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Theorem mgmplusf 18663
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 2737 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2mgmcl 18656 . . . 4 ((𝑀 ∈ Mgm ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
433expb 1121 . . 3 ((𝑀 ∈ Mgm ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
54ralrimivva 3202 . 2 (𝑀 ∈ Mgm → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) ∈ 𝐵)
6 mgmplusf.2 . . . 4 = (+𝑓𝑀)
71, 2, 6plusffval 18659 . . 3 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝑀)𝑦))
87fmpo 8093 . 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 1540  wcel 2108  wral 3061   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  +𝑓cplusf 18650  Mgmcmgm 18651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-plusf 18652  df-mgm 18653
This theorem is referenced by:  mgmb1mgm1  18668  mndplusf  18765  mgmplusfreseq  48081
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