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Mirrors > Home > MPE Home > Th. List > mgmb1mgm1 | Structured version Visualization version GIF version |
Description: The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.) |
Ref | Expression |
---|---|
mgmb1mgm1.b | ⊢ 𝐵 = (Base‘𝑀) |
mgmb1mgm1.p | ⊢ + = (+g‘𝑀) |
Ref | Expression |
---|---|
mgmb1mgm1 | ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgmb1mgm1.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
2 | mgmb1mgm1.p | . . . . . 6 ⊢ + = (+g‘𝑀) | |
3 | eqid 2821 | . . . . . 6 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀) | |
4 | 1, 2, 3 | plusfeq 17854 | . . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝑀) = + ) |
5 | 1, 3 | mgmplusf 17856 | . . . . . 6 ⊢ (𝑀 ∈ Mgm → (+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵) |
6 | feq1 6490 | . . . . . 6 ⊢ ((+𝑓‘𝑀) = + → ((+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵 ↔ + :(𝐵 × 𝐵)⟶𝐵)) | |
7 | 5, 6 | syl5ib 246 | . . . . 5 ⊢ ((+𝑓‘𝑀) = + → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵)) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵)) |
9 | 8 | impcom 410 | . . 3 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵) |
10 | 9 | 3adant2 1127 | . 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵) |
11 | simp2 1133 | . 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵) | |
12 | intopsn 17858 | . 2 ⊢ (( + :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) | |
13 | 10, 11, 12 | syl2anc 586 | 1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {csn 4561 〈cop 4567 × cxp 5548 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 Basecbs 16477 +gcplusg 16559 +𝑓cplusf 17843 Mgmcmgm 17844 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-plusf 17845 df-mgm 17846 |
This theorem is referenced by: srg1zr 19273 |
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