Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgmplusfreseq Structured version   Visualization version   GIF version

Theorem mgmplusfreseq 47140
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 18595 . . . 4 (𝑀 ∈ Mgm → :(𝐵 × 𝐵)⟶𝐵)
4 frn 6723 . . . 4 ( :(𝐵 × 𝐵)⟶𝐵 → ran 𝐵)
5 ssel 3971 . . . . 5 (ran 𝐵 → (∅ ∈ ran → ∅ ∈ 𝐵))
65nelcon3d 3045 . . . 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 47139 . 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 1534  wcel 2099  wnel 3041  wss 3944  c0 4318   × cxp 5670  ran crn 5673  cres 5674  wf 6538  cfv 6542  Basecbs 17165  +gcplusg 17218  +𝑓cplusf 18582  Mgmcmgm 18583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-plusf 18584  df-mgm 18585
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator