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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 38025 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → (𝑅 + 𝑆) = (𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))) |
8 | 7 | fveq1d 6665 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → ((𝑅 + 𝑆)‘𝐺) = ((𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))‘𝐺)) |
9 | 8 | 3adantr3 1163 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) → ((𝑅 + 𝑆)‘𝐺) = ((𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓)))‘𝐺)) |
10 | simpr3 1188 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) → 𝐺 ∈ 𝑇) | |
11 | fveq2 6663 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑅‘𝑓) = (𝑅‘𝐺)) | |
12 | fveq2 6663 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑆‘𝑓) = (𝑆‘𝐺)) | |
13 | 11, 12 | coeq12d 5728 | . . . 4 ⊢ (𝑓 = 𝐺 → ((𝑅‘𝑓) ∘ (𝑆‘𝑓)) = ((𝑅‘𝐺) ∘ (𝑆‘𝐺))) |
14 | eqid 2818 | . . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑅‘𝑓) ∘ (𝑆‘𝑓))) | |
15 | fvex 6676 | . . . . 5 ⊢ (𝑅‘𝐺) ∈ V | |
16 | fvex 6676 | . . . . 5 ⊢ (𝑆‘𝐺) ∈ V | |
17 | 15, 16 | coex 7624 | . . . 4 ⊢ ((𝑅‘𝐺) ∘ (𝑆‘𝐺)) ∈ V |
18 | 13, 14, 17 | fvmpt 6761 | . . 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 2853 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7145 +gcplusg 16553 Scalarcsca 16556 LHypclh 37000 LTrncltrn 37117 TEndoctendo 37768 DVecAcdveca 38018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-edring 37773 df-dveca 38019 |
This theorem is referenced by: dvalveclem 38041 |
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