Theorem lflnegl 38543

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 38613, 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 6906	. . 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 2728	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		lflnegcl.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
9	6, 7, 1, 8	lflf 38530	. . 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 6906	. . 3 ⊢ 0 ∈ V
13	12	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
14		lflnegcl.i	. . . 4 ⊢ 𝐼 = (invg‘𝑅)
15	6	lmodring 20745	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16		ringgrp 20172	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
17	4, 15, 16	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	7, 14, 17	grpinvf1o 18959	. . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19		f1of 6834	. . 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 18940	. . 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 7710	1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 }))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3470  {csn 4625   ↦ cmpt 5226   × cxp 5671  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7415   ∘_f cof 7678  Basecbs 17174  +_gcplusg 17227  Scalarcsca 17230  0_gc0g 17415  Grpcgrp 18884  inv_gcminusg 18885  Ringcrg 20167  LModclmod 20737  LFnlclfn 38524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7680  df-map 8841  df-0g 17417  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-grp 18887  df-minusg 18888  df-ring 20169  df-lmod 20739  df-lfl 38525

This theorem is referenced by:  ldualgrplem  38612