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Theorem gcdcllem2 15146
 Description: Lemma for gcdn0cl 15148, gcddvds 15149 and dvdslegcd 15150. (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 4616 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑛𝑥𝑛))
21ralbidv 2980 . . . . 5 (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
3 gcdcllem2.1 . . . . 5 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛}
42, 3elrab2 3348 . . . 4 (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
5 breq2 4617 . . . . . 6 (𝑛 = 𝑀 → (𝑥𝑛𝑥𝑀))
6 breq2 4617 . . . . . 6 (𝑛 = 𝑁 → (𝑥𝑛𝑥𝑁))
75, 6ralprg 4205 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛 ↔ (𝑥𝑀𝑥𝑁)))
87anbi2d 739 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛) ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
94, 8syl5bb 272 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
10 breq1 4616 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑀𝑥𝑀))
11 breq1 4616 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑁𝑥𝑁))
1210, 11anbi12d 746 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑀𝑧𝑁) ↔ (𝑥𝑀𝑥𝑁)))
13 gcdcllem2.2 . . . 4 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧𝑀𝑧𝑁)}
1412, 13elrab2 3348 . . 3 (𝑥𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁)))
159, 14syl6rbbr 279 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑅𝑥𝑆))
1615eqrdv 2619 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911  {cpr 4150   class class class wbr 4613  ℤcz 11321   ∥ cdvds 14907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614 This theorem is referenced by:  gcdcllem3  15147
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