Theorem mgmplusf 12597

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 2165	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	mgmcl 12590	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
4	3	3expb 1194	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
5	4	ralrimivva 2548	. . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
6		eqid 2165	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦))
7	6	fmpo 6169	. . 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 121	. 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 12593	. . 3 ⊢ (𝑀 ∈ Mgm → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)))
11	10	feq1d 5324	. 2 ⊢ (𝑀 ∈ Mgm → ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦)):(𝐵 × 𝐵)⟶𝐵))
12	8, 11	mpbird 166	1 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  ∀wral 2444   × cxp 4602  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  Basecbs 12394  +_gcplusg 12457  +_𝑓cplusf 12584  Mgmcmgm 12585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-inn 8858  df-2 8916  df-ndx 12397  df-slot 12398  df-base 12400  df-plusg 12470  df-plusf 12586  df-mgm 12587

This theorem is referenced by:  mgmb1mgm1  12599