Theorem gcdval 16456

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 2741	. . . 4 ⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
2	1	anbi1d 632	. . 3 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑦 = 0)))
3		breq2 5090	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑛 ∥ 𝑥 ↔ 𝑛 ∥ 𝑀))
4	3	anbi1d 632	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)))
5	4	rabbidv 3397	. . . 4 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)})
6	5	supeq1d 9352	. . 3 ⊢ (𝑥 = 𝑀 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ))
7	2, 6	ifbieq2d 4494	. 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 2741	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
9	8	anbi2d 631	. . 3 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0)))
10		breq2 5090	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑛 ∥ 𝑦 ↔ 𝑛 ∥ 𝑁))
11	10	anbi2d 631	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)))
12	11	rabbidv 3397	. . . 4 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
13	12	supeq1d 9352	. . 3 ⊢ (𝑦 = 𝑁 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
14	9, 13	ifbieq2d 4494	. 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 16455	. 2 ⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
16		c0ex 11129	. . 3 ⊢ 0 ∈ V
17		ltso 11217	. . . 4 ⊢ < Or ℝ
18	17	supex 9370	. . 3 ⊢ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∈ V
19	16, 18	ifex 4518	. 2 ⊢ if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) ∈ V
20	7, 14, 15, 19	ovmpo 7520	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 1542   ∈ wcel 2114  {crab 3390  ifcif 4467   class class class wbr 5086  (class class class)co 7360  supcsup 9346  ℝcr 11028  0cc0 11029   < clt 11170  ℤcz 12515   ∥ cdvds 16212   gcd cgcd 16454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-mulcl 11091  ax-i2m1 11097  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-pnf 11172  df-mnf 11173  df-ltxr 11175  df-gcd 16455

This theorem is referenced by:  gcd0val  16457  gcdn0val  16458  gcdf  16472  gcdcom  16473  dfgcd2  16506  gcdass  16507