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Mirrors > Home > MPE Home > Th. List > modfsummodslem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for modfsummods 15837. (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 4685 | . . 3 ⊢ 𝑧 ∈ {𝑧} | |
2 | elun2 4200 | . . 3 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧}) |
4 | rspcsbela 4457 | . 2 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) | |
5 | 3, 4 | mpan 689 | 1 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 ∀wral 3063 ⦋csb 3915 ∪ cun 3968 {csn 4648 ℤcz 12635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-sn 4649 |
This theorem is referenced by: modfsummods 15837 |
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