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Theorem mgmplusf 18676
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 = (+𝑓𝑀)
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.
StepHypRef Expression
1 mgmplusf.1 . . . . 5 𝐵 = (Base‘𝑀)
2 eqid 2735 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2mgmcl 18669 . . . 4 ((𝑀 ∈ Mgm ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
433expb 1119 . . 3 ((𝑀 ∈ Mgm ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
54ralrimivva 3200 . 2 (𝑀 ∈ Mgm → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) ∈ 𝐵)
6 mgmplusf.2 . . . 4 = (+𝑓𝑀)
71, 2, 6plusffval 18672 . . 3 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝑀)𝑦))
87fmpo 8092 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) ∈ 𝐵 :(𝐵 × 𝐵)⟶𝐵)
95, 8sylib 218 1 (𝑀 ∈ Mgm → :(𝐵 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  +𝑓cplusf 18663  Mgmcmgm 18664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-plusf 18665  df-mgm 18666
This theorem is referenced by:  mgmb1mgm1  18681  mndplusf  18778  mgmplusfreseq  48009
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