Theorem plusfval 18438

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 7358	. 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 18437	. 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
6		ovex 7382	. 2 ⊢ (𝑋 + 𝑌) ∈ V
7	1, 5, 6	ovmpoa 7502	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⨣ 𝑌) = (𝑋 + 𝑌))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ‘cfv 6491  (class class class)co 7349  Basecbs 17017  +_gcplusg 17067  +_𝑓cplusf 18428

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7662

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7911  df-2nd 7912  df-plusf 18430

This theorem is referenced by:  mndpfo  18513  lmodfopne  20274  cnmpt1plusg  23351  cnmpt2plusg  23352  tmdcn2  23353  tsmsadd  23411  mhmhmeotmd  32242  plusfreseq  45748