Theorem plusfreseq 45366

Description: If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily 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
plusfreseq	⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Proof of Theorem plusfreseq

Dummy variables 𝑝 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plusfreseq.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		plusfreseq.3	. . . . 5 ⊢ ⨣ = (+𝑓‘𝑀)
3	1, 2	plusffn 18363	. . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵)
4		fnfun 6552	. . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → Fun ⨣ )
5	3, 4	ax-mp 5	. . 3 ⊢ Fun ⨣
6	5	a1i 11	. 2 ⊢ (∅ ∉ ran ⨣ → Fun ⨣ )
7		id 22	. 2 ⊢ (∅ ∉ ran ⨣ → ∅ ∉ ran ⨣ )
8		plusfreseq.2	. . . . . . 7 ⊢ + = (+g‘𝑀)
9	1, 8, 2	plusfval 18361	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦))
10	9	eqcomd 2739	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
11	10	rgen2 3188	. . . 4 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)
12	11	a1i 11	. . 3 ⊢ (∅ ∉ ran ⨣ → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
13		fveq2 6792	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14		df-ov 7298	. . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
15	13, 14	eqtr4di 2791	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = (𝑥 + 𝑦))
16		fveq2 6792	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
17		df-ov 7298	. . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
18	16, 17	eqtr4di 2791	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦))
19	15, 18	eqeq12d 2749	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)))
20	19	ralxp 5754	. . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
21	12, 20	sylibr 233	. 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝))
22		fndm 6555	. . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵))
23	22	eqcomd 2739	. . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ )
24	3, 23	ax-mp 5	. . 3 ⊢ (𝐵 × 𝐵) = dom ⨣
25	24	fveqressseq 6977	. 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
26	6, 7, 21, 25	syl3anc 1369	1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2101   ∉ wnel 3044  ∀wral 3059  ∅c0 4259  ⟨cop 4570   × cxp 5589  dom cdm 5591  ran crn 5592   ↾ cres 5593  Fun wfun 6441   Fn wfn 6442  ‘cfv 6447  (class class class)co 7295  Basecbs 16940  +_gcplusg 16990  +_𝑓cplusf 18351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-1st 7851  df-2nd 7852  df-plusf 18353

This theorem is referenced by:  mgmplusfreseq  45367