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Mirrors > Home > MPE Home > Th. List > modfsummodslem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for modfsummods 14366. (Contributed by Alexander van der Vekens, 1-Sep-2018.) |
Ref | Expression |
---|---|
modfsummodslem1 | ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vsnid 4156 | . . 3 ⊢ 𝑧 ∈ {𝑧} | |
2 | elun2 3743 | . . 3 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧}) |
4 | rspcsbela 3958 | . 2 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) | |
5 | 3, 4 | mpan 702 | 1 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∀wral 2896 ⦋csb 3499 ∪ cun 3538 {csn 4125 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 |
This theorem is referenced by: modfsummods 14366 |
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