Theorem gcdcllem2 16676

Description: Lemma for gcdn0cl 16678, gcddvds 16679 and dvdslegcd 16680. (Contributed by Paul Chapman, 21-Mar-2011.)

Hypotheses

Ref	Expression
gcdcllem2.1	⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛}
gcdcllem2.2	⊢ 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)}

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.

Step	Hyp	Ref	Expression
1		breq1 5105	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑀 ↔ 𝑥 ∥ 𝑀))
2		breq1 5105	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑁 ↔ 𝑥 ∥ 𝑁))
3	1, 2	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁) ↔ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))
4		gcdcllem2.2	. . . 4 ⊢ 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)}
5	3, 4	elrab2 3648	. . 3 ⊢ (𝑥 ∈ 𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))
6		breq1 5105	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑛 ↔ 𝑥 ∥ 𝑛))
7	6	ralbidv 3185	. . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛))
8		gcdcllem2.1	. . . . 5 ⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛}
9	7, 8	elrab2 3648	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛))
10		breq2 5106	. . . . . 6 ⊢ (𝑛 = 𝑀 → (𝑥 ∥ 𝑛 ↔ 𝑥 ∥ 𝑀))
11		breq2 5106	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑥 ∥ 𝑛 ↔ 𝑥 ∥ 𝑁))
12	10, 11	ralprg 4653	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛 ↔ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))
13	12	anbi2d 642	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛) ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁))))
14	9, 13	bitrid 286	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁))))
15	5, 14	bitr4id 293	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ 𝑆))
16	15	eqrdv 2758	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  {cpr 4586   class class class wbr 5102  ℤcz 12697   ∥ cdvds 16428

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103

This theorem is used by:  gcdcllem3  16677