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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfladdass | Structured version Visualization version GIF version |
Description: Associativity of functional addition. (Contributed by NM, 19-Oct-2014.) |
Ref | Expression |
---|---|
lfladdcl.r | ⊢ 𝑅 = (Scalar‘𝑊) |
lfladdcl.p | ⊢ + = (+g‘𝑅) |
lfladdcl.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lfladdcl.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lfladdcl.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lfladdcl.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
lfladdass.i | ⊢ (𝜑 → 𝐼 ∈ 𝐹) |
Ref | Expression |
---|---|
lfladdass | ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f + 𝐼) = (𝐺 ∘f + (𝐻 ∘f + 𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6921 | . 2 ⊢ (𝜑 → (Base‘𝑊) ∈ V) | |
2 | lfladdcl.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lfladdcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
4 | lfladdcl.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
5 | eqid 2734 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2734 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | lfladdcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
8 | 4, 5, 6, 7 | lflf 39044 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
9 | 2, 3, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
10 | lfladdcl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
11 | 4, 5, 6, 7 | lflf 39044 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
12 | 2, 10, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
13 | lfladdass.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐹) | |
14 | 4, 5, 6, 7 | lflf 39044 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝐹) → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
15 | 2, 13, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
16 | 4 | lmodring 20882 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
17 | ringgrp 20255 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
18 | 2, 16, 17 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
19 | lfladdcl.p | . . . 4 ⊢ + = (+g‘𝑅) | |
20 | 5, 19 | grpass 18972 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
21 | 18, 20 | sylan 580 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
22 | 1, 9, 12, 15, 21 | caofass 7735 | 1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f + 𝐼) = (𝐺 ∘f + (𝐻 ∘f + 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 Basecbs 17244 +gcplusg 17297 Scalarcsca 17300 Grpcgrp 18963 Ringcrg 20250 LModclmod 20874 LFnlclfn 39038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-map 8866 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-ring 20252 df-lmod 20876 df-lfl 39039 |
This theorem is referenced by: ldualgrplem 39126 |
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