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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 2778 | . . . . . 6 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀) | |
4 | 1, 2, 3 | plusfeq 17646 | . . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝑀) = + ) |
5 | 1, 3 | mgmplusf 17648 | . . . . . 6 ⊢ (𝑀 ∈ Mgm → (+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵) |
6 | feq1 6274 | . . . . . 6 ⊢ ((+𝑓‘𝑀) = + → ((+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵 ↔ + :(𝐵 × 𝐵)⟶𝐵)) | |
7 | 5, 6 | syl5ib 236 | . . . . 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 398 | . . 3 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵) |
10 | 9 | 3adant2 1122 | . 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵) |
11 | simp2 1128 | . 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵) | |
12 | intopsn 17650 | . 2 ⊢ (( + :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) | |
13 | 10, 11, 12 | syl2anc 579 | 1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 {csn 4398 〈cop 4404 × cxp 5355 Fn wfn 6132 ⟶wf 6133 ‘cfv 6137 Basecbs 16266 +gcplusg 16349 +𝑓cplusf 17636 Mgmcmgm 17637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-plusf 17638 df-mgm 17639 |
This theorem is referenced by: srg1zr 18927 |
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