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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 6203 | . 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 2622 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2622 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | lfladdcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
8 | 4, 5, 6, 7 | lflf 34350 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
9 | 2, 3, 8 | syl2anc 693 | . 2 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
10 | lfladdcl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
11 | 4, 5, 6, 7 | lflf 34350 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
12 | 2, 10, 11 | syl2anc 693 | . 2 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
13 | lfladdass.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐹) | |
14 | 4, 5, 6, 7 | lflf 34350 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝐹) → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
15 | 2, 13, 14 | syl2anc 693 | . 2 ⊢ (𝜑 → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
16 | 4 | lmodring 18871 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
17 | ringgrp 18552 | . . . 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 17431 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
21 | 18, 20 | sylan 488 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
22 | 1, 9, 12, 15, 21 | caofass 6931 | 1 ⊢ (𝜑 → ((𝐺 ∘𝑓 + 𝐻) ∘𝑓 + 𝐼) = (𝐺 ∘𝑓 + (𝐻 ∘𝑓 + 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 +gcplusg 15941 Scalarcsca 15944 Grpcgrp 17422 Ringcrg 18547 LModclmod 18863 LFnlclfn 34344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-map 7859 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ring 18549 df-lmod 18865 df-lfl 34345 |
This theorem is referenced by: ldualgrplem 34432 |
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