Theorem mgmplusfreseq 49369

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 18833	. . . 4 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
4		frn 6720	. . . 4 ⊢ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 → ran ⨣ ⊆ 𝐵)
5		ssel 3903	. . . . 5 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∈ ran ⨣ → ∅ ∈ 𝐵))
6	5	nelcon3d 3065	. . . 4 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ ))
7	3, 4, 6	3syl 19	. . 3 ⊢ (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ ))
8	7	imp 412	. 2 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran ⨣ )
9		plusfreseq.2	. . 3 ⊢ + = (+g‘𝑀)
10	1, 9, 2	plusfreseq 49368	. 2 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
11	8, 10	syl 18	1 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∉ wnel 3061   ⊆ wss 3896  ∅c0 4277   × cxp 5648  ran crn 5651   ↾ cres 5652  ⟶wf 6537  ‘cfv 6541  Basecbs 17394  +_gcplusg 17435  +_𝑓cplusf 18820  Mgmcmgm 18821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-plusf 18822  df-mgm 18823

This theorem is used by: (None)