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Theorem gcdval 16529
Description: The value of the gcd operator. (𝑀 gcd 𝑁) is the greatest common divisor of 𝑀 and 𝑁. If 𝑀 and 𝑁 are both 0, the result is defined conventionally as 0. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
gcdval ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )))
Distinct variable groups:   𝑛,𝑀   𝑛,𝑁

Proof of Theorem gcdval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2738 . . . 4 (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
21anbi1d 631 . . 3 (𝑥 = 𝑀 → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑦 = 0)))
3 breq2 5151 . . . . . 6 (𝑥 = 𝑀 → (𝑛𝑥𝑛𝑀))
43anbi1d 631 . . . . 5 (𝑥 = 𝑀 → ((𝑛𝑥𝑛𝑦) ↔ (𝑛𝑀𝑛𝑦)))
54rabbidv 3440 . . . 4 (𝑥 = 𝑀 → {𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑦)})
65supeq1d 9483 . . 3 (𝑥 = 𝑀 → sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑦)}, ℝ, < ))
72, 6ifbieq2d 4556 . 2 (𝑥 = 𝑀 → if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < )) = if((𝑀 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑦)}, ℝ, < )))
8 eqeq1 2738 . . . 4 (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
98anbi2d 630 . . 3 (𝑦 = 𝑁 → ((𝑀 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0)))
10 breq2 5151 . . . . . 6 (𝑦 = 𝑁 → (𝑛𝑦𝑛𝑁))
1110anbi2d 630 . . . . 5 (𝑦 = 𝑁 → ((𝑛𝑀𝑛𝑦) ↔ (𝑛𝑀𝑛𝑁)))
1211rabbidv 3440 . . . 4 (𝑦 = 𝑁 → {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)})
1312supeq1d 9483 . . 3 (𝑦 = 𝑁 → sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < ))
149, 13ifbieq2d 4556 . 2 (𝑦 = 𝑁 → if((𝑀 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑦)}, ℝ, < )) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )))
15 df-gcd 16528 . 2 gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < )))
16 c0ex 11252 . . 3 0 ∈ V
17 ltso 11338 . . . 4 < Or ℝ
1817supex 9500 . . 3 sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < ) ∈ V
1916, 18ifex 4580 . 2 if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) ∈ V
207, 14, 15, 19ovmpo 7592 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  {crab 3432  ifcif 4530   class class class wbr 5147  (class class class)co 7430  supcsup 9477  cr 11151  0cc0 11152   < clt 11292  cz 12610  cdvds 16286   gcd cgcd 16527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-mulcl 11214  ax-i2m1 11220  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-ltxr 11297  df-gcd 16528
This theorem is referenced by:  gcd0val  16530  gcdn0val  16531  gcdf  16545  gcdcom  16546  dfgcd2  16579  gcdass  16580
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