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Theorem plusffval 18463
Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
Hypotheses
Ref Expression
plusffval.1 𝐵 = (Base‘𝐺)
plusffval.2 + = (+g𝐺)
plusffval.3 = (+𝑓𝐺)
Assertion
Ref Expression
plusffval = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, + ,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem plusffval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 plusffval.3 . 2 = (+𝑓𝐺)
2 fveq2 6839 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 plusffval.1 . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6839 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 plusffval.2 . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2795 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7368 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
94, 4, 8mpoeq123dv 7426 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
10 df-plusf 18456 . . . 4 +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)))
113fvexi 6853 . . . . 5 𝐵 ∈ V
126fvexi 6853 . . . . . . 7 + ∈ V
1312rnex 7841 . . . . . 6 ran + ∈ V
14 p0ex 5337 . . . . . 6 {∅} ∈ V
1513, 14unex 7672 . . . . 5 (ran + ∪ {∅}) ∈ V
16 df-ov 7354 . . . . . . 7 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17 fvrn0 6869 . . . . . . 7 ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
1816, 17eqeltri 2834 . . . . . 6 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
1918rgen2w 3067 . . . . 5 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
2011, 11, 15, 19mpoexw 8003 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) ∈ V
219, 10, 20fvmpt 6945 . . 3 (𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
22 fvprc 6831 . . . 4 𝐺 ∈ V → (+𝑓𝐺) = ∅)
23 fvprc 6831 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23eqtrid 2789 . . . . . 6 𝐺 ∈ V → 𝐵 = ∅)
2524olcd 872 . . . . 5 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26 0mpo0 7434 . . . . 5 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = ∅)
2725, 26syl 17 . . . 4 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = ∅)
2822, 27eqtr4d 2780 . . 3 𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
2921, 28pm2.61i 182 . 2 (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
301, 29eqtri 2765 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1541  wcel 2106  Vcvv 3443  cun 3906  c0 4280  {csn 4584  cop 4590  ran crn 5632  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:  plusfval  18464  plusfeq  18465  plusffn  18466  mgmplusf  18467  rlmscaf  20631  istgp2  23394  oppgtmd  23400  submtmd  23407  prdstmdd  23427  ressplusf  31643  pl1cn  32348
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