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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 36442, 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 6659 | . . 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 2798 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | lflnegcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 36359 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅)) |
10 | 4, 5, 9 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑅)) |
11 | lflnegl.o | . . . 4 ⊢ 0 = (0g‘𝑅) | |
12 | 11 | fvexi 6659 | . . 3 ⊢ 0 ∈ V |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ V) |
14 | lflnegcl.i | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
15 | 6 | lmodring 19635 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
16 | ringgrp 19295 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
17 | 4, 15, 16 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
18 | 7, 14, 17 | grpinvf1o 18161 | . . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
19 | f1of 6590 | . . 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 18144 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 ) |
25 | 17, 24 | sylan 583 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐼‘𝑦) + 𝑦) = 0 ) |
26 | 3, 10, 13, 20, 22, 25 | caofinvl 7416 | 1 ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 ↦ cmpt 5110 × cxp 5517 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 Ringcrg 19290 LModclmod 19627 LFnlclfn 36353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-map 8391 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-ring 19292 df-lmod 19629 df-lfl 36354 |
This theorem is referenced by: ldualgrplem 36441 |
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