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Theorem gcdcllem2 15430
Description: Lemma for gcdn0cl 15432, gcddvds 15433 and dvdslegcd 15434. (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 4789 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑛𝑥𝑛))
21ralbidv 3135 . . . . 5 (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
3 gcdcllem2.1 . . . . 5 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧𝑛}
42, 3elrab2 3518 . . . 4 (𝑥𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛))
5 breq2 4790 . . . . . 6 (𝑛 = 𝑀 → (𝑥𝑛𝑥𝑀))
6 breq2 4790 . . . . . 6 (𝑛 = 𝑁 → (𝑥𝑛𝑥𝑁))
75, 6ralprg 4371 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥𝑛 ↔ (𝑥𝑀𝑥𝑁)))
87anbi2d 614 . . . 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 4789 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑀𝑥𝑀))
11 breq1 4789 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑁𝑥𝑁))
1210, 11anbi12d 616 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑀𝑧𝑁) ↔ (𝑥𝑀𝑥𝑁)))
13 gcdcllem2.2 . . . 4 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧𝑀𝑧𝑁)}
1412, 13elrab2 3518 . . 3 (𝑥𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥𝑀𝑥𝑁)))
159, 14syl6rbbr 279 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥𝑅𝑥𝑆))
1615eqrdv 2769 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  {cpr 4318   class class class wbr 4786  cz 11579  cdvds 15189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787
This theorem is referenced by:  gcdcllem3  15431
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