Theorem gcdval 16555

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 2767	. . . 4 ⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
2	1	anbi1d 642	. . 3 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑦 = 0)))
3		breq2 5114	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑛 ∥ 𝑥 ↔ 𝑛 ∥ 𝑀))
4	3	anbi1d 642	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)))
5	4	rabbidv 3423	. . . 4 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)})
6	5	supeq1d 9407	. . 3 ⊢ (𝑥 = 𝑀 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ))
7	2, 6	ifbieq2d 4515	. 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 2767	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
9	8	anbi2d 641	. . 3 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∧ 𝑦 = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0)))
10		breq2 5114	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑛 ∥ 𝑦 ↔ 𝑛 ∥ 𝑁))
11	10	anbi2d 641	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦) ↔ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)))
12	11	rabbidv 3423	. . . 4 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
13	12	supeq1d 9407	. . 3 ⊢ (𝑦 = 𝑁 → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
14	9, 13	ifbieq2d 4515	. 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 16554	. 2 ⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
16		c0ex 11201	. . 3 ⊢ 0 ∈ V
17		ltso 11291	. . . 4 ⊢ < Or ℝ
18	17	supex 9425	. . 3 ⊢ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∈ V
19	16, 18	ifex 4539	. 2 ⊢ if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) ∈ V
20	7, 14, 15, 19	ovmpo 7572	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 400   = wceq 1570   ∈ wcel 2143  {crab 3416  ifcif 4488   class class class wbr 5110  (class class class)co 7412  supcsup 9401  ℝcr 11100  0cc0 11101   < clt 11244  ℤcz 12592   ∥ cdvds 16311   gcd cgcd 16553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-mulcl 11163  ax-i2m1 11169  ax-pre-lttri 11175  ax-pre-lttrn 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-sup 9403  df-pnf 11246  df-mnf 11247  df-ltxr 11249  df-gcd 16554

This theorem is referenced by:  gcd0val  16556  gcdn0val  16557  gcdf  16571  gcdcom  16572  dfgcd2  16605  gcdass  16606