Theorem mgmplusfreseq 42798

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 17641	. . . 4 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
4		frn 6299	. . . 4 ⊢ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 → ran ⨣ ⊆ 𝐵)
5		ssel 3815	. . . . 5 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∈ ran ⨣ → ∅ ∈ 𝐵))
6	5	nelcon3d 3087	. . . 4 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ ))
7	3, 4, 6	3syl 18	. . 3 ⊢ (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ ))
8	7	imp 397	. 2 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran ⨣ )
9		plusfreseq.2	. . 3 ⊢ + = (+g‘𝑀)
10	1, 9, 2	plusfreseq 42797	. 2 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
11	8, 10	syl 17	1 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ∉ wnel 3075   ⊆ wss 3792  ∅c0 4141   × cxp 5355  ran crn 5358   ↾ cres 5359  ⟶wf 6133  ‘cfv 6137  Basecbs 16259  +_gcplusg 16342  +_𝑓cplusf 17629  Mgmcmgm 17630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-plusf 17631  df-mgm 17632

This theorem is referenced by: (None)