Theorem mgmplusf 18741

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 2760	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	mgmcl 18734	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
4	3	3expb 1138	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
5	4	ralrimivva 3205	. 2 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
6		mgmplusf.2	. . . 4 ⊢ ⨣ = (+𝑓‘𝑀)
7	1, 2, 6	plusffval 18737	. . 3 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝑀)𝑦))
8	7	fmpo 8066	. 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) ∈ 𝐵 ↔ ⨣ :(𝐵 × 𝐵)⟶𝐵)
9	5, 8	sylib 221	1 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076   × cxp 5653  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  +_gcplusg 17343  +_𝑓cplusf 18728  Mgmcmgm 18729

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-plusf 18730  df-mgm 18731

This theorem is used by:  mgmn0plusgplusf  18743  mgmb1mgm1  18748  mgmidpfod  18771  mndplusf  18856  mgmplusfreseq  49081