Theorem gcdcllem2 15839

Description: Lemma for gcdn0cl 15841, gcddvds 15842 and dvdslegcd 15843. (Contributed by Paul Chapman, 21-Mar-2011.)

Hypotheses

Ref	Expression
gcdcllem2.1	⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛}
gcdcllem2.2	⊢ 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)}

Assertion

Ref	Expression
gcdcllem2	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)

Distinct variable groups:   𝑧,𝑛,𝑀   𝑛,𝑁,𝑧

Allowed substitution hints:   𝑅(𝑧,𝑛)   𝑆(𝑧,𝑛)

Proof of Theorem gcdcllem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5033	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑀 ↔ 𝑥 ∥ 𝑀))
2		breq1 5033	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑁 ↔ 𝑥 ∥ 𝑁))
3	1, 2	anbi12d 633	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁) ↔ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))
4		gcdcllem2.2	. . . 4 ⊢ 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)}
5	3, 4	elrab2 3631	. . 3 ⊢ (𝑥 ∈ 𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))
6		breq1 5033	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑛 ↔ 𝑥 ∥ 𝑛))
7	6	ralbidv 3162	. . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛))
8		gcdcllem2.1	. . . . 5 ⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛}
9	7, 8	elrab2 3631	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛))
10		breq2 5034	. . . . . 6 ⊢ (𝑛 = 𝑀 → (𝑥 ∥ 𝑛 ↔ 𝑥 ∥ 𝑀))
11		breq2 5034	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑥 ∥ 𝑛 ↔ 𝑥 ∥ 𝑁))
12	10, 11	ralprg 4592	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛 ↔ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))
13	12	anbi2d 631	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛) ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁))))
14	9, 13	syl5bb 286	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁))))
15	5, 14	bitr4id 293	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ 𝑆))
16	15	eqrdv 2796	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  {cpr 4527   class class class wbr 5030  ℤcz 11969   ∥ cdvds 15599

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-sbc 3721  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031

This theorem is referenced by:  gcdcllem3  15840