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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvaplusgv | Structured version Visualization version GIF version |
Description: Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.) |
Ref | Expression |
---|---|
dvafplus.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvafplus.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dvafplus.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dvafplus.u | ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) |
dvafplus.f | ⊢ 𝐹 = (Scalar‘𝑈) |
dvafplus.p | ⊢ + = (+g‘𝐹) |
Ref | Expression |
---|---|
dvaplusgv | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) → ((𝑅 + 𝑆)‘𝐺) = ((𝑅‘𝐺) ∘ (𝑆‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvafplus.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dvafplus.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | dvafplus.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | dvafplus.u | . . . . 5 ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) | |
5 | dvafplus.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑈) | |
6 | dvafplus.p | . . . . 5 ⊢ + = (+g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | dvaplusg 40953 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → (𝑅 + 𝑆) = (𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))) |
8 | 7 | fveq1d 6903 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → ((𝑅 + 𝑆)‘𝐺) = ((𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))‘𝐺)) |
9 | 8 | 3adantr3 1169 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) → ((𝑅 + 𝑆)‘𝐺) = ((𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))‘𝐺)) |
10 | simpr3 1194 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) → 𝐺 ∈ 𝑇) | |
11 | fveq2 6901 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑅‘𝑓) = (𝑅‘𝐺)) | |
12 | fveq2 6901 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑆‘𝑓) = (𝑆‘𝐺)) | |
13 | 11, 12 | coeq12d 5872 | . . . 4 ⊢ (𝑓 = 𝐺 → ((𝑅‘𝑓) ∘ (𝑆‘𝑓)) = ((𝑅‘𝐺) ∘ (𝑆‘𝐺))) |
14 | eqid 2733 | . . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓))) | |
15 | fvex 6914 | . . . . 5 ⊢ (𝑅‘𝐺) ∈ V | |
16 | fvex 6914 | . . . . 5 ⊢ (𝑆‘𝐺) ∈ V | |
17 | 15, 16 | coex 7947 | . . . 4 ⊢ ((𝑅‘𝐺) ∘ (𝑆‘𝐺)) ∈ V |
18 | 13, 14, 17 | fvmpt 7010 | . . 3 ⊢ (𝐺 ∈ 𝑇 → ((𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))‘𝐺) = ((𝑅‘𝐺) ∘ (𝑆‘𝐺))) |
19 | 10, 18 | syl 17 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) → ((𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))‘𝐺) = ((𝑅‘𝐺) ∘ (𝑆‘𝐺))) |
20 | 9, 19 | eqtrd 2773 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) → ((𝑅 + 𝑆)‘𝐺) = ((𝑅‘𝐺) ∘ (𝑆‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 ↦ cmpt 5232 ∘ ccom 5687 ‘cfv 6558 (class class class)co 7425 +gcplusg 17287 Scalarcsca 17290 LHypclh 39928 LTrncltrn 40045 TEndoctendo 40696 DVecAcdveca 40946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-n0 12518 df-z 12605 df-uz 12870 df-fz 13538 df-struct 17170 df-slot 17205 df-ndx 17217 df-base 17235 df-plusg 17300 df-mulr 17301 df-sca 17303 df-vsca 17304 df-edring 40701 df-dveca 40947 |
This theorem is referenced by: dvalveclem 40969 |
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