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Mirrors > Home > ILE Home > Th. List > modfsummodlem1 | GIF version |
Description: Lemma for modfsummod 11461. (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 3624 | . . 3 ⊢ 𝑧 ∈ {𝑧} | |
2 | elun2 3303 | . . 3 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧}) |
4 | rspcsbela 3116 | . 2 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) | |
5 | 3, 4 | mpan 424 | 1 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ∀wral 2455 ⦋csb 3057 ∪ cun 3127 {csn 3592 ℤcz 9251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 |
This theorem is referenced by: modfsummodlemstep 11460 |
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