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Mirrors > Home > ILE Home > Th. List > modfsummodlem1 | GIF version |
Description: Lemma for modfsummod 11234. (Contributed by Alexander van der Vekens, 1-Sep-2018.) |
Ref | Expression |
---|---|
modfsummodlem1 | ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vsnid 3557 | . . 3 ⊢ 𝑧 ∈ {𝑧} | |
2 | elun2 3244 | . . 3 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧}) |
4 | rspcsbela 3059 | . 2 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) | |
5 | 3, 4 | mpan 420 | 1 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∀wral 2416 ⦋csb 3003 ∪ cun 3069 {csn 3527 ℤcz 9061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 |
This theorem is referenced by: modfsummodlemstep 11233 |
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