| 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 | fvexd 6921 | . 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 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 6 | eqid 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 7 | lfladdcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 8 | 4, 5, 6, 7 | lflf 39064 | . . 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 39064 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
| 12 | 2, 10, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅)) |
| 13 | 4 | lmodring 20866 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
| 14 | ringabl 20278 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
| 15 | 2, 13, 14 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Abel) |
| 16 | 15 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑅 ∈ Abel) |
| 17 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅)) | |
| 18 | simprr 773 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅)) | |
| 19 | lfladdcl.p | . . . 4 ⊢ + = (+g‘𝑅) | |
| 20 | 5, 19 | ablcom 19817 | . . 3 ⊢ ((𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
| 21 | 16, 17, 18, 20 | syl3anc 1373 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
| 22 | 1, 9, 12, 21 | caofcom 7734 | 1 ⊢ (𝜑 → (𝐺 ∘f + 𝐻) = (𝐻 ∘f + 𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 Abelcabl 19799 Ringcrg 20230 LModclmod 20858 LFnlclfn 39058 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-ur 20179 df-ring 20232 df-lmod 20860 df-lfl 39059 |
| This theorem is referenced by: ldualvaddcom 39141 |
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