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Theorem mgmplusfreseq 48054
Description: If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
Hypotheses
Ref Expression
plusfreseq.1 𝐵 = (Base‘𝑀)
plusfreseq.2 + = (+g𝑀)
plusfreseq.3 = (+𝑓𝑀)
Assertion
Ref Expression
mgmplusfreseq ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = )

Proof of Theorem mgmplusfreseq
StepHypRef Expression
1 plusfreseq.1 . . . . 5 𝐵 = (Base‘𝑀)
2 plusfreseq.3 . . . . 5 = (+𝑓𝑀)
31, 2mgmplusf 18659 . . . 4 (𝑀 ∈ Mgm → :(𝐵 × 𝐵)⟶𝐵)
4 frn 6741 . . . 4 ( :(𝐵 × 𝐵)⟶𝐵 → ran 𝐵)
5 ssel 3976 . . . . 5 (ran 𝐵 → (∅ ∈ ran → ∅ ∈ 𝐵))
65nelcon3d 3049 . . . 4 (ran 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ))
73, 4, 63syl 18 . . 3 (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ))
87imp 406 . 2 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran )
9 plusfreseq.2 . . 3 + = (+g𝑀)
101, 9, 2plusfreseq 48053 . 2 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
118, 10syl 17 1 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wnel 3045  wss 3950  c0 4332   × cxp 5681  ran crn 5684  cres 5685  wf 6555  cfv 6559  Basecbs 17243  +gcplusg 17293  +𝑓cplusf 18646  Mgmcmgm 18647
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 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-plusf 18648  df-mgm 18649
This theorem is referenced by: (None)
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