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Theorem lflnegl 38440
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 38510, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
lflnegcl.v 𝑉 = (Baseβ€˜π‘Š)
lflnegcl.r 𝑅 = (Scalarβ€˜π‘Š)
lflnegcl.i 𝐼 = (invgβ€˜π‘…)
lflnegcl.n 𝑁 = (π‘₯ ∈ 𝑉 ↦ (πΌβ€˜(πΊβ€˜π‘₯)))
lflnegcl.f 𝐹 = (LFnlβ€˜π‘Š)
lflnegcl.w (πœ‘ β†’ π‘Š ∈ LMod)
lflnegcl.g (πœ‘ β†’ 𝐺 ∈ 𝐹)
lflnegl.p + = (+gβ€˜π‘…)
lflnegl.o 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
lflnegl (πœ‘ β†’ (𝑁 ∘f + 𝐺) = (𝑉 Γ— { 0 }))
Distinct variable groups:   π‘₯,𝐺   π‘₯,𝐼   π‘₯,𝑅   π‘₯,𝑉   π‘₯,π‘Š   πœ‘,π‘₯
Allowed substitution hints:   + (π‘₯)   𝐹(π‘₯)   𝑁(π‘₯)   0 (π‘₯)

Proof of Theorem lflnegl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 lflnegcl.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
21fvexi 6896 . . 3 𝑉 ∈ V
32a1i 11 . 2 (πœ‘ β†’ 𝑉 ∈ V)
4 lflnegcl.w . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
5 lflnegcl.g . . 3 (πœ‘ β†’ 𝐺 ∈ 𝐹)
6 lflnegcl.r . . . 4 𝑅 = (Scalarβ€˜π‘Š)
7 eqid 2724 . . . 4 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
8 lflnegcl.f . . . 4 𝐹 = (LFnlβ€˜π‘Š)
96, 7, 1, 8lflf 38427 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ 𝐺:π‘‰βŸΆ(Baseβ€˜π‘…))
104, 5, 9syl2anc 583 . 2 (πœ‘ β†’ 𝐺:π‘‰βŸΆ(Baseβ€˜π‘…))
11 lflnegl.o . . . 4 0 = (0gβ€˜π‘…)
1211fvexi 6896 . . 3 0 ∈ V
1312a1i 11 . 2 (πœ‘ β†’ 0 ∈ V)
14 lflnegcl.i . . . 4 𝐼 = (invgβ€˜π‘…)
156lmodring 20706 . . . . 5 (π‘Š ∈ LMod β†’ 𝑅 ∈ Ring)
16 ringgrp 20135 . . . . 5 (𝑅 ∈ Ring β†’ 𝑅 ∈ Grp)
174, 15, 163syl 18 . . . 4 (πœ‘ β†’ 𝑅 ∈ Grp)
187, 14, 17grpinvf1o 18930 . . 3 (πœ‘ β†’ 𝐼:(Baseβ€˜π‘…)–1-1-ontoβ†’(Baseβ€˜π‘…))
19 f1of 6824 . . 3 (𝐼:(Baseβ€˜π‘…)–1-1-ontoβ†’(Baseβ€˜π‘…) β†’ 𝐼:(Baseβ€˜π‘…)⟢(Baseβ€˜π‘…))
2018, 19syl 17 . 2 (πœ‘ β†’ 𝐼:(Baseβ€˜π‘…)⟢(Baseβ€˜π‘…))
21 lflnegcl.n . . 3 𝑁 = (π‘₯ ∈ 𝑉 ↦ (πΌβ€˜(πΊβ€˜π‘₯)))
2221a1i 11 . 2 (πœ‘ β†’ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (πΌβ€˜(πΊβ€˜π‘₯))))
23 lflnegl.p . . . 4 + = (+gβ€˜π‘…)
247, 23, 11, 14grplinv 18911 . . 3 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ ((πΌβ€˜π‘¦) + 𝑦) = 0 )
2517, 24sylan 579 . 2 ((πœ‘ ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ ((πΌβ€˜π‘¦) + 𝑦) = 0 )
263, 10, 13, 20, 22, 25caofinvl 7694 1 (πœ‘ β†’ (𝑁 ∘f + 𝐺) = (𝑉 Γ— { 0 }))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3466  {csn 4621   ↦ cmpt 5222   Γ— cxp 5665  βŸΆwf 6530  β€“1-1-ontoβ†’wf1o 6533  β€˜cfv 6534  (class class class)co 7402   ∘f cof 7662  Basecbs 17145  +gcplusg 17198  Scalarcsca 17201  0gc0g 17386  Grpcgrp 18855  invgcminusg 18856  Ringcrg 20130  LModclmod 20698  LFnlclfn 38421
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-map 8819  df-0g 17388  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-ring 20132  df-lmod 20700  df-lfl 38422
This theorem is referenced by:  ldualgrplem  38509
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