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Theorem sqpweven 11235
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 11234 . . . . . . 7 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5250 . . . . . . 7 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5237 . . . . . . 7 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . 6 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelrni 5417 . . . . 5 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 5919 . . . . 5 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 8836 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 8547 . . . . 5 2 ∈ ℕ
1211a1i 9 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 8837 . . 3 (𝐴 ∈ ℕ → 2 ∈ ℤ)
14 dvdsmul2 10901 . . 3 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1510, 13, 14syl2anc 403 . 2 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
16 xp1st 5918 . . . . . . . . . 10 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
177, 16syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
18 breq2 3841 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
1918notbid 627 . . . . . . . . . . 11 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2019, 1elrab2 2772 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 ↔ ((1st ‘(𝐹𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2120simplbi 268 . . . . . . . . 9 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → (1st ‘(𝐹𝐴)) ∈ ℕ)
2217, 21syl 14 . . . . . . . 8 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℕ)
2322nnsqcld 10072 . . . . . . 7 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℕ)
2420simprbi 269 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
2517, 24syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
26 2prm 11191 . . . . . . . . . 10 2 ∈ ℙ
2722nnzd 8837 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
28 euclemma 11207 . . . . . . . . . . 11 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ (2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴)))))
29 oridm 709 . . . . . . . . . . 11 ((2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴)))
3028, 29syl6bb 194 . . . . . . . . . 10 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3126, 27, 27, 30mp3an2i 1278 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3225, 31mtbird 633 . . . . . . . 8 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3322nncnd 8408 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3433sqvald 10048 . . . . . . . . 9 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3534breq2d 3849 . . . . . . . 8 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
3632, 35mtbird 633 . . . . . . 7 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
37 breq2 3841 . . . . . . . . 9 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3837notbid 627 . . . . . . . 8 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3938, 1elrab2 2772 . . . . . . 7 (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(𝐹𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4023, 36, 39sylanbrc 408 . . . . . 6 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ 𝐽)
4112nnnn0d 8696 . . . . . . 7 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
429, 41nn0mulcld 8701 . . . . . 6 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
43 opelxp 4457 . . . . . 6 (⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0))
4440, 42, 43sylanbrc 408 . . . . 5 (𝐴 ∈ ℕ → ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0))
4512nncnd 8408 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℂ)
4645, 41, 9expmuld 10054 . . . . . . . 8 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
4746oveq1d 5649 . . . . . . 7 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
4812, 42nnexpcld 10073 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℕ)
4948, 23nnmulcld 8442 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
50 oveq2 5642 . . . . . . . . 9 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
51 oveq2 5642 . . . . . . . . . 10 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(𝐹𝐴)) · 2)))
5251oveq1d 5649 . . . . . . . . 9 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
5350, 52, 2ovmpt2g 5761 . . . . . . . 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 1174 . . . . . . 7 (𝐴 ∈ ℕ → (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
55 f1ocnvfv2 5539 . . . . . . . . . . . . 13 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
563, 55mpan 415 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
57 1st2nd2 5927 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
587, 57syl 14 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
5958fveq2d 5293 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
6056, 59eqtr3d 2122 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
61 df-ov 5637 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6260, 61syl6eqr 2138 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
6312, 9nnexpcld 10073 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
6463, 22nnmulcld 8442 . . . . . . . . . . 11 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
65 oveq2 5642 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
66 oveq2 5642 . . . . . . . . . . . . 13 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
6766oveq1d 5649 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
6865, 67, 2ovmpt2g 5761 . . . . . . . . . . 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 1174 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7062, 69eqtrd 2120 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7170oveq1d 5649 . . . . . . . 8 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
7263nncnd 8408 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
7372, 33sqmuld 10063 . . . . . . . 8 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7471, 73eqtrd 2120 . . . . . . 7 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7547, 54, 743eqtr4rd 2131 . . . . . 6 (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)))
76 df-ov 5637 . . . . . 6 (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩)
7775, 76syl6req 2137 . . . . 5 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = (𝐴↑2))
78 f1ocnvfv 5540 . . . . . 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 415 . . . . 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 61 . . . 4 (𝐴 ∈ ℕ → (𝐹‘(𝐴↑2)) = ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩)
8180fveq2d 5293 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩))
82 op2ndg 5904 . . . 4 ((((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0) → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = ((2nd ‘(𝐹𝐴)) · 2))
8340, 42, 82syl2anc 403 . . 3 (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = ((2nd ‘(𝐹𝐴)) · 2))
8481, 83eqtrd 2120 . 2 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = ((2nd ‘(𝐹𝐴)) · 2))
8515, 84breqtrrd 3863 1 (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(𝐹‘(𝐴↑2))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 664  w3a 924   = wceq 1289  wcel 1438  {crab 2363  cop 3444   class class class wbr 3837   × cxp 4426  ccnv 4427  wf 4998  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  cmpt2 5636  1st c1st 5891  2nd c2nd 5892   · cmul 7334  cn 8394  2c2 8444  0cn0 8643  cz 8720  cexp 9919  cdvds 10878  cprime 11171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-1o 6163  df-2o 6164  df-er 6272  df-en 6438  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879  df-gcd 11021  df-prm 11172
This theorem is referenced by:  sqne2sq  11237
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