Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isnmgm | Structured version Visualization version GIF version |
Description: A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.) |
Ref | Expression |
---|---|
mgmcl.b | ⊢ 𝐵 = (Base‘𝑀) |
mgmcl.o | ⊢ ⚬ = (+g‘𝑀) |
Ref | Expression |
---|---|
isnmgm | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgmcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
2 | mgmcl.o | . . . . . 6 ⊢ ⚬ = (+g‘𝑀) | |
3 | 1, 2 | mgmcl 17854 | . . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
4 | 3 | 3expib 1118 | . . . 4 ⊢ (𝑀 ∈ Mgm → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)) |
5 | 4 | com12 32 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀 ∈ Mgm → (𝑋 ⚬ 𝑌) ∈ 𝐵)) |
6 | 5 | nelcon3d 3135 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ⚬ 𝑌) ∉ 𝐵 → 𝑀 ∉ Mgm)) |
7 | 6 | 3impia 1113 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Mgmcmgm 17849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5209 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-mgm 17851 |
This theorem is referenced by: oddinmgm 44081 |
Copyright terms: Public domain | W3C validator |