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Theorem plusffval 17854
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 6649 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 plusffval.1 . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2854 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6649 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 plusffval.2 . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2854 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7156 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
94, 4, 8mpoeq123dv 7212 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
10 df-plusf 17847 . . . 4 +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)))
113fvexi 6663 . . . . 5 𝐵 ∈ V
126fvexi 6663 . . . . . . 7 + ∈ V
1312rnex 7603 . . . . . 6 ran + ∈ V
14 p0ex 5253 . . . . . 6 {∅} ∈ V
1513, 14unex 7453 . . . . 5 (ran + ∪ {∅}) ∈ V
16 df-ov 7142 . . . . . . 7 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17 fvrn0 6677 . . . . . . 7 ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
1816, 17eqeltri 2889 . . . . . 6 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
1918rgen2w 3122 . . . . 5 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
2011, 11, 15, 19mpoexw 7763 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) ∈ V
219, 10, 20fvmpt 6749 . . 3 (𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
22 fvprc 6642 . . . 4 𝐺 ∈ V → (+𝑓𝐺) = ∅)
23 fvprc 6642 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23syl5eq 2848 . . . . . 6 𝐺 ∈ V → 𝐵 = ∅)
2524olcd 871 . . . . 5 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26 0mpo0 7220 . . . . 5 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = ∅)
2725, 26syl 17 . . . 4 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = ∅)
2822, 27eqtr4d 2839 . . 3 𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
2921, 28pm2.61i 185 . 2 (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
301, 29eqtri 2824 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 844   = wceq 1538  wcel 2112  Vcvv 3444  cun 3882  c0 4246  {csn 4528  cop 4534  ran crn 5524  cfv 6328  (class class class)co 7139  cmpo 7141  Basecbs 16479  +gcplusg 16561  +𝑓cplusf 17845
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-plusf 17847
This theorem is referenced by:  plusfval  17855  plusfeq  17856  plusffn  17857  mgmplusf  17858  rlmscaf  19978  istgp2  22700  oppgtmd  22706  submtmd  22713  prdstmdd  22733  ressplusf  30667  pl1cn  31312
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