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Theorem sqpweven 10760
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 10759 . . . . . . 7 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5190 . . . . . . 7 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5177 . . . . . . 7 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . 6 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelrni 5353 . . . . 5 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 5844 . . . . 5 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 8600 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 8312 . . . . 5 2 ∈ ℕ
1211a1i 9 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 8601 . . 3 (𝐴 ∈ ℕ → 2 ∈ ℤ)
14 dvdsmul2 10426 . . 3 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1510, 13, 14syl2anc 403 . 2 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
16 xp1st 5843 . . . . . . . . . 10 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
177, 16syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
18 breq2 3809 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
1918notbid 625 . . . . . . . . . . 11 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2019, 1elrab2 2760 . . . . . . . . . 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 9775 . . . . . . 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 10716 . . . . . . . . . 10 2 ∈ ℙ
2722nnzd 8601 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
28 euclemma 10732 . . . . . . . . . . 11 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ (2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴)))))
29 oridm 707 . . . . . . . . . . 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 1274 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3225, 31mtbird 631 . . . . . . . 8 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3322nncnd 8172 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3433sqvald 9751 . . . . . . . . 9 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3534breq2d 3817 . . . . . . . 8 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
3632, 35mtbird 631 . . . . . . 7 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
37 breq2 3809 . . . . . . . . 9 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3837notbid 625 . . . . . . . 8 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3938, 1elrab2 2760 . . . . . . 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 8460 . . . . . . 7 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
429, 41nn0mulcld 8465 . . . . . 6 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
43 opelxp 4420 . . . . . 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 8172 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℂ)
4645, 41, 9expmuld 9757 . . . . . . . 8 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
4746oveq1d 5578 . . . . . . 7 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
4812, 42nnexpcld 9776 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℕ)
4948, 23nnmulcld 8206 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
50 oveq2 5571 . . . . . . . . 9 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
51 oveq2 5571 . . . . . . . . . 10 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(𝐹𝐴)) · 2)))
5251oveq1d 5578 . . . . . . . . 9 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
5350, 52, 2ovmpt2g 5686 . . . . . . . 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 1170 . . . . . . 7 (𝐴 ∈ ℕ → (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
55 f1ocnvfv2 5469 . . . . . . . . . . . . 13 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
563, 55mpan 415 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
57 1st2nd2 5852 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
587, 57syl 14 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
5958fveq2d 5233 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
6056, 59eqtr3d 2117 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
61 df-ov 5566 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6260, 61syl6eqr 2133 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
6312, 9nnexpcld 9776 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
6463, 22nnmulcld 8206 . . . . . . . . . . 11 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
65 oveq2 5571 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
66 oveq2 5571 . . . . . . . . . . . . 13 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
6766oveq1d 5578 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
6865, 67, 2ovmpt2g 5686 . . . . . . . . . . 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 1170 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7062, 69eqtrd 2115 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7170oveq1d 5578 . . . . . . . 8 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
7263nncnd 8172 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
7372, 33sqmuld 9766 . . . . . . . 8 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7471, 73eqtrd 2115 . . . . . . 7 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7547, 54, 743eqtr4rd 2126 . . . . . 6 (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)))
76 df-ov 5566 . . . . . 6 (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩)
7775, 76syl6req 2132 . . . . 5 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = (𝐴↑2))
78 f1ocnvfv 5470 . . . . . 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 5233 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩))
82 op2ndg 5829 . . . 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 2115 . 2 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = ((2nd ‘(𝐹𝐴)) · 2))
8515, 84breqtrrd 3831 1 (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(𝐹‘(𝐴↑2))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  {crab 2357  cop 3419   class class class wbr 3805   × cxp 4389  ccnv 4390  wf 4948  1-1-ontowf1o 4951  cfv 4952  (class class class)co 5563  cmpt2 5565  1st c1st 5816  2nd c2nd 5817   · cmul 7100  cn 8158  2c2 8208  0cn0 8407  cz 8484  cexp 9624  cdvds 10403  cprime 10696
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-1o 6085  df-2o 6086  df-er 6193  df-en 6309  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546  df-prm 10697
This theorem is referenced by:  sqne2sq  10762
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