Theorem plusfeq 18616

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
plusfeq	⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + )

Proof of Theorem plusfeq

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

Step	Hyp	Ref	Expression
1		plusffval.1	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		plusffval.2	. . 3 ⊢ + = (+g‘𝐺)
3		plusffval.3	. . 3 ⊢ ⨣ = (+𝑓‘𝐺)
4	1, 2, 3	plusffval 18614	. 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
5		fnov 7498	. . 3 ⊢ ( + Fn (𝐵 × 𝐵) ↔ + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
6	5	biimpi 216	. 2 ⊢ ( + Fn (𝐵 × 𝐵) → + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
7	4, 6	eqtr4id 2790	1 ⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + )

Syntax hints:   → wi 4   = wceq 1542   × cxp 5629   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  +_gcplusg 17220  +_𝑓cplusf 18605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-plusf 18607

This theorem is referenced by:  mgmb1mgm1  18623  mndfo  18726  cnfldplusf  21379  efmndtmd  24066