Theorem lflnegl 37941

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 38011, 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 6905	. . 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 2732	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		lflnegcl.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
9	6, 7, 1, 8	lflf 37928	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
10	4, 5, 9	syl2anc 584	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑅))
11		lflnegl.o	. . . 4 ⊢ 0 = (0g‘𝑅)
12	11	fvexi 6905	. . 3 ⊢ 0 ∈ V
13	12	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
14		lflnegcl.i	. . . 4 ⊢ 𝐼 = (invg‘𝑅)
15	6	lmodring 20478	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16		ringgrp 20060	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
17	4, 15, 16	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	7, 14, 17	grpinvf1o 18892	. . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19		f1of 6833	. . 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 18873	. . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
25	17, 24	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
26	3, 10, 13, 20, 22, 25	caofinvl 7699	1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 }))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4628   ↦ cmpt 5231   × cxp 5674  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7408   ∘_f cof 7667  Basecbs 17143  +_gcplusg 17196  Scalarcsca 17199  0_gc0g 17384  Grpcgrp 18818  inv_gcminusg 18819  Ringcrg 20055  LModclmod 20470  LFnlclfn 37922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-map 8821  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-ring 20057  df-lmod 20472  df-lfl 37923

This theorem is referenced by:  ldualgrplem  38010