Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfladdcom | Structured version Visualization version GIF version |
Description: Commutativity 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) |
Ref | Expression |
---|---|
lfladdcom | ⊢ (𝜑 → (𝐺 ∘𝑓 + 𝐻) = (𝐻 ∘𝑓 + 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6113 | . . 3 ⊢ (Base‘𝑊) ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (Base‘𝑊) ∈ V) |
3 | lfladdcl.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | lfladdcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
5 | lfladdcl.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
6 | eqid 2610 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2610 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
8 | lfladdcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 5, 6, 7, 8 | lflf 33368 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
10 | 3, 4, 9 | syl2anc 691 | . 2 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑅)) |
11 | lfladdcl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
12 | 5, 6, 7, 8 | lflf 33368 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
13 | 3, 11, 12 | syl2anc 691 | . 2 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
14 | 5 | lmodring 18694 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
15 | ringabl 18403 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
16 | 3, 14, 15 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Abel) |
17 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑅 ∈ Abel) |
18 | simprl 790 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅)) | |
19 | simprr 792 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅)) | |
20 | lfladdcl.p | . . . 4 ⊢ + = (+g‘𝑅) | |
21 | 6, 20 | ablcom 18033 | . . 3 ⊢ ((𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
22 | 17, 18, 19, 21 | syl3anc 1318 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
23 | 2, 10, 13, 22 | caofcom 6827 | 1 ⊢ (𝜑 → (𝐺 ∘𝑓 + 𝐻) = (𝐻 ∘𝑓 + 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 Abelcabl 18017 Ringcrg 18370 LModclmod 18686 LFnlclfn 33362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lfl 33363 |
This theorem is referenced by: ldualvaddcom 33445 |
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