Theorem plusfeqg 12647

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 12645	. . 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 5973	. . 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 2211	1 ⊢ ((𝐺 ∈ 𝑉 ∧ + Fn (𝐵 × 𝐵)) → ⨣ = + )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2146   × cxp 4618   Fn wfn 5203  ‘cfv 5208  (class class class)co 5865   ∈ cmpo 5867  Basecbs 12427  +_gcplusg 12491  +_𝑓cplusf 12636

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-inn 8891  df-ndx 12430  df-slot 12431  df-base 12433  df-plusf 12638

This theorem is referenced by:  mgmb1mgm1  12651  mndfo  12704