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Theorem lflnegl 36080
 Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 36150, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
lflnegcl.v 𝑉 = (Base‘𝑊)
lflnegcl.r 𝑅 = (Scalar‘𝑊)
lflnegcl.i 𝐼 = (invg𝑅)
lflnegcl.n 𝑁 = (𝑥𝑉 ↦ (𝐼‘(𝐺𝑥)))
lflnegcl.f 𝐹 = (LFnl‘𝑊)
lflnegcl.w (𝜑𝑊 ∈ LMod)
lflnegcl.g (𝜑𝐺𝐹)
lflnegl.p + = (+g𝑅)
lflnegl.o 0 = (0g𝑅)
Assertion
Ref Expression
lflnegl (𝜑 → (𝑁f + 𝐺) = (𝑉 × { 0 }))
Distinct variable groups:   𝑥,𝐺   𝑥,𝐼   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊   𝜑,𝑥
Allowed substitution hints:   + (𝑥)   𝐹(𝑥)   𝑁(𝑥)   0 (𝑥)

Proof of Theorem lflnegl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 lflnegcl.v . . . 4 𝑉 = (Base‘𝑊)
21fvexi 6680 . . 3 𝑉 ∈ V
32a1i 11 . 2 (𝜑𝑉 ∈ V)
4 lflnegcl.w . . 3 (𝜑𝑊 ∈ LMod)
5 lflnegcl.g . . 3 (𝜑𝐺𝐹)
6 lflnegcl.r . . . 4 𝑅 = (Scalar‘𝑊)
7 eqid 2824 . . . 4 (Base‘𝑅) = (Base‘𝑅)
8 lflnegcl.f . . . 4 𝐹 = (LFnl‘𝑊)
96, 7, 1, 8lflf 36067 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
104, 5, 9syl2anc 584 . 2 (𝜑𝐺:𝑉⟶(Base‘𝑅))
11 lflnegl.o . . . 4 0 = (0g𝑅)
1211fvexi 6680 . . 3 0 ∈ V
1312a1i 11 . 2 (𝜑0 ∈ V)
14 lflnegcl.i . . . 4 𝐼 = (invg𝑅)
156lmodring 19564 . . . . 5 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16 ringgrp 19224 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
174, 15, 163syl 18 . . . 4 (𝜑𝑅 ∈ Grp)
187, 14, 17grpinvf1o 18101 . . 3 (𝜑𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19 f1of 6611 . . 3 (𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝐼:(Base‘𝑅)⟶(Base‘𝑅))
2018, 19syl 17 . 2 (𝜑𝐼:(Base‘𝑅)⟶(Base‘𝑅))
21 lflnegcl.n . . 3 𝑁 = (𝑥𝑉 ↦ (𝐼‘(𝐺𝑥)))
2221a1i 11 . 2 (𝜑𝑁 = (𝑥𝑉 ↦ (𝐼‘(𝐺𝑥))))
23 lflnegl.p . . . 4 + = (+g𝑅)
247, 23, 11, 14grplinv 18084 . . 3 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼𝑦) + 𝑦) = 0 )
2517, 24sylan 580 . 2 ((𝜑𝑦 ∈ (Base‘𝑅)) → ((𝐼𝑦) + 𝑦) = 0 )
263, 10, 13, 20, 22, 25caofinvl 7429 1 (𝜑 → (𝑁f + 𝐺) = (𝑉 × { 0 }))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2106  Vcvv 3499  {csn 4563   ↦ cmpt 5142   × cxp 5551  ⟶wf 6347  –1-1-onto→wf1o 6350  ‘cfv 6351  (class class class)co 7151   ∘f cof 7400  Basecbs 16475  +gcplusg 16557  Scalarcsca 16560  0gc0g 16705  Grpcgrp 18035  invgcminusg 18036  Ringcrg 19219  LModclmod 19556  LFnlclfn 36061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-map 8401  df-0g 16707  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038  df-minusg 18039  df-ring 19221  df-lmod 19558  df-lfl 36062 This theorem is referenced by:  ldualgrplem  36149
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