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Mirrors > Home > MPE Home > Th. List > gcdcllem2 | Structured version Visualization version GIF version |
Description: Lemma for gcdn0cl 15559, gcddvds 15560 and dvdslegcd 15561. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcdcllem2.1 | ⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛} |
gcdcllem2.2 | ⊢ 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)} |
Ref | Expression |
---|---|
gcdcllem2 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4846 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑛 ↔ 𝑥 ∥ 𝑛)) | |
2 | 1 | ralbidv 3167 | . . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛 ↔ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛)) |
3 | gcdcllem2.1 | . . . . 5 ⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ {𝑀, 𝑁}𝑧 ∥ 𝑛} | |
4 | 2, 3 | elrab2 3560 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛)) |
5 | breq2 4847 | . . . . . 6 ⊢ (𝑛 = 𝑀 → (𝑥 ∥ 𝑛 ↔ 𝑥 ∥ 𝑀)) | |
6 | breq2 4847 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑥 ∥ 𝑛 ↔ 𝑥 ∥ 𝑁)) | |
7 | 5, 6 | ralprg 4424 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛 ↔ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁))) |
8 | 7 | anbi2d 623 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 ∈ ℤ ∧ ∀𝑛 ∈ {𝑀, 𝑁}𝑥 ∥ 𝑛) ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))) |
9 | 4, 8 | syl5bb 275 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁)))) |
10 | breq1 4846 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑀 ↔ 𝑥 ∥ 𝑀)) | |
11 | breq1 4846 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∥ 𝑁 ↔ 𝑥 ∥ 𝑁)) | |
12 | 10, 11 | anbi12d 625 | . . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁) ↔ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁))) |
13 | gcdcllem2.2 | . . . 4 ⊢ 𝑅 = {𝑧 ∈ ℤ ∣ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)} | |
14 | 12, 13 | elrab2 3560 | . . 3 ⊢ (𝑥 ∈ 𝑅 ↔ (𝑥 ∈ ℤ ∧ (𝑥 ∥ 𝑀 ∧ 𝑥 ∥ 𝑁))) |
15 | 9, 14 | syl6rbbr 282 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ 𝑆)) |
16 | 15 | eqrdv 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 ℤcz 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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