Theorem plusffng 13631

Description: The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
plusffn.1	⊢ 𝐵 = (Base‘𝐺)
plusffn.2	⊢ ⨣ = (+𝑓‘𝐺)

Assertion

Ref	Expression
plusffng	⊢ (𝐺 ∈ 𝑉 → ⨣ Fn (𝐵 × 𝐵))

Proof of Theorem plusffng

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

Step	Hyp	Ref	Expression
1		vex 2818	. . . . 5 ⊢ 𝑥 ∈ V
2		plusgslid 13412	. . . . . 6 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
3	2	slotex 13326	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (+g‘𝐺) ∈ V)
4		vex 2818	. . . . . 6 ⊢ 𝑦 ∈ V
5	4	a1i 9	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ V)
6		ovexg 6092	. . . . 5 ⊢ ((𝑥 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑦 ∈ V) → (𝑥(+g‘𝐺)𝑦) ∈ V)
7	1, 3, 5, 6	mp3an2ani 1381	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ V)
8	7	ralrimivva 2626	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) ∈ V)
9		eqid 2234	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦))
10	9	fnmpo 6411	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦)) Fn (𝐵 × 𝐵))
11	8, 10	syl 14	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦)) Fn (𝐵 × 𝐵))
12		plusffn.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
13		eqid 2234	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		plusffn.2	. . . 4 ⊢ ⨣ = (+𝑓‘𝐺)
15	12, 13, 14	plusffvalg 13628	. . 3 ⊢ (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦)))
16	15	fneq1d 5451	. 2 ⊢ (𝐺 ∈ 𝑉 → ( ⨣ Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦)) Fn (𝐵 × 𝐵)))
17	11, 16	mpbird 167	1 ⊢ (𝐺 ∈ 𝑉 → ⨣ Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   × cxp 4752   Fn wfn 5352  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060  Basecbs 13299  +_gcplusg 13377  +_𝑓cplusf 13619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9258  df-2 9316  df-ndx 13302  df-slot 13303  df-base 13305  df-plusg 13390  df-plusf 13621

This theorem is referenced by:  lmodfopnelem1  14601