MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plusfval Structured version   Visualization version   GIF version

Theorem plusfval 17850
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 + = (+g𝐺)
plusffval.3 = (+𝑓𝐺)
Assertion
Ref Expression
plusfval ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + 𝑌))

Proof of Theorem plusfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 7149 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 + 𝑦) = (𝑋 + 𝑌))
2 plusffval.1 . . 3 𝐵 = (Base‘𝐺)
3 plusffval.2 . . 3 + = (+g𝐺)
4 plusffval.3 . . 3 = (+𝑓𝐺)
52, 3, 4plusffval 17849 . 2 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
6 ovex 7173 . 2 (𝑋 + 𝑌) ∈ V
71, 5, 6ovmpoa 7289 1 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  +𝑓cplusf 17840
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-plusf 17842
This theorem is referenced by:  mndpfo  17925  lmodfopne  19663  cnmpt1plusg  22690  cnmpt2plusg  22691  tmdcn2  22692  tsmsadd  22750  mhmhmeotmd  31244  plusfreseq  44331
  Copyright terms: Public domain W3C validator