MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gcdcllem2 Structured version   Visualization version   GIF version

Theorem gcdcllem2 16538
Description: Lemma for gcdn0cl 16540, gcddvds 16541 and dvdslegcd 16542. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
gcdcllem2.1 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛}
gcdcllem2.2 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧𝑀𝑧𝑁)}
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.
StepHypRef Expression
1 breq1 5145 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑀𝑥𝑀))
2 breq1 5145 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑁𝑥𝑁))
31, 2anbi12d 632 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑀𝑧𝑁) ↔ (𝑥𝑀𝑥𝑁)))
4 gcdcllem2.2 . . . 4 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧𝑀𝑧𝑁)}
53, 4elrab2 3694 . . 3 (𝑥𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁)))
6 breq1 5145 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑛𝑥𝑛))
76ralbidv 3177 . . . . 5 (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
8 gcdcllem2.1 . . . . 5 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛}
97, 8elrab2 3694 . . . 4 (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
10 breq2 5146 . . . . . 6 (𝑛 = 𝑀 → (𝑥𝑛𝑥𝑀))
11 breq2 5146 . . . . . 6 (𝑛 = 𝑁 → (𝑥𝑛𝑥𝑁))
1210, 11ralprg 4695 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛 ↔ (𝑥𝑀𝑥𝑁)))
1312anbi2d 630 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛) ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
149, 13bitrid 283 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
155, 14bitr4id 290 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑅𝑥𝑆))
1615eqrdv 2734 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  {cpr 4627   class class class wbr 5142  cz 12615  cdvds 16291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143
This theorem is referenced by:  gcdcllem3  16539
  Copyright terms: Public domain W3C validator