Theorem mgmplusf 13439

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.)

Hypotheses

Ref	Expression
mgmplusf.1	⊢ 𝐵 = (Base‘𝑀)
mgmplusf.2	⊢ ⨣ = (+𝑓‘𝑀)

Assertion

Ref	Expression
mgmplusf	⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)

Proof of Theorem mgmplusf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgmplusf.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
2		eqid 2229	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	mgmcl 13432	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
4	3	3expb 1228	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
5	4	ralrimivva 2612	. . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
6		eqid 2229	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦))
7	6	fmpo 6361	. . 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 13435	. . 3 ⊢ (𝑀 ∈ Mgm → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)))
11	10	feq1d 5466	. 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)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  ∀wral 2508   × cxp 4721  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013   ∈ cmpo 6015  Basecbs 13072  +_gcplusg 13150  +_𝑓cplusf 13426  Mgmcmgm 13427

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-plusf 13428  df-mgm 13429

This theorem is referenced by:  mgmb1mgm1  13441  mndplusf  13506