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Theorem plusffval 18659
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 6906 . . . . . 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 6906 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 plusffval.2 . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2795 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7448 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
94, 4, 8mpoeq123dv 7508 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
10 df-plusf 18652 . . . 4 +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)))
113fvexi 6920 . . . . 5 𝐵 ∈ V
126fvexi 6920 . . . . . . 7 + ∈ V
1312rnex 7932 . . . . . 6 ran + ∈ V
14 p0ex 5384 . . . . . 6 {∅} ∈ V
1513, 14unex 7764 . . . . 5 (ran + ∪ {∅}) ∈ V
16 df-ov 7434 . . . . . . 7 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17 fvrn0 6936 . . . . . . 7 ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
1816, 17eqeltri 2837 . . . . . 6 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
1918rgen2w 3066 . . . . 5 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
2011, 11, 15, 19mpoexw 8103 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) ∈ V
219, 10, 20fvmpt 7016 . . 3 (𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
22 fvprc 6898 . . . 4 𝐺 ∈ V → (+𝑓𝐺) = ∅)
23 fvprc 6898 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23eqtrid 2789 . . . . . 6 𝐺 ∈ V → 𝐵 = ∅)
2524olcd 875 . . . . 5 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26 0mpo0 7516 . . . . 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 848   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  c0 4333  {csn 4626  cop 4632  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  +𝑓cplusf 18650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-plusf 18652
This theorem is referenced by:  plusfval  18660  plusfeq  18661  plusffn  18662  mgmplusf  18663  rlmscaf  21214  istgp2  24099  oppgtmd  24105  submtmd  24112  prdstmdd  24132  ressplusf  32948  pl1cn  33954
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