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Theorem plusfeq 17450
 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 fnov 6933 . . 3 ( + Fn (𝐵 × 𝐵) ↔ + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
21biimpi 206 . 2 ( + Fn (𝐵 × 𝐵) → + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
3 plusffval.1 . . 3 𝐵 = (Base‘𝐺)
4 plusffval.2 . . 3 + = (+g𝐺)
5 plusffval.3 . . 3 = (+𝑓𝐺)
63, 4, 5plusffval 17448 . 2 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
72, 6syl6reqr 2813 1 ( + Fn (𝐵 × 𝐵) → = + )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   × cxp 5264   Fn wfn 6044  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815  Basecbs 16059  +gcplusg 16143  +𝑓cplusf 17440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-plusf 17442 This theorem is referenced by:  mgmb1mgm1  17455  mndfo  17516  cnfldplusf  19975  symgtgp  22106
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