Theorem plusfeqg 13446

Description: If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)

Hypotheses

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

Assertion

Ref	Expression
plusfeqg	⊢ ((𝐺 ∈ 𝑉 ∧ + Fn (𝐵 × 𝐵)) → ⨣ = + )

Proof of Theorem plusfeqg

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

Step	Hyp	Ref	Expression
1		plusffval.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		plusffval.2	. . . 4 ⊢ + = (+g‘𝐺)
3		plusffval.3	. . . 4 ⊢ ⨣ = (+𝑓‘𝐺)
4	1, 2, 3	plusffvalg 13444	. . 3 ⊢ (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
5	4	adantr 276	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ + Fn (𝐵 × 𝐵)) → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
6		fnovim 6129	. . 3 ⊢ ( + Fn (𝐵 × 𝐵) → + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
7	6	adantl 277	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ + Fn (𝐵 × 𝐵)) → + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
8	5, 7	eqtr4d 2267	1 ⊢ ((𝐺 ∈ 𝑉 ∧ + Fn (𝐵 × 𝐵)) → ⨣ = + )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202   × cxp 4723   Fn wfn 5321  ‘cfv 5326  (class class class)co 6017   ∈ cmpo 6019  Basecbs 13081  +_gcplusg 13159  +_𝑓cplusf 13435

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 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

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 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-plusf 13437

This theorem is referenced by:  mgmb1mgm1  13450  mndfo  13521  cnfldplusf  14587