Theorem plusffvalg 13416

Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

Ref	Expression
plusffval.1	⊢ 𝐵 = (Base‘𝐺)
plusffval.2	⊢ + = (+g‘𝐺)
plusffval.3	⊢ ⨣ = (+𝑓‘𝐺)

Assertion

Ref	Expression
plusffvalg	⊢ (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   ⨣ (𝑥,𝑦)

Proof of Theorem plusffvalg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plusffval.3	. 2 ⊢ ⨣ = (+𝑓‘𝐺)
2		df-plusf 13409	. . 3 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
3		fveq2 5632	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		plusffval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2280	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6		fveq2 5632	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		plusffval.2	. . . . . 6 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2280	. . . . 5 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9	8	oveqd 6027	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
10	5, 5, 9	mpoeq123dv 6075	. . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
11		elex 2811	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
12		basfn 13112	. . . . . 6 ⊢ Base Fn V
13		funfvex 5649	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
14	13	funfni 5426	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
15	12, 11, 14	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
16	4, 15	eqeltrid 2316	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
17		mpoexga 6369	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V)
18	16, 16, 17	syl2anc 411	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V)
19	2, 10, 11, 18	fvmptd3 5733	. 2 ⊢ (𝐺 ∈ 𝑉 → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
20	1, 19	eqtrid 2274	1 ⊢ (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799   Fn wfn 5316  ‘cfv 5321  (class class class)co 6010   ∈ cmpo 6012  Basecbs 13053  +_gcplusg 13131  +_𝑓cplusf 13407

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-cnex 8106  ax-resscn 8107  ax-1re 8109  ax-addrcl 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-inn 9127  df-ndx 13056  df-slot 13057  df-base 13059  df-plusf 13409

This theorem is referenced by:  plusfvalg  13417  plusfeqg  13418  plusffng  13419  mgmplusf  13420