Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 6678 | . 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 2818 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2818 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | lfladdcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
8 | 4, 5, 6, 7 | lflf 36079 | . . 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 36079 | . . 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 36079 | . . 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 19571 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
17 | ringgrp 19231 | . . . 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 18050 | . . 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 7432 | 1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f + 𝐼) = (𝐺 ∘f + (𝐻 ∘f + 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 Grpcgrp 18041 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-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-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-map 8397 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-ring 19228 df-lmod 19565 df-lfl 36074 |
This theorem is referenced by: ldualgrplem 36161 |
Copyright terms: Public domain | W3C validator |