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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflnegl | Structured version Visualization version GIF version |
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 39128, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.) |
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) |
Ref | Expression |
---|---|
lflnegl | ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflnegcl.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6921 | . . 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 2735 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | lflnegcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 39045 | . . 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 6921 | . . 3 ⊢ 0 ∈ V |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ V) |
14 | lflnegcl.i | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
15 | 6 | lmodring 20883 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
16 | ringgrp 20256 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
17 | 4, 15, 16 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
18 | 7, 14, 17 | grpinvf1o 19040 | . . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
19 | f1of 6849 | . . 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 19020 | . . 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 7729 | 1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ↦ cmpt 5231 × cxp 5687 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 0gc0g 17486 Grpcgrp 18964 invgcminusg 18965 Ringcrg 20251 LModclmod 20875 LFnlclfn 39039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-map 8867 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-ring 20253 df-lmod 20877 df-lfl 39040 |
This theorem is referenced by: ldualgrplem 39127 |
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