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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 6840 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 plusffval.1 . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2796 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6840 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 plusffval.2 . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2796 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7369 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
94, 4, 8mpoeq123dv 7427 . . . 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 6854 . . . . 5 𝐵 ∈ V
126fvexi 6854 . . . . . . 7 + ∈ V
1312rnex 7842 . . . . . 6 ran + ∈ V
14 p0ex 5338 . . . . . 6 {∅} ∈ V
1513, 14unex 7673 . . . . 5 (ran + ∪ {∅}) ∈ V
16 df-ov 7355 . . . . . . 7 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17 fvrn0 6870 . . . . . . 7 ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
1816, 17eqeltri 2835 . . . . . 6 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
1918rgen2w 3068 . . . . 5 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
2011, 11, 15, 19mpoexw 8004 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) ∈ V
219, 10, 20fvmpt 6946 . . 3 (𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
22 fvprc 6832 . . . 4 𝐺 ∈ V → (+𝑓𝐺) = ∅)
23 fvprc 6832 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23eqtrid 2790 . . . . . 6 𝐺 ∈ V → 𝐵 = ∅)
2524olcd 873 . . . . 5 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26 0mpo0 7435 . . . . 5 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = ∅)
2725, 26syl 17 . . . 4 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = ∅)
2822, 27eqtr4d 2781 . . 3 𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
2921, 28pm2.61i 182 . 2 (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
301, 29eqtri 2766 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 846   = wceq 1542  wcel 2107  Vcvv 3444  cun 3907  c0 4281  {csn 4585  cop 4591  ran crn 5633  cfv 6494  (class class class)co 7352  cmpo 7354  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 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7914  df-2nd 7915  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  31642  pl1cn  32340
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