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Theorem mgmplusfreseq 46153
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 18512 . . . 4 (𝑀 ∈ Mgm → :(𝐵 × 𝐵)⟶𝐵)
4 frn 6676 . . . 4 ( :(𝐵 × 𝐵)⟶𝐵 → ran 𝐵)
5 ssel 3938 . . . . 5 (ran 𝐵 → (∅ ∈ ran → ∅ ∈ 𝐵))
65nelcon3d 3058 . . . 4 (ran 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ))
73, 4, 63syl 18 . . 3 (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ))
87imp 408 . 2 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran )
9 plusfreseq.2 . . 3 + = (+g𝑀)
101, 9, 2plusfreseq 46152 . 2 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
118, 10syl 17 1 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wnel 3046  wss 3911  c0 4283   × cxp 5632  ran crn 5635  cres 5636  wf 6493  cfv 6497  Basecbs 17088  +gcplusg 17138  +𝑓cplusf 18499  Mgmcmgm 18500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-plusf 18501  df-mgm 18502
This theorem is referenced by: (None)
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