂ
66 pncan 10885 . . . . . . . . . . . . . . . . . . . . 21 (((2 · 𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑘) + 1) − 1) = (2 · 𝑘))
6764, 65, 66sylancl 589 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (((2 · 𝑘) + 1) − 1) = (2 · 𝑘))
6859, 67eqtr3d 2838 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (𝑛 − 1) = (2 · 𝑘))
6968oveq1d 7154 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((𝑛 − 1) / 2) = ((2 · 𝑘) / 2))
70 2cnd 11707 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 2 ∈ ℂ)
71 2ne0 11733 . . . . . . . . . . . . . . . . . . . 20 2 ≠ 0
7271a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 2 ≠ 0)
7362, 70, 72divcan3d 11414 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((2 · 𝑘) / 2) = 𝑘)
7469, 73eqtrd 2836 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((𝑛 − 1) / 2) = 𝑘)
7557, 74breqtrd 5059 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 0 ≤ 𝑘)
76 elnn0z 11986 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
7751, 75, 76sylanbrc 586 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 𝑘 ∈ ℕ0)
7877ex 416 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛) → 𝑘 ∈ ℕ0))
79 simpr 488 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛) → ((2 · 𝑘) + 1) = 𝑛)
8079eqcomd 2807 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛) → 𝑛 = ((2 · 𝑘) + 1))
8178, 80jca2 517 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛) → (𝑘 ∈ ℕ0𝑛 = ((2 · 𝑘) + 1))))
8281reximdv2 3233 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = 𝑛 → ∃𝑘 ∈ ℕ0 𝑛 = ((2 · 𝑘) + 1)))
8350, 82sylbid 243 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛 → ∃𝑘 ∈ ℕ0 𝑛 = ((2 · 𝑘) + 1)))
84 iserodd.f . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → 𝐶 ∈ ℂ)
85 iserodd.h . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑘) + 1) → 𝐵 = 𝐶)
8685eleq1d 2877 . . . . . . . . . . . . . 14 (𝑛 = ((2 · 𝑘) + 1) → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
8784, 86syl5ibrcom 250 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ0) → (𝑛 = ((2 · 𝑘) + 1) → 𝐵 ∈ ℂ))
8887rexlimdva 3246 . . . . . . . . . . . 12 (𝜑 → (∃𝑘 ∈ ℕ0 𝑛 = ((2 · 𝑘) + 1) → 𝐵 ∈ ℂ))
8988adantr 484 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (∃𝑘 ∈ ℕ0 𝑛 = ((2 · 𝑘) + 1) → 𝐵 ∈ ℂ))
9083, 89syld 47 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛𝐵 ∈ ℂ))
9190imp 410 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝐵 ∈ ℂ)
9246, 91ifclda 4462 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → if(2 ∥ 𝑛, 0, 𝐵) ∈ ℂ)
93 eqid 2801 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵)) = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))
9493fvmpt2 6760 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ if(2 ∥ 𝑛, 0, 𝐵) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = if(2 ∥ 𝑛, 0, 𝐵))
9545, 92, 94syl2anc 587 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = if(2 ∥ 𝑛, 0, 𝐵))
9644, 95sylan2 595 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = if(2 ∥ 𝑛, 0, 𝐵))
97 eldif 3894 . . . . . . . 8 (𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))))
98 oveq2 7147 . . . . . . . . . . . . . . 15 (𝑚 = 𝑘 → (2 · 𝑚) = (2 · 𝑘))
9998oveq1d 7154 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1))
10099cbvmptv 5136 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)) = (𝑘 ∈ ℕ0 ↦ ((2 · 𝑘) + 1))
101100elrnmpt 5796 . . . . . . . . . . . 12 (𝑛 ∈ V → (𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)) ↔ ∃𝑘 ∈ ℕ0 𝑛 = ((2 · 𝑘) + 1)))
102101elv 3449 . . . . . . . . . . 11 (𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)) ↔ ∃𝑘 ∈ ℕ0 𝑛 = ((2 · 𝑘) + 1))
10383, 102syl6ibr 255 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))))
104103con1d 147 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (¬ 𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)) → 2 ∥ 𝑛))
105104impr 458 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))) → 2 ∥ 𝑛)
10697, 105sylan2b 596 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))) → 2 ∥ 𝑛)
107106iftrued 4436 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))) → if(2 ∥ 𝑛, 0, 𝐵) = 0)
10896, 107eqtrd 2836 . . . . 5 ((𝜑𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0)
109108ralrimiva 3152 . . . 4 (𝜑 → ∀𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0)
110 nfv 1915 . . . . 5 𝑗((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0
111 nffvmpt1 6660 . . . . . 6 𝑛((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗)
112111nfeq1 2973 . . . . 5 𝑛((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) = 0
113 fveqeq2 6658 . . . . 5 (𝑛 = 𝑗 → (((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0 ↔ ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) = 0))
114110, 112, 113cbvralw 3390 . . . 4 (∀𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0 ↔ ∀𝑗 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) = 0)
115109, 114sylib 221 . . 3 (𝜑 → ∀𝑗 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) = 0)
116115r19.21bi 3176 . 2 ((𝜑𝑗 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1)))) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) = 0)
11792fmpttd 6860 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵)):ℕ⟶ℂ)
