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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 39147, 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 6920 | . . 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 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 8 | lflnegcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 9 | 6, 7, 1, 8 | lflf 39064 | . . 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 6920 | . . 3 ⊢ 0 ∈ V | 
| 13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ V) | 
| 14 | lflnegcl.i | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
| 15 | 6 | lmodring 20866 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) | 
| 16 | ringgrp 20235 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
| 17 | 4, 15, 16 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) | 
| 18 | 7, 14, 17 | grpinvf1o 19027 | . . 3 ⊢ (𝜑 → 𝐼:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) | 
| 19 | f1of 6848 | . . 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 19007 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 ↦ cmpt 5225 × cxp 5683 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 0gc0g 17484 Grpcgrp 18951 invgcminusg 18952 Ringcrg 20230 LModclmod 20858 LFnlclfn 39058 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-map 8868 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-ring 20232 df-lmod 20860 df-lfl 39059 | 
| This theorem is referenced by: ldualgrplem 39146 | 
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