Theorem plusfreseq 48152

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 18576	. . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵)
4		fnfun 6618	. . . 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 18574	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦))
10	9	eqcomd 2735	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
11	10	rgen2 3177	. . . 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 6858	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14		df-ov 7390	. . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
15	13, 14	eqtr4di 2782	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = (𝑥 + 𝑦))
16		fveq2 6858	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
17		df-ov 7390	. . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
18	16, 17	eqtr4di 2782	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦))
19	15, 18	eqeq12d 2745	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)))
20	19	ralxp 5805	. . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
21	12, 20	sylibr 234	. 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝))
22		fndm 6621	. . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵))
23	22	eqcomd 2735	. . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ )
24	3, 23	ax-mp 5	. . 3 ⊢ (𝐵 × 𝐵) = dom ⨣
25	24	fveqressseq 7051	. 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
26	6, 7, 21, 25	syl3anc 1373	1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  ∀wral 3044  ∅c0 4296  ⟨cop 4595   × cxp 5636  dom cdm 5638  ran crn 5639   ↾ cres 5640  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  +_𝑓cplusf 18564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-plusf 18566

This theorem is referenced by:  mgmplusfreseq  48153