118117ffvelrnda 6832 . 2 ((𝜑𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) ∈ ℂ)
119 simpr 488 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
120 eqid 2801 . . . . . . . 8 (𝑘 ∈ ℕ0𝐶) = (𝑘 ∈ ℕ0𝐶)
121120fvmpt2 6760 . . . . . . 7 ((𝑘 ∈ ℕ0𝐶 ∈ ℂ) → ((𝑘 ∈ ℕ0𝐶)‘𝑘) = 𝐶)
122119, 84, 121syl2anc 587 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → ((𝑘 ∈ ℕ0𝐶)‘𝑘) = 𝐶)
123 ovex 7172 . . . . . . . . . 10 ((2 · 𝑘) + 1) ∈ V
12499, 33, 123fvmpt 6749 . . . . . . . . 9 (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘) = ((2 · 𝑘) + 1))
125124adantl 485 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → ((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘) = ((2 · 𝑘) + 1))
126125fveq2d 6653 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘)) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((2 · 𝑘) + 1)))
127 breq2 5037 . . . . . . . . 9 (𝑛 = ((2 · 𝑘) + 1) → (2 ∥ 𝑛 ↔ 2 ∥ ((2 · 𝑘) + 1)))
128127, 85ifbieq2d 4453 . . . . . . . 8 (𝑛 = ((2 · 𝑘) + 1) → if(2 ∥ 𝑛, 0, 𝐵) = if(2 ∥ ((2 · 𝑘) + 1), 0, 𝐶))
129 nn0mulcl 11925 . . . . . . . . . 10 ((2 ∈ ℕ0𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
1306, 129sylan 583 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
131 nn0p1nn 11928 . . . . . . . . 9 ((2 · 𝑘) ∈ ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ)
132130, 131syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℕ)
133 2z 12006 . . . . . . . . . . . 12 2 ∈ ℤ
134 nn0z 11997 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
135134adantl 485 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
136 dvdsmul1 15626 . . . . . . . . . . . 12 ((2 ∈ ℤ ∧ 𝑘 ∈ ℤ) → 2 ∥ (2 · 𝑘))
137133, 135, 136sylancr 590 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ0) → 2 ∥ (2 · 𝑘))
138130nn0zd 12077 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℤ)
139 2nn 11702 . . . . . . . . . . . . 13 2 ∈ ℕ
140139a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → 2 ∈ ℕ)
141 1lt2 11800 . . . . . . . . . . . . 13 1 < 2
142141a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → 1 < 2)
143 ndvdsp1 15755 . . . . . . . . . . . 12 (((2 · 𝑘) ∈ ℤ ∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ (2 · 𝑘) → ¬ 2 ∥ ((2 · 𝑘) + 1)))
144138, 140, 142, 143syl3anc 1368 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ0) → (2 ∥ (2 · 𝑘) → ¬ 2 ∥ ((2 · 𝑘) + 1)))
145137, 144mpd 15 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ0) → ¬ 2 ∥ ((2 · 𝑘) + 1))
146145iffalsed 4439 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ0) → if(2 ∥ ((2 · 𝑘) + 1), 0, 𝐶) = 𝐶)
147146, 84eqeltrd 2893 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → if(2 ∥ ((2 · 𝑘) + 1), 0, 𝐶) ∈ ℂ)
14893, 128, 132, 147fvmptd3 6772 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((2 · 𝑘) + 1)) = if(2 ∥ ((2 · 𝑘) + 1), 0, 𝐶))
149126, 148, 1463eqtrd 2840 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘)) = 𝐶)
150122, 149eqtr4d 2839 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → ((𝑘 ∈ ℕ0𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘)))
151150ralrimiva 3152 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ0 ((𝑘 ∈ ℕ0𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘)))
152 nfv 1915 . . . . 5 𝑖((𝑘 ∈ ℕ0𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘))
153 nffvmpt1 6660 . . . . . 6 𝑘((𝑘 ∈ ℕ0𝐶)‘𝑖)
154153nfeq1 2973 . . . . 5 𝑘((𝑘 ∈ ℕ0𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖))
155 fveq2 6649 . . . . . 6 (𝑘 = 𝑖 → ((𝑘 ∈ ℕ0𝐶)‘𝑘) = ((𝑘 ∈ ℕ0𝐶)‘𝑖))
156 2fveq3 6654 . . . . . 6 (𝑘 = 𝑖 → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘)) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖)))
157155, 156eqeq12d 2817 . . . . 5 (𝑘 = 𝑖 → (((𝑘 ∈ ℕ0𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘)) ↔ ((𝑘 ∈ ℕ0𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖))))
158152, 154, 157cbvralw 3390 . . . 4 (∀𝑘 ∈ ℕ0 ((𝑘 ∈ ℕ0𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘)) ↔ ∀𝑖 ∈ ℕ0 ((𝑘 ∈ ℕ0𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖)))
159151, 158sylib 221 . . 3 (𝜑 → ∀𝑖 ∈ ℕ0 ((𝑘 ∈ ℕ0𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖)))
160159r19.21bi 3176 . 2 ((𝜑𝑖 ∈ ℕ0) → ((𝑘 ∈ ℕ0𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖)))
1611, 2, 3, 4, 11, 43, 116, 118, 160isercoll2 15020 1 (𝜑 → (seq0( + , (𝑘 ∈ ℕ0𝐶)) ⇝ 𝐴 ↔ seq1( + , (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))) ⇝ 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  Vcvv 3444   ∖ cdif 3881  ifcif 4428   class class class wbr 5033   ↦ cmpt 5113  ran crn 5524  ‘cfv 6328  (class class class)co 7139  ℂcc 10528  ℝcr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535   < clt 10668   ≤ cle 10669   − cmin 10863   / cdiv 11290  ℕcn 11629  2c2 11684  ℕ0cn0 11889  ℤcz 11973  ℝ+crp 12381  seqcseq 13368   ⇝ cli 14836   ∥ cdvds 15602 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-seq 13369  df-exp 13430  df-hash 13691  df-shft 14421  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-dvds 15603 This theorem is referenced by:  atantayl3  25528  leibpilem2  25530
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