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Theorem mgmplusfreseq 44386
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 17858 . . . 4 (𝑀 ∈ Mgm → :(𝐵 × 𝐵)⟶𝐵)
4 frn 6497 . . . 4 ( :(𝐵 × 𝐵)⟶𝐵 → ran 𝐵)
5 ssel 3911 . . . . 5 (ran 𝐵 → (∅ ∈ ran → ∅ ∈ 𝐵))
65nelcon3d 3106 . . . 4 (ran 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ))
73, 4, 63syl 18 . . 3 (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ))
87imp 410 . 2 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran )
9 plusfreseq.2 . . 3 + = (+g𝑀)
101, 9, 2plusfreseq 44385 . 2 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
118, 10syl 17 1 ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wnel 3094  wss 3884  c0 4246   × cxp 5521  ran crn 5524  cres 5525  wf 6324  cfv 6328  Basecbs 16479  +gcplusg 16561  +𝑓cplusf 17845  Mgmcmgm 17846
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-plusf 17847  df-mgm 17848
This theorem is referenced by: (None)
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