ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sqpweven GIF version

Theorem sqpweven 11759
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 11758 . . . . . . 7 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5346 . . . . . . 7 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5333 . . . . . . 7 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . 6 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelrni 5520 . . . . 5 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 6030 . . . . 5 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 9125 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 8835 . . . . 5 2 ∈ ℕ
1211a1i 9 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 9126 . . 3 (𝐴 ∈ ℕ → 2 ∈ ℤ)
14 dvdsmul2 11423 . . 3 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1510, 13, 14syl2anc 406 . 2 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
16 xp1st 6029 . . . . . . . . . 10 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
177, 16syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
18 breq2 3901 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
1918notbid 639 . . . . . . . . . . 11 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2019, 1elrab2 2814 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 ↔ ((1st ‘(𝐹𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2120simplbi 270 . . . . . . . . 9 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → (1st ‘(𝐹𝐴)) ∈ ℕ)
2217, 21syl 14 . . . . . . . 8 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℕ)
2322nnsqcld 10396 . . . . . . 7 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℕ)
2420simprbi 271 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
2517, 24syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
26 2prm 11715 . . . . . . . . . 10 2 ∈ ℙ
2722nnzd 9126 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
28 euclemma 11731 . . . . . . . . . . 11 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ (2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴)))))
29 oridm 729 . . . . . . . . . . 11 ((2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴)))
3028, 29syl6bb 195 . . . . . . . . . 10 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3126, 27, 27, 30mp3an2i 1303 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3225, 31mtbird 645 . . . . . . . 8 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3322nncnd 8694 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3433sqvald 10372 . . . . . . . . 9 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3534breq2d 3909 . . . . . . . 8 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
3632, 35mtbird 645 . . . . . . 7 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
37 breq2 3901 . . . . . . . . 9 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3837notbid 639 . . . . . . . 8 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
3938, 1elrab2 2814 . . . . . . 7 (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(𝐹𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4023, 36, 39sylanbrc 411 . . . . . 6 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ 𝐽)
4112nnnn0d 8984 . . . . . . 7 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
429, 41nn0mulcld 8989 . . . . . 6 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
43 opelxp 4537 . . . . . 6 (⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0))
4440, 42, 43sylanbrc 411 . . . . 5 (𝐴 ∈ ℕ → ⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0))
4512nncnd 8694 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℂ)
4645, 41, 9expmuld 10378 . . . . . . . 8 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
4746oveq1d 5755 . . . . . . 7 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
4812, 42nnexpcld 10397 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℕ)
4948, 23nnmulcld 8729 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
50 oveq2 5748 . . . . . . . . 9 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
51 oveq2 5748 . . . . . . . . . 10 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(𝐹𝐴)) · 2)))
5251oveq1d 5755 . . . . . . . . 9 (𝑦 = ((2nd ‘(𝐹𝐴)) · 2) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
5350, 52, 2ovmpog 5871 . . . . . . . 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 1199 . . . . . . 7 (𝐴 ∈ ℕ → (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · ((1st ‘(𝐹𝐴))↑2)))
55 f1ocnvfv2 5645 . . . . . . . . . . . . 13 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
563, 55mpan 418 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
57 1st2nd2 6039 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
587, 57syl 14 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
5958fveq2d 5391 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
6056, 59eqtr3d 2150 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
61 df-ov 5743 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6260, 61syl6eqr 2166 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
6312, 9nnexpcld 10397 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
6463, 22nnmulcld 8729 . . . . . . . . . . 11 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
65 oveq2 5748 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
66 oveq2 5748 . . . . . . . . . . . . 13 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
6766oveq1d 5755 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
6865, 67, 2ovmpog 5871 . . . . . . . . . . 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 1199 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7062, 69eqtrd 2148 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
7170oveq1d 5755 . . . . . . . 8 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
7263nncnd 8694 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
7372, 33sqmuld 10387 . . . . . . . 8 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7471, 73eqtrd 2148 . . . . . . 7 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
7547, 54, 743eqtr4rd 2159 . . . . . 6 (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)))
76 df-ov 5743 . . . . . 6 (((1st ‘(𝐹𝐴))↑2)𝐹((2nd ‘(𝐹𝐴)) · 2)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩)
7775, 76syl6req 2165 . . . . 5 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = (𝐴↑2))
78 f1ocnvfv 5646 . . . . . 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 418 . . . . 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 5391 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩))
82 op2ndg 6015 . . . 4 ((((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0) → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = ((2nd ‘(𝐹𝐴)) · 2))
8340, 42, 82syl2anc 406 . . 3 (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), ((2nd ‘(𝐹𝐴)) · 2)⟩) = ((2nd ‘(𝐹𝐴)) · 2))
8481, 83eqtrd 2148 . 2 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(𝐴↑2))) = ((2nd ‘(𝐹𝐴)) · 2))
8515, 84breqtrrd 3924 1 (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(𝐹‘(𝐴↑2))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 680  w3a 945   = wceq 1314  wcel 1463  {crab 2395  cop 3498   class class class wbr 3897   × cxp 4505  ccnv 4506  wf 5087  1-1-ontowf1o 5090  cfv 5091  (class class class)co 5740  cmpo 5742  1st c1st 6002  2nd c2nd 6003   · cmul 7589  cn 8680  2c2 8731  0cn0 8931  cz 9008  cexp 10243  cdvds 11400  cprime 11695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-1o 6279  df-2o 6280  df-er 6395  df-en 6601  df-sup 6837  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-fl 9994  df-mod 10047  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-dvds 11401  df-gcd 11543  df-prm 11696
This theorem is referenced by:  sqne2sq  11761
  Copyright terms: Public domain W3C validator