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Theorem lflnegl 39077
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 39147, 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 6920 . . 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 2737 . . . 4 (Base‘𝑅) = (Base‘𝑅)
8 lflnegcl.f . . . 4 𝐹 = (LFnl‘𝑊)
96, 7, 1, 8lflf 39064 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
104, 5, 9syl2anc 584 . 2 (𝜑𝐺:𝑉⟶(Base‘𝑅))
11 lflnegl.o . . . 4 0 = (0g𝑅)
1211fvexi 6920 . . 3 0 ∈ V
1312a1i 11 . 2 (𝜑0 ∈ V)
14 lflnegcl.i . . . 4 𝐼 = (invg𝑅)
156lmodring 20866 . . . . 5 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16 ringgrp 20235 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
174, 15, 163syl 18 . . . 4 (𝜑𝑅 ∈ Grp)
187, 14, 17grpinvf1o 19027 . . 3 (𝜑𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19 f1of 6848 . . 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 19007 . . 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 7729 1 (𝜑 → (𝑁f + 𝐺) = (𝑉 × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cmpt 5225   × cxp 5683  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  f cof 7695  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952  Ringcrg 20230  LModclmod 20858  LFnlclfn 39058
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-rep 5279  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-rmo 3380  df-reu 3381  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-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-ring 20232  df-lmod 20860  df-lfl 39059
This theorem is referenced by:  ldualgrplem  39146
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