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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 36162, 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 6677 | . . 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 2818 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | lflnegcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 36079 | . . 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 6677 | . . 3 ⊢ 0 ∈ V |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ V) |
14 | lflnegcl.i | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
15 | 6 | lmodring 19571 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
16 | ringgrp 19231 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
17 | 4, 15, 16 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
18 | 7, 14, 17 | grpinvf1o 18107 | . . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
19 | f1of 6608 | . . 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 18090 | . . 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 7425 | 1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 ↦ cmpt 5137 × cxp 5546 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 0gc0g 16701 Grpcgrp 18041 invgcminusg 18042 Ringcrg 19226 LModclmod 19563 LFnlclfn 36073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-map 8397 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-ring 19228 df-lmod 19565 df-lfl 36074 |
This theorem is referenced by: ldualgrplem 36161 |
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