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Theorem lflnegl 35151
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 35221, 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 6447 . . 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 2825 . . . 4 (Base‘𝑅) = (Base‘𝑅)
8 lflnegcl.f . . . 4 𝐹 = (LFnl‘𝑊)
96, 7, 1, 8lflf 35138 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
104, 5, 9syl2anc 581 . 2 (𝜑𝐺:𝑉⟶(Base‘𝑅))
11 lflnegl.o . . . 4 0 = (0g𝑅)
1211fvexi 6447 . . 3 0 ∈ V
1312a1i 11 . 2 (𝜑0 ∈ V)
14 lflnegcl.i . . . 4 𝐼 = (invg𝑅)
156lmodring 19227 . . . . 5 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16 ringgrp 18906 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
174, 15, 163syl 18 . . . 4 (𝜑𝑅 ∈ Grp)
187, 14, 17grpinvf1o 17839 . . 3 (𝜑𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19 f1of 6378 . . 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 17822 . . 3 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼𝑦) + 𝑦) = 0 )
2517, 24sylan 577 . 2 ((𝜑𝑦 ∈ (Base‘𝑅)) → ((𝐼𝑦) + 𝑦) = 0 )
263, 10, 13, 20, 22, 25caofinvl 7184 1 (𝜑 → (𝑁𝑓 + 𝐺) = (𝑉 × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  Vcvv 3414  {csn 4397  cmpt 4952   × cxp 5340  wf 6119  1-1-ontowf1o 6122  cfv 6123  (class class class)co 6905  𝑓 cof 7155  Basecbs 16222  +gcplusg 16305  Scalarcsca 16308  0gc0g 16453  Grpcgrp 17776  invgcminusg 17777  Ringcrg 18901  LModclmod 19219  LFnlclfn 35132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-map 8124  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-ring 18903  df-lmod 19221  df-lfl 35133
This theorem is referenced by:  ldualgrplem  35220
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