Theorem gcdval 16404

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 631	. . 3 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑦 = 0)))
3		breq2 5095	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑛 ∥ 𝑥 ↔ 𝑛 ∥ 𝑀))
4	3	anbi1d 631	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)))
5	4	rabbidv 3402	. . . 4 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)})
6	5	supeq1d 9330	. . 3 ⊢ (𝑥 = 𝑀 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ))
7	2, 6	ifbieq2d 4502	. 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 630	. . 3 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0)))
10		breq2 5095	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑛 ∥ 𝑦 ↔ 𝑛 ∥ 𝑁))
11	10	anbi2d 630	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)))
12	11	rabbidv 3402	. . . 4 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
13	12	supeq1d 9330	. . 3 ⊢ (𝑦 = 𝑁 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
14	9, 13	ifbieq2d 4502	. 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 16403	. 2 ⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
16		c0ex 11103	. . 3 ⊢ 0 ∈ V
17		ltso 11190	. . . 4 ⊢ < Or ℝ
18	17	supex 9348	. . 3 ⊢ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∈ V
19	16, 18	ifex 4526	. 2 ⊢ if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) ∈ V
20	7, 14, 15, 19	ovmpo 7506	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 1541   ∈ wcel 2111  {crab 3395  ifcif 4475   class class class wbr 5091  (class class class)co 7346  supcsup 9324  ℝcr 11002  0cc0 11003   < clt 11143  ℤcz 12465   ∥ cdvds 16160   gcd cgcd 16402

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-mulcl 11065  ax-i2m1 11071  ax-pre-lttri 11077  ax-pre-lttrn 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-po 5524  df-so 5525  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-pnf 11145  df-mnf 11146  df-ltxr 11148  df-gcd 16403

This theorem is referenced by:  gcd0val  16405  gcdn0val  16406  gcdf  16420  gcdcom  16421  dfgcd2  16454  gcdass  16455