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Theorem sqpweven 12158
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 12157 . . . . . . 7 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5470 . . . . . . 7 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5457 . . . . . . 7 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . 6 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelcdmi 5646 . . . . 5 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 6161 . . . . 5 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 9362 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 9069 . . . . 5 2 ∈ ℕ
1211a1i 9 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 9363 . . 3 (𝐴 ∈ ℕ → 2 ∈ ℤ)
14 dvdsmul2 11805 . . 3 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1510, 13, 14syl2anc 411 . 2 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
16 xp1st 6160 . . . . . . . . . 10 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
177, 16syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
18 breq2 4004 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
1918notbid 667 . . . . . . . . . . 11 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2019, 1elrab2 2896 . . . . . . . . . 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 10660 . . . . . . 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 12110 . . . . . . . . . 10 2 ∈ ℙ
2722nnzd 9363 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
28 euclemma 12129 . . . . . . . . . . 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 8922 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3433sqvald 10636 . . . . . . . . 9 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3534breq2d 4012 . . . . . . . 8 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
3632, 35mtbird 673 . . . . . . 7 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
37 breq2 4004 . . . . . . . . 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 2896 . . . . . . 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 9218 . . . . . . 7 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
429, 41nn0mulcld 9223 . . . . . 6 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
43 opelxp 4653 . . . . . 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 8922 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℂ)
4645, 41, 9expmuld 10642 . . . . . . . 8 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
4746oveq1d 5884 . . . . . . 7 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
4812, 42nnexpcld 10661 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℕ)
4948, 23nnmulcld 8957 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
50 oveq2 5877 . . . . . . . . 9 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
51 oveq2 5877 . . . . . . . . . 10 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(𝐹𝐴)) · 2)))
5251oveq1d 5884 . . . . . . . . 9 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
5350, 52, 2ovmpog 6003 . . . . . . . 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 5773 . . . . . . . . . . . . 13 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
563, 55mpan 424 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
57 1st2nd2 6170 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
587, 57syl 14 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
5958fveq2d 5515 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
6056, 59eqtr3d 2212 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
61 df-ov 5872 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6260, 61eqtr4di 2228 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
6312, 9nnexpcld 10661 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
6463, 22nnmulcld 8957 . . . . . . . . . . 11 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
65 oveq2 5877 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
66 oveq2 5877 . . . . . . . . . . . . 13 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
6766oveq1d 5884 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
6865, 67, 2ovmpog 6003 . . . . . . . . . . 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 2210 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7170oveq1d 5884 . . . . . . . 8 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
7263nncnd 8922 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
7372, 33sqmuld 10651 . . . . . . . 8 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7471, 73eqtrd 2210 . . . . . . 7 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7547, 54, 743eqtr4rd 2221 . . . . . 6 (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)))
76 df-ov 5872 . . . . . 6 (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩)
7775, 76eqtr2di 2227 . . . . 5 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = (𝐴↑2))
78 f1ocnvfv 5774 . . . . . 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 5515 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩))
82 op2ndg 6146 . . . 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 2210 . 2 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = ((2nd ‘(𝐹𝐴)) · 2))
8515, 84breqtrrd 4028 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 2148  {crab 2459  cop 3594   class class class wbr 4000   × cxp 4621  ccnv 4622  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  cmpo 5871  1st c1st 6133  2nd c2nd 6134   · cmul 7807  cn 8908  2c2 8959  0cn0 9165  cz 9242  cexp 10505  cdvds 11778  cprime 12090
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091
This theorem is referenced by:  sqne2sq  12160
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