Theorem plusfreseq 48667

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 18612	. . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵)
4		fnfun 6588	. . . 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 18610	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦))
10	9	eqcomd 2747	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
11	10	rgen2 3181	. . . 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 6830	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14		df-ov 7362	. . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
15	13, 14	eqtr4di 2794	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = (𝑥 + 𝑦))
16		fveq2 6830	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
17		df-ov 7362	. . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
18	16, 17	eqtr4di 2794	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦))
19	15, 18	eqeq12d 2757	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)))
20	19	ralxp 5785	. . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
21	12, 20	sylibr 236	. 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝))
22		fndm 6591	. . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵))
23	22	eqcomd 2747	. . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ )
24	3, 23	ax-mp 5	. . 3 ⊢ (𝐵 × 𝐵) = dom ⨣
25	24	fveqressseq 7023	. 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
26	6, 7, 21, 25	syl3anc 1380	1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ∉ wnel 3040  ∀wral 3055  ∅c0 4263  ⟨cop 4563   × cxp 5618  dom cdm 5620  ran crn 5621   ↾ cres 5622  Fun wfun 6482   Fn wfn 6483  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  +_gcplusg 17215  +_𝑓cplusf 18600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-plusf 18602

This theorem is referenced by:  mgmplusfreseq  48668