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Mirrors > Home > ILE Home > Th. List > mgmplusf | GIF version |
Description: The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.) |
Ref | Expression |
---|---|
mgmplusf.1 | ⊢ 𝐵 = (Base‘𝑀) |
mgmplusf.2 | ⊢ ⨣ = (+𝑓‘𝑀) |
Ref | Expression |
---|---|
mgmplusf | ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgmplusf.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
2 | eqid 2187 | . . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | mgmcl 12796 | . . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵) |
4 | 3 | 3expb 1205 | . . . 4 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵) |
5 | 4 | ralrimivva 2569 | . . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵) |
6 | eqid 2187 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)) | |
7 | 6 | fmpo 6215 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)):(𝐵 × 𝐵)⟶𝐵) |
8 | 5, 7 | sylib 122 | . 2 ⊢ (𝑀 ∈ Mgm → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)):(𝐵 × 𝐵)⟶𝐵) |
9 | mgmplusf.2 | . . . 4 ⊢ ⨣ = (+𝑓‘𝑀) | |
10 | 1, 2, 9 | plusffvalg 12799 | . . 3 ⊢ (𝑀 ∈ Mgm → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦))) |
11 | 10 | feq1d 5364 | . 2 ⊢ (𝑀 ∈ Mgm → ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)):(𝐵 × 𝐵)⟶𝐵)) |
12 | 8, 11 | mpbird 167 | 1 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 ∀wral 2465 × cxp 4636 ⟶wf 5224 ‘cfv 5228 (class class class)co 5888 ∈ cmpo 5890 Basecbs 12475 +gcplusg 12550 +𝑓cplusf 12790 Mgmcmgm 12791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-inn 8933 df-2 8991 df-ndx 12478 df-slot 12479 df-base 12481 df-plusg 12563 df-plusf 12792 df-mgm 12793 |
This theorem is referenced by: mgmb1mgm1 12805 mndplusf 12853 |
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