Theorem mgmplusfreseq 48289

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	⊢ + = (+g‘𝑀)
plusfreseq.3	⊢ ⨣ = (+𝑓‘𝑀)

Assertion

Ref	Expression
mgmplusfreseq	⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Proof of Theorem mgmplusfreseq

Step	Hyp	Ref	Expression
1		plusfreseq.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		plusfreseq.3	. . . . 5 ⊢ ⨣ = (+𝑓‘𝑀)
3	1, 2	mgmplusf 18560	. . . 4 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
4		frn 6663	. . . 4 ⊢ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 → ran ⨣ ⊆ 𝐵)
5		ssel 3924	. . . . 5 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∈ ran ⨣ → ∅ ∈ 𝐵))
6	5	nelcon3d 3037	. . . 4 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ ))
7	3, 4, 6	3syl 18	. . 3 ⊢ (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ ))
8	7	imp 406	. 2 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran ⨣ )
9		plusfreseq.2	. . 3 ⊢ + = (+g‘𝑀)
10	1, 9, 2	plusfreseq 48288	. 2 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
11	8, 10	syl 17	1 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∉ wnel 3033   ⊆ wss 3898  ∅c0 4282   × cxp 5617  ran crn 5620   ↾ cres 5621  ⟶wf 6482  ‘cfv 6486  Basecbs 17122  +_gcplusg 17163  +_𝑓cplusf 18547  Mgmcmgm 18548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-plusf 18549  df-mgm 18550

This theorem is referenced by: (None)