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Theorem sqpweven 12140
Description: The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
Hypotheses
Ref Expression
oddpwdc.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
oddpwdc.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
Assertion
Ref Expression
sqpweven (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(𝐹‘(𝐴↑2))))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝐽,𝑦   𝑥,𝐴,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧
Allowed substitution hint:   𝐽(𝑧)

Proof of Theorem sqpweven
StepHypRef Expression
1 oddpwdc.j . . . . . . . 8 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
2 oddpwdc.f . . . . . . . 8 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
31, 2oddpwdc 12139 . . . . . . 7 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5466 . . . . . . 7 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5453 . . . . . . 7 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . 6 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelcdmi 5642 . . . . 5 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 6157 . . . . 5 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 9344 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 9051 . . . . 5 2 ∈ ℕ
1211a1i 9 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 9345 . . 3 (𝐴 ∈ ℕ → 2 ∈ ℤ)
14 dvdsmul2 11787 . . 3 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1510, 13, 14syl2anc 411 . 2 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
16 xp1st 6156 . . . . . . . . . 10 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
177, 16syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
18 breq2 4002 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
1918notbid 667 . . . . . . . . . . 11 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2019, 1elrab2 2894 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 ↔ ((1st ‘(𝐹𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2120simplbi 274 . . . . . . . . 9 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → (1st ‘(𝐹𝐴)) ∈ ℕ)
2217, 21syl 14 . . . . . . . 8 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℕ)
2322nnsqcld 10642 . . . . . . 7 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℕ)
2420simprbi 275 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
2517, 24syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
26 2prm 12092 . . . . . . . . . 10 2 ∈ ℙ
2722nnzd 9345 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
28 euclemma 12111 . . . . . . . . . . 11 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ (2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴)))))
29 oridm 757 . . . . . . . . . . 11 ((2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴)))
3028, 29bitrdi 196 . . . . . . . . . 10 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3126, 27, 27, 30mp3an2i 1342 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3225, 31mtbird 673 . . . . . . . 8 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3322nncnd 8904 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3433sqvald 10618 . . . . . . . . 9 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3534breq2d 4010 . . . . . . . 8 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
3632, 35mtbird 673 . . . . . . 7 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
37 breq2 4002 . . . . . . . . 9 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3837notbid 667 . . . . . . . 8 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3938, 1elrab2 2894 . . . . . . 7 (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(𝐹𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4023, 36, 39sylanbrc 417 . . . . . 6 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ 𝐽)
4112nnnn0d 9200 . . . . . . 7 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
429, 41nn0mulcld 9205 . . . . . 6 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
43 opelxp 4650 . . . . . 6 (⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0))
4440, 42, 43sylanbrc 417 . . . . 5 (𝐴 ∈ ℕ → ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0))
4512nncnd 8904 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℂ)
4645, 41, 9expmuld 10624 . . . . . . . 8 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
4746oveq1d 5880 . . . . . . 7 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
4812, 42nnexpcld 10643 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℕ)
4948, 23nnmulcld 8939 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
50 oveq2 5873 . . . . . . . . 9 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
51 oveq2 5873 . . . . . . . . . 10 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(𝐹𝐴)) · 2)))
5251oveq1d 5880 . . . . . . . . 9 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
5350, 52, 2ovmpog 5999 . . . . . . . 8 ((((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0 ∧ ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ) → (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
5440, 42, 49, 53syl3anc 1238 . . . . . . 7 (𝐴 ∈ ℕ → (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
55 f1ocnvfv2 5769 . . . . . . . . . . . . 13 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
563, 55mpan 424 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
57 1st2nd2 6166 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
587, 57syl 14 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
5958fveq2d 5511 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
6056, 59eqtr3d 2210 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
61 df-ov 5868 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6260, 61eqtr4di 2226 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
6312, 9nnexpcld 10643 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
6463, 22nnmulcld 8939 . . . . . . . . . . 11 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
65 oveq2 5873 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
66 oveq2 5873 . . . . . . . . . . . . 13 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
6766oveq1d 5880 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
6865, 67, 2ovmpog 5999 . . . . . . . . . . 11 (((1st ‘(𝐹𝐴)) ∈ 𝐽 ∧ (2nd ‘(𝐹𝐴)) ∈ ℕ0 ∧ ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ) → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
6917, 9, 64, 68syl3anc 1238 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7062, 69eqtrd 2208 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7170oveq1d 5880 . . . . . . . 8 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
7263nncnd 8904 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
7372, 33sqmuld 10633 . . . . . . . 8 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7471, 73eqtrd 2208 . . . . . . 7 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7547, 54, 743eqtr4rd 2219 . . . . . 6 (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)))
76 df-ov 5868 . . . . . 6 (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩)
7775, 76eqtr2di 2225 . . . . 5 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = (𝐴↑2))
78 f1ocnvfv 5770 . . . . . 6 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0)) → ((𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = (𝐴↑2) → (𝐹‘(𝐴↑2)) = ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩))
793, 78mpan 424 . . . . 5 (⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0) → ((𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = (𝐴↑2) → (𝐹‘(𝐴↑2)) = ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩))
8044, 77, 79sylc 62 . . . 4 (𝐴 ∈ ℕ → (𝐹‘(𝐴↑2)) = ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩)
8180fveq2d 5511 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩))
82 op2ndg 6142 . . . 4 ((((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0) → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = ((2nd ‘(𝐹𝐴)) · 2))
8340, 42, 82syl2anc 411 . . 3 (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = ((2nd ‘(𝐹𝐴)) · 2))
8481, 83eqtrd 2208 . 2 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = ((2nd ‘(𝐹𝐴)) · 2))
8515, 84breqtrrd 4026 1 (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(𝐹‘(𝐴↑2))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2146  {crab 2457  cop 3592   class class class wbr 3998   × cxp 4618  ccnv 4619  wf 5204  1-1-ontowf1o 5207  cfv 5208  (class class class)co 5865  cmpo 5867  1st c1st 6129  2nd c2nd 6130   · cmul 7791  cn 8890  2c2 8941  0cn0 9147  cz 9224  cexp 10487  cdvds 11760  cprime 12072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-2o 6408  df-er 6525  df-en 6731  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909  df-prm 12073
This theorem is referenced by:  sqne2sq  12142
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