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Theorem eucalgval 11642
Description: Euclid's Algorithm eucalg 11647 computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0.

The value of the step function 𝐸 for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Hypothesis
Ref Expression
eucalgval.1 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
Assertion
Ref Expression
eucalgval (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
Distinct variable group:   𝑥,𝑦,𝑋
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem eucalgval
StepHypRef Expression
1 df-ov 5743 . . 3 ((1st𝑋)𝐸(2nd𝑋)) = (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩)
2 xp1st 6029 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
3 xp2nd 6030 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
4 eucalgval.1 . . . . 5 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
54eucalgval2 11641 . . . 4 (((1st𝑋) ∈ ℕ0 ∧ (2nd𝑋) ∈ ℕ0) → ((1st𝑋)𝐸(2nd𝑋)) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
62, 3, 5syl2anc 406 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ((1st𝑋)𝐸(2nd𝑋)) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
71, 6syl5eqr 2162 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
8 1st2nd2 6039 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
98fveq2d 5391 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩))
108fveq2d 5391 . . . . 5 (𝑋 ∈ (ℕ0 × ℕ0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
11 df-ov 5743 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
1210, 11syl6eqr 2166 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
1312opeq2d 3680 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ⟨(2nd𝑋), ( mod ‘𝑋)⟩ = ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩)
148, 13ifeq12d 3459 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
157, 9, 143eqtr4d 2158 1 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  ifcif 3442  cop 3498   × cxp 4505  cfv 5091  (class class class)co 5740  cmpo 5742  1st c1st 6002  2nd c2nd 6003  0cc0 7584  0cn0 8931   mod cmo 10046
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-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  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
This theorem depends on definitions:  df-bi 116  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-id 4183  df-po 4186  df-iso 4187  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-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  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-n0 8932  df-z 9009  df-q 9364  df-rp 9394  df-fl 9994  df-mod 10047
This theorem is referenced by:  eucalginv  11644  eucalglt  11645
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