Theorem lflnegl 39850

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 39920, 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 6895	. . 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 2763	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		lflnegcl.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
9	6, 7, 1, 8	lflf 39837	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅))
10	4, 5, 9	syl2anc 595	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑅))
11		lflnegl.o	. . . 4 ⊢ 0 = (0g‘𝑅)
12	11	fvexi 6895	. . 3 ⊢ 0 ∈ V
13	12	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
14		lflnegcl.i	. . . 4 ⊢ 𝐼 = (invg‘𝑅)
15	6	lmodring 20989	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16		ringgrp 20315	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
17	4, 15, 16	3syl 19	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	7, 14, 17	grpinvf1o 19070	. . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19		f1of 6820	. . 3 ⊢ (𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝐼:(Base‘𝑅)⟶(Base‘𝑅))
20	18, 19	syl 18	. 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 19051	. . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
25	17, 24	sylan 591	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 )
26	3, 10, 13, 20, 22, 25	caofinvl 7706	1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 }))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4589   ↦ cmpt 5192   × cxp 5659  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7672  Basecbs 17264  +_gcplusg 17305  Scalarcsca 17308  0_gc0g 17487  Grpcgrp 18995  inv_gcminusg 18996  Ringcrg 20310  LModclmod 20981  LFnlclfn 39831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-map 8822  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-minusg 18999  df-ring 20312  df-lmod 20983  df-lfl 39832

This theorem is referenced by:  ldualgrplem  39919