Theorem gcdval 16442

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 ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, 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.

Step	Hyp	Ref	Expression
1		eqeq1 2735	. . . 4 ⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
2	1	anbi1d 629	. . 3 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑦 = 0)))
3		breq2 5152	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑛 ∥ 𝑥 ↔ 𝑛 ∥ 𝑀))
4	3	anbi1d 629	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)))
5	4	rabbidv 3439	. . . 4 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)})
6	5	supeq1d 9444	. . 3 ⊢ (𝑥 = 𝑀 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ))
7	2, 6	ifbieq2d 4554	. 2 ⊢ (𝑥 = 𝑀 → if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )) = if((𝑀 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
8		eqeq1 2735	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
9	8	anbi2d 628	. . 3 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0)))
10		breq2 5152	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑛 ∥ 𝑦 ↔ 𝑛 ∥ 𝑁))
11	10	anbi2d 628	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)))
12	11	rabbidv 3439	. . . 4 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
13	12	supeq1d 9444	. . 3 ⊢ (𝑦 = 𝑁 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
14	9, 13	ifbieq2d 4554	. 2 ⊢ (𝑦 = 𝑁 → if((𝑀 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )))
15		df-gcd 16441	. 2 ⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
16		c0ex 11213	. . 3 ⊢ 0 ∈ V
17		ltso 11299	. . . 4 ⊢ < Or ℝ
18	17	supex 9461	. . 3 ⊢ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∈ V
19	16, 18	ifex 4578	. 2 ⊢ if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) ∈ V
20	7, 14, 15, 19	ovmpo 7571	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {crab 3431  ifcif 4528   class class class wbr 5148  (class class class)co 7412  supcsup 9438  ℝcr 11112  0cc0 11113   < clt 11253  ℤcz 12563   ∥ cdvds 16202   gcd cgcd 16440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-mulcl 11175  ax-i2m1 11181  ax-pre-lttri 11187  ax-pre-lttrn 11188

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9440  df-pnf 11255  df-mnf 11256  df-ltxr 11258  df-gcd 16441

This theorem is referenced by:  gcd0val  16443  gcdn0val  16444  gcdf  16458  gcdcom  16459  dfgcd2  16493  gcdass  16494