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Theorem plusffval 17468
Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
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 6353 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 plusffval.1 . . . . . 6 𝐵 = (Base‘𝐺)
42, 3syl6eqr 2812 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6353 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 plusffval.2 . . . . . . 7 + = (+g𝐺)
75, 6syl6eqr 2812 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 6831 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
94, 4, 8mpt2eq123dv 6883 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
10 df-plusf 17462 . . . 4 +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)))
11 df-ov 6817 . . . . . . . 8 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
12 fvrn0 6378 . . . . . . . 8 ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
1311, 12eqeltri 2835 . . . . . . 7 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
1413rgen2w 3063 . . . . . 6 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
15 eqid 2760 . . . . . . 7 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
1615fmpt2 7406 . . . . . 6 (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) ↔ (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)):(𝐵 × 𝐵)⟶(ran + ∪ {∅}))
1714, 16mpbi 220 . . . . 5 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)):(𝐵 × 𝐵)⟶(ran + ∪ {∅})
18 fvex 6363 . . . . . . 7 (Base‘𝐺) ∈ V
193, 18eqeltri 2835 . . . . . 6 𝐵 ∈ V
2019, 19xpex 7128 . . . . 5 (𝐵 × 𝐵) ∈ V
21 fvex 6363 . . . . . . . 8 (+g𝐺) ∈ V
226, 21eqeltri 2835 . . . . . . 7 + ∈ V
2322rnex 7266 . . . . . 6 ran + ∈ V
24 p0ex 5002 . . . . . 6 {∅} ∈ V
2523, 24unex 7122 . . . . 5 (ran + ∪ {∅}) ∈ V
26 fex2 7287 . . . . 5 (((𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)):(𝐵 × 𝐵)⟶(ran + ∪ {∅}) ∧ (𝐵 × 𝐵) ∈ V ∧ (ran + ∪ {∅}) ∈ V) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) ∈ V)
2717, 20, 25, 26mp3an 1573 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) ∈ V
289, 10, 27fvmpt 6445 . . 3 (𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
29 fvprc 6347 . . . . 5 𝐺 ∈ V → (+𝑓𝐺) = ∅)
30 mpt20 6891 . . . . 5 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)) = ∅
3129, 30syl6eqr 2812 . . . 4 𝐺 ∈ V → (+𝑓𝐺) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)))
32 fvprc 6347 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
333, 32syl5eq 2806 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
34 mpt2eq12 6881 . . . . 5 ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)))
3533, 33, 34syl2anc 696 . . . 4 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)))
3631, 35eqtr4d 2797 . . 3 𝐺 ∈ V → (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦)))
3728, 36pm2.61i 176 . 2 (+𝑓𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
381, 37eqtri 2782 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cun 3713  c0 4058  {csn 4321  cop 4327   × cxp 5264  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  Basecbs 16079  +gcplusg 16163  +𝑓cplusf 17460
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 7115
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 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-plusf 17462
This theorem is referenced by:  plusfval  17469  plusfeq  17470  plusffn  17471  mgmplusf  17472  rlmscaf  19430  istgp2  22116  oppgtmd  22122  submtmd  22129  prdstmdd  22148  ressplusf  29980  pl1cn  30331
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