Theorem lflnegl 39042

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 39112, 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 6854	. . 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 2729	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		lflnegcl.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
9	6, 7, 1, 8	lflf 39029	. . 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 6854	. . 3 ⊢ 0 ∈ V
13	12	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
14		lflnegcl.i	. . . 4 ⊢ 𝐼 = (invg‘𝑅)
15	6	lmodring 20750	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
16		ringgrp 20123	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
17	4, 15, 16	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	7, 14, 17	grpinvf1o 18917	. . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
19		f1of 6782	. . 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 18897	. . 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 7665	1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 }))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444  {csn 4585   ↦ cmpt 5183   × cxp 5629  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631  Basecbs 17155  +_gcplusg 17196  Scalarcsca 17199  0_gc0g 17378  Grpcgrp 18841  inv_gcminusg 18842  Ringcrg 20118  LModclmod 20742  LFnlclfn 39023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-map 8778  df-0g 17380  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-ring 20120  df-lmod 20744  df-lfl 39024

This theorem is referenced by:  ldualgrplem  39111