Theorem plusfreseq 48087

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 18625	. . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵)
4		fnfun 6637	. . . 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 18623	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦))
10	9	eqcomd 2741	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
11	10	rgen2 3184	. . . 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 6875	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14		df-ov 7406	. . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
15	13, 14	eqtr4di 2788	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = (𝑥 + 𝑦))
16		fveq2 6875	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
17		df-ov 7406	. . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
18	16, 17	eqtr4di 2788	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦))
19	15, 18	eqeq12d 2751	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)))
20	19	ralxp 5821	. . 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 6640	. . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵))
23	22	eqcomd 2741	. . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ )
24	3, 23	ax-mp 5	. . 3 ⊢ (𝐵 × 𝐵) = dom ⨣
25	24	fveqressseq 7068	. 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 2108   ∉ wnel 3036  ∀wral 3051  ∅c0 4308  ⟨cop 4607   × cxp 5652  dom cdm 5654  ran crn 5655   ↾ cres 5656  Fun wfun 6524   Fn wfn 6525  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  +_gcplusg 17269  +_𝑓cplusf 18613

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-plusf 18615

This theorem is referenced by:  mgmplusfreseq  48088