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Theorem gcdcllem2 15557
Description: Lemma for gcdn0cl 15559, gcddvds 15560 and dvdslegcd 15561. (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 4846 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑛𝑥𝑛))
21ralbidv 3167 . . . . 5 (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
3 gcdcllem2.1 . . . . 5 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛}
42, 3elrab2 3560 . . . 4 (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
5 breq2 4847 . . . . . 6 (𝑛 = 𝑀 → (𝑥𝑛𝑥𝑀))
6 breq2 4847 . . . . . 6 (𝑛 = 𝑁 → (𝑥𝑛𝑥𝑁))
75, 6ralprg 4424 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛 ↔ (𝑥𝑀𝑥𝑁)))
87anbi2d 623 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛) ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
94, 8syl5bb 275 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
10 breq1 4846 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑀𝑥𝑀))
11 breq1 4846 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑁𝑥𝑁))
1210, 11anbi12d 625 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑀𝑧𝑁) ↔ (𝑥𝑀𝑥𝑁)))
13 gcdcllem2.2 . . . 4 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧𝑀𝑧𝑁)}
1412, 13elrab2 3560 . . 3 (𝑥𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁)))
159, 14syl6rbbr 282 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑅𝑥𝑆))
1615eqrdv 2797 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  {crab 3093  {cpr 4370   class class class wbr 4843  cz 11666  cdvds 15319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844
This theorem is referenced by:  gcdcllem3  15558
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