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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 18631 | . . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
4 | 3 | 3expib 1119 | . . . 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 3040 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ⚬ 𝑌) ∉ 𝐵 → 𝑀 ∉ Mgm)) |
7 | 6 | 3impia 1114 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∉ wnel 3036 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 +gcplusg 17261 Mgmcmgm 18626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5303 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-nel 3037 df-ral 3052 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-iota 6498 df-fv 6554 df-ov 7419 df-mgm 18628 |
This theorem is referenced by: oddinmgm 47588 |
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