Theorem mgmplusf 18575

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	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		eqid 2736	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	mgmcl 18568	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
4	3	3expb 1120	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
5	4	ralrimivva 3179	. 2 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
6		mgmplusf.2	. . . 4 ⊢ ⨣ = (+𝑓‘𝑀)
7	1, 2, 6	plusffval 18571	. . 3 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦))
8	7	fmpo 8012	. 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵 ↔ ⨣ :(𝐵 × 𝐵)⟶𝐵)
9	5, 8	sylib 218	1 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3051   × cxp 5622  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  +_𝑓cplusf 18562  Mgmcmgm 18563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-plusf 18564  df-mgm 18565

This theorem is referenced by:  mgmb1mgm1  18580  mndplusf  18677  mgmplusfreseq  48411