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Theorem plusfeq 18465
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 + = (+g𝐺)
plusffval.3 = (+𝑓𝐺)
Assertion
Ref Expression
plusfeq ( + Fn (𝐵 × 𝐵) → = + )

Proof of Theorem plusfeq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plusffval.1 . . 3 𝐵 = (Base‘𝐺)
2 plusffval.2 . . 3 + = (+g𝐺)
3 plusffval.3 . . 3 = (+𝑓𝐺)
41, 2, 3plusffval 18463 . 2 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
5 fnov 7481 . . 3 ( + Fn (𝐵 × 𝐵) ↔ + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
65biimpi 215 . 2 ( + Fn (𝐵 × 𝐵) → + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
74, 6eqtr4id 2796 1 ( + Fn (𝐵 × 𝐵) → = + )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541   × cxp 5629   Fn wfn 6488  cfv 6493  (class class class)co 7351  cmpo 7353  Basecbs 17043  +gcplusg 17093  +𝑓cplusf 18454
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 7664
This theorem depends on definitions:  df-bi 206  df-an 397  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 5529  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 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-plusf 18456
This theorem is referenced by:  mgmb1mgm1  18470  mndfo  18540  cnfldplusf  20777  efmndtmd  23404
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