Theorem plusffng 13450

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 2805	. . . . 5 ⊢ 𝑥 ∈ V
2		plusgslid 13197	. . . . . 6 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
3	2	slotex 13111	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (+g‘𝐺) ∈ V)
4		vex 2805	. . . . . 6 ⊢ 𝑦 ∈ V
5	4	a1i 9	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ V)
6		ovexg 6052	. . . . 5 ⊢ ((𝑥 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑦 ∈ V) → (𝑥(+g‘𝐺)𝑦) ∈ V)
7	1, 3, 5, 6	mp3an2ani 1380	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ V)
8	7	ralrimivva 2614	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) ∈ V)
9		eqid 2231	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦))
10	9	fnmpo 6367	. . 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 2231	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		plusffn.2	. . . 4 ⊢ ⨣ = (+𝑓‘𝐺)
15	12, 13, 14	plusffvalg 13447	. . 3 ⊢ (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦)))
16	15	fneq1d 5420	. 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 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   × cxp 4723   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018   ∈ cmpo 6020  Basecbs 13084  +_gcplusg 13162  +_𝑓cplusf 13438

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-inn 9144  df-2 9202  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-plusf 13440

This theorem is referenced by:  lmodfopnelem1  14341