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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 6660 | . 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 2798 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2798 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | lfladdcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
8 | 4, 5, 6, 7 | lflf 36359 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
9 | 2, 3, 8 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
10 | lfladdcl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
11 | 4, 5, 6, 7 | lflf 36359 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
12 | 2, 10, 11 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
13 | lfladdass.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐹) | |
14 | 4, 5, 6, 7 | lflf 36359 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝐹) → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
15 | 2, 13, 14 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
16 | 4 | lmodring 19635 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
17 | ringgrp 19295 | . . . 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 18104 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
21 | 18, 20 | sylan 583 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
22 | 1, 9, 12, 15, 21 | caofass 7423 | 1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f + 𝐼) = (𝐺 ∘f + (𝐻 ∘f + 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 Grpcgrp 18095 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-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-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-map 8391 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-ring 19292 df-lmod 19629 df-lfl 36354 |
This theorem is referenced by: ldualgrplem 36441 |
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