Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > tendospdi1 | Structured version Visualization version GIF version |
Description: Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.) |
Ref | Expression |
---|---|
tendosp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendosp.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendosp.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tendospdi1 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑈‘(𝐹 ∘ 𝐺)) = ((𝑈‘𝐹) ∘ (𝑈‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 790 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐾 ∈ 𝑉) | |
2 | simplr 792 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝑊 ∈ 𝐻) | |
3 | simpr1 1067 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝑈 ∈ 𝐸) | |
4 | simpr2 1068 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐹 ∈ 𝑇) | |
5 | simpr3 1069 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐺 ∈ 𝑇) | |
6 | tendosp.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | tendosp.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | tendosp.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
9 | 6, 7, 8 | tendovalco 36053 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑈 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑈‘(𝐹 ∘ 𝐺)) = ((𝑈‘𝐹) ∘ (𝑈‘𝐺))) |
10 | 1, 2, 3, 4, 5, 9 | syl32anc 1334 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑈‘(𝐹 ∘ 𝐺)) = ((𝑈‘𝐹) ∘ (𝑈‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∘ ccