Theorem lflnegl 34363

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 34433, 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 6201	. . . 4 ⊢ (Base‘𝑊) ∈ V
3	1, 2	eqeltri 2697	. . 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 2622	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		lflnegcl.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
10	7, 8, 1, 9	lflf 34350	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
11	5, 6, 10	syl2anc 693	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑅))
12		lflnegl.o	. . . 4 ⊢ 0 = (0g‘𝑅)
13		fvex 6201	. . . 4 ⊢ (0g‘𝑅) ∈ V
14	12, 13	eqeltri 2697	. . 3 ⊢ 0 ∈ V
15	14	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
16		lflnegcl.i	. . . 4 ⊢ 𝐼 = (invg‘𝑅)
17	7	lmodring 18871	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
18		ringgrp 18552	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
19	5, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
20	8, 16, 19	grpinvf1o 17485	. . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
21		f1of 6137	. . 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 17468	. . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
27	19, 26	sylan 488	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
28	4, 11, 15, 22, 24, 27	caofinvl 6924	1 ⊢ (𝜑 → (𝑁 ∘𝑓 + 𝐺) = (𝑉 × { 0 }))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177   ↦ cmpt 4729   × cxp 5112  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  Basecbs 15857  +_gcplusg 15941  Scalarcsca 15944  0_gc0g 16100  Grpcgrp 17422  inv_gcminusg 17423  Ringcrg 18547  LModclmod 18863  LFnlclfn 34344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ring 18549  df-lmod 18865  df-lfl 34345

This theorem is referenced by:  ldualgrplem  34432