Theorem lflnegl 33381

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 33451, 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		fvex 6113	. . . 4 ⊢ (Base‘𝑊) ∈ V
3	1, 2	eqeltri 2684	. . 3 ⊢ 𝑉 ∈ V
4	3	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
5		lflnegcl.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		lflnegcl.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
7		lflnegcl.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑊)
8		eqid 2610	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		lflnegcl.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
10	7, 8, 1, 9	lflf 33368	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
11	5, 6, 10	syl2anc 691	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑅))
12		lflnegl.o	. . . 4 ⊢ 0 = (0g‘𝑅)
13		fvex 6113	. . . 4 ⊢ (0g‘𝑅) ∈ V
14	12, 13	eqeltri 2684	. . 3 ⊢ 0 ∈ V
15	14	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
16		lflnegcl.i	. . . 4 ⊢ 𝐼 = (invg‘𝑅)
17	7	lmodring 18694	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
18		ringgrp 18375	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
19	5, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
20	8, 16, 19	grpinvf1o 17308	. . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
21		f1of 6050	. . 3 ⊢ (𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝐼:(Base‘𝑅)⟶(Base‘𝑅))
22	20, 21	syl 17	. 2 ⊢ (𝜑 → 𝐼:(Base‘𝑅)⟶(Base‘𝑅))
23		lflnegcl.n	. . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥)))
24	23	a1i 11	. 2 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥))))
25		lflnegl.p	. . . 4 ⊢ + = (+g‘𝑅)
26	8, 25, 12, 16	grplinv 17291	. . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
27	19, 26	sylan 487	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
28	4, 11, 15, 22, 24, 27	caofinvl 6822	1 ⊢ (𝜑 → (𝑁 ∘𝑓 + 𝐺) = (𝑉 × { 0 }))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  Basecbs 15695  +_gcplusg 15768  Scalarcsca 15771  0_gc0g 15923  Grpcgrp 17245  inv_gcminusg 17246  Ringcrg 18370  LModclmod 18686  LFnlclfn 33362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-ring 18372  df-lmod 18688  df-lfl 33363

This theorem is referenced by:  ldualgrplem  33450