Theorem plusffvalg 13665

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 13658	. . 3 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
3		fveq2 5693	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		plusffval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2289	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6		fveq2 5693	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		plusffval.2	. . . . . 6 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2289	. . . . 5 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9	8	oveqd 6096	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
10	5, 5, 9	mpoeq123dv 6144	. . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
11		elex 2833	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
12		basfn 13394	. . . . . 6 ⊢ Base Fn V
13		funfvex 5710	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
14	13	funfni 5481	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
15	12, 11, 14	sylancr 418	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
16	4, 15	eqeltrid 2325	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
17		mpoexga 6442	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V)
18	16, 16, 17	syl2anc 415	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V)
19	2, 10, 11, 18	fvmptd3 5796	. 2 ⊢ (𝐺 ∈ 𝑉 → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
20	1, 19	eqtrid 2283	1 ⊢ (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ∈ cmpo 6081  Basecbs 13335  +_gcplusg 13414  +_𝑓cplusf 13656

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-plusf 13658

This theorem is referenced by:  plusfvalg  13666  plusfeqg  13667  plusffng  13668  mgmplusf  13669