Theorem plusfval 18552

Description: The group addition operation as a function. (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
plusfval	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⨣ 𝑌) = (𝑋 + 𝑌))

Proof of Theorem plusfval

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

Step	Hyp	Ref	Expression
1		oveq12 7355	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 + 𝑦) = (𝑋 + 𝑌))
2		plusffval.1	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		plusffval.2	. . 3 ⊢ + = (+g‘𝐺)
4		plusffval.3	. . 3 ⊢ ⨣ = (+𝑓‘𝐺)
5	2, 3, 4	plusffval 18551	. 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
6		ovex 7379	. 2 ⊢ (𝑋 + 𝑌) ∈ V
7	1, 5, 6	ovmpoa 7501	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⨣ 𝑌) = (𝑋 + 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  +_𝑓cplusf 18542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-plusf 18544

This theorem is referenced by:  mndpfo  18662  lmodfopne  20831  cnmpt1plusg  24000  cnmpt2plusg  24001  tmdcn2  24002  tsmsadd  24060  mhmhmeotmd  33935  plusfreseq  48194