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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 6687 | . 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 2823 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2823 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | lfladdcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
8 | 4, 5, 6, 7 | lflf 36201 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
9 | 2, 3, 8 | syl2anc 586 | . 2 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
10 | lfladdcl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
11 | 4, 5, 6, 7 | lflf 36201 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
12 | 2, 10, 11 | syl2anc 586 | . 2 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
13 | lfladdass.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐹) | |
14 | 4, 5, 6, 7 | lflf 36201 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝐹) → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
15 | 2, 13, 14 | syl2anc 586 | . 2 ⊢ (𝜑 → 𝐼:(Base‘𝑊)⟶(Base‘𝑅)) |
16 | 4 | lmodring 19644 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
17 | ringgrp 19304 | . . . 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 18114 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
21 | 18, 20 | sylan 582 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
22 | 1, 9, 12, 15, 21 | caofass 7445 | 1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f + 𝐼) = (𝐺 ∘f + (𝐻 ∘f + 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 Basecbs 16485 +gcplusg 16567 Scalarcsca 16570 Grpcgrp 18105 Ringcrg 19299 LModclmod 19636 LFnlclfn 36195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-map 8410 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-ring 19301 df-lmod 19638 df-lfl 36196 |
This theorem is referenced by: ldualgrplem 36283 |
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