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Theorem mgmplusfreseq 45962
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 18461 . . . 4 (𝑀 ∈ Mgm → :(𝐵 × 𝐵)⟶𝐵)
4 frn 6672 . . . 4 ( :(𝐵 × 𝐵)⟶𝐵 → ran 𝐵)
5 ssel 3935 . . . . 5 (ran 𝐵 → (∅ ∈ ran → ∅ ∈ 𝐵))
65nelcon3d 3059 . . . 4 (ran 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ))
73, 4, 63syl 18 . . 3 (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ))
87imp 407 . 2 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran )
9 plusfreseq.2 . . 3 + = (+g𝑀)
101, 9, 2plusfreseq 45961 . 2 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
118, 10syl 17 1 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wnel 3047  wss 3908  c0 4280   × cxp 5629  ran crn 5632  cres 5633  wf 6489  cfv 6493  Basecbs 17037  +gcplusg 17087  +𝑓cplusf 18448  Mgmcmgm 18449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-plusf 18450  df-mgm 18451
This theorem is referenced by: (None)
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