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Theorem lflnegl 37588
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 37658, 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 6860 . . 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 2733 . . . 4 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
8 lflnegcl.f . . . 4 𝐹 = (LFnlβ€˜π‘Š)
96, 7, 1, 8lflf 37575 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ 𝐺:π‘‰βŸΆ(Baseβ€˜π‘…))
104, 5, 9syl2anc 585 . 2 (πœ‘ β†’ 𝐺:π‘‰βŸΆ(Baseβ€˜π‘…))
11 lflnegl.o . . . 4 0 = (0gβ€˜π‘…)
1211fvexi 6860 . . 3 0 ∈ V
1312a1i 11 . 2 (πœ‘ β†’ 0 ∈ V)
14 lflnegcl.i . . . 4 𝐼 = (invgβ€˜π‘…)
156lmodring 20373 . . . . 5 (π‘Š ∈ LMod β†’ 𝑅 ∈ Ring)
16 ringgrp 19977 . . . . 5 (𝑅 ∈ Ring β†’ 𝑅 ∈ Grp)
174, 15, 163syl 18 . . . 4 (πœ‘ β†’ 𝑅 ∈ Grp)
187, 14, 17grpinvf1o 18825 . . 3 (πœ‘ β†’ 𝐼:(Baseβ€˜π‘…)–1-1-ontoβ†’(Baseβ€˜π‘…))
19 f1of 6788 . . 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 18808 . . 3 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ ((πΌβ€˜π‘¦) + 𝑦) = 0 )
2517, 24sylan 581 . 2 ((πœ‘ ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ ((πΌβ€˜π‘¦) + 𝑦) = 0 )
263, 10, 13, 20, 22, 25caofinvl 7651 1 (πœ‘ β†’ (𝑁 ∘f + 𝐺) = (𝑉 Γ— { 0 }))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3447  {csn 4590   ↦ cmpt 5192   Γ— cxp 5635  βŸΆwf 6496  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361   ∘f cof 7619  Basecbs 17091  +gcplusg 17141  Scalarcsca 17144  0gc0g 17329  Grpcgrp 18756  invgcminusg 18757  Ringcrg 19972  LModclmod 20365  LFnlclfn 37569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-map 8773  df-0g 17331  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-minusg 18760  df-ring 19974  df-lmod 20367  df-lfl 37570
This theorem is referenced by:  ldualgrplem  37657
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