Theorem lflnegl 37017

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 37087, 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 = (invg‘𝑅)
lflnegcl.n	⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥)))
lflnegcl.f	⊢ 𝐹 = (LFnl‘𝑊)
lflnegcl.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lflnegcl.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lflnegl.p	⊢ + = (+g‘𝑅)
lflnegl.o	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
lflnegl	⊢ (𝜑 → (𝑁 ∘f + 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.

Step	Hyp	Ref	Expression
1		lflnegcl.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2	1	fvexi 6770	. . 3 ⊢ 𝑉 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
4		lflnegcl.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		lflnegcl.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
6		lflnegcl.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑊)
7		eqid 2738	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		lflnegcl.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
9	6, 7, 1, 8	lflf 37004	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
10	4, 5, 9	syl2anc 583	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑅))
11		lflnegl.o	. . . 4 ⊢ 0 = (0g‘𝑅)
12	11	fvexi 6770	. . 3 ⊢ 0 ∈ V
13	12	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
14		lflnegcl.i	. . . 4 ⊢ 𝐼 = (invg‘𝑅)
15	6	lmodring 20046	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16		ringgrp 19703	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
17	4, 15, 16	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	7, 14, 17	grpinvf1o 18560	. . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19		f1of 6700	. . 3 ⊢ (𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝐼:(Base‘𝑅)⟶(Base‘𝑅))
20	18, 19	syl 17	. 2 ⊢ (𝜑 → 𝐼:(Base‘𝑅)⟶(Base‘𝑅))
21		lflnegcl.n	. . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥)))
22	21	a1i 11	. 2 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥))))
23		lflnegl.p	. . . 4 ⊢ + = (+g‘𝑅)
24	7, 23, 11, 14	grplinv 18543	. . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
25	17, 24	sylan 579	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
26	3, 10, 13, 20, 22, 25	caofinvl 7541	1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 }))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558   ↦ cmpt 5153   × cxp 5578  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  Basecbs 16840  +_gcplusg 16888  Scalarcsca 16891  0_gc0g 17067  Grpcgrp 18492  inv_gcminusg 18493  Ringcrg 19698  LModclmod 20038  LFnlclfn 36998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-map 8575  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-ring 19700  df-lmod 20040  df-lfl 36999

This theorem is referenced by:  ldualgrplem  37086