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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendopl | Structured version Visualization version GIF version |
Description: Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.) |
Ref | Expression |
---|---|
tendoplcbv.p | ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) |
tendopl2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tendopl | ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6498 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑢‘𝑔) = (𝑈‘𝑔)) | |
2 | 1 | coeq1d 5582 | . . 3 ⊢ (𝑢 = 𝑈 → ((𝑢‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑣‘𝑔))) |
3 | 2 | mpteq2dv 5023 | . 2 ⊢ (𝑢 = 𝑈 → (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑣‘𝑔)))) |
4 | fveq1 6498 | . . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘𝑔) = (𝑉‘𝑔)) | |
5 | 4 | coeq2d 5583 | . . 3 ⊢ (𝑣 = 𝑉 → ((𝑈‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) |
6 | 5 | mpteq2dv 5023 | . 2 ⊢ (𝑣 = 𝑉 → (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑣‘𝑔))) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔)))) |
7 | tendoplcbv.p | . . 3 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) | |
8 | 7 | tendoplcbv 37362 | . 2 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))) |
9 | tendopl2.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | 9 | fvexi 6513 | . . 3 ⊢ 𝑇 ∈ V |
11 | 10 | mptex 6812 | . 2 ⊢ (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) ∈ V |
12 | 3, 6, 8, 11 | ovmpo 7126 | 1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ↦ cmpt 5008 ∘ ccom 5411 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 LTrncltrn 36688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 |
This theorem is referenced by: tendopl2 37364 tendoplcl 37368 erngplus 37390 erngplus-rN 37398 dvaplusg 37596 |
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