Theorem plusfreseq 47412

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 6655	. . . 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 2731	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))
11	10	rgen2 3187	. . . 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 6896	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14		df-ov 7422	. . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
15	13, 14	eqtr4di 2783	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( + ‘𝑝) = (𝑥 + 𝑦))
16		fveq2 6896	. . . . . 6 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
17		df-ov 7422	. . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
18	16, 17	eqtr4di 2783	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦))
19	15, 18	eqeq12d 2741	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)))
20	19	ralxp 5844	. . 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 6658	. . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵))
23	22	eqcomd 2731	. . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ )
24	3, 23	ax-mp 5	. . 3 ⊢ (𝐵 × 𝐵) = dom ⨣
25	24	fveqressseq 7088	. 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
26	6, 7, 21, 25	syl3anc 1368	1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∉ wnel 3035  ∀wral 3050  ∅c0 4322  ⟨cop 4636   × cxp 5676  dom cdm 5678  ran crn 5679   ↾ cres 5680  Fun wfun 6543   Fn wfn 6544  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  +_gcplusg 17236  +_𝑓cplusf 18600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-plusf 18602

This theorem is referenced by:  mgmplusfreseq  47413