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Theorem gcdcllem2 15837
Description: Lemma for gcdn0cl 15839, gcddvds 15840 and dvdslegcd 15841. (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 5060 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑛𝑥𝑛))
21ralbidv 3194 . . . . 5 (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
3 gcdcllem2.1 . . . . 5 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛}
42, 3elrab2 3680 . . . 4 (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
5 breq2 5061 . . . . . 6 (𝑛 = 𝑀 → (𝑥𝑛𝑥𝑀))
6 breq2 5061 . . . . . 6 (𝑛 = 𝑁 → (𝑥𝑛𝑥𝑁))
75, 6ralprg 4624 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛 ↔ (𝑥𝑀𝑥𝑁)))
87anbi2d 628 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛) ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
94, 8syl5bb 284 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁))))
10 breq1 5060 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑀𝑥𝑀))
11 breq1 5060 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑁𝑥𝑁))
1210, 11anbi12d 630 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑀𝑧𝑁) ↔ (𝑥𝑀𝑥𝑁)))
13 gcdcllem2.2 . . . 4 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧𝑀𝑧𝑁)}
1412, 13elrab2 3680 . . 3 (𝑥𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁)))
159, 14syl6rbbr 291 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑅𝑥𝑆))
1615eqrdv 2816 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  {cpr 4559   class class class wbr 5057  cz 11969  cdvds 15595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058
This theorem is referenced by:  gcdcllem3  15838
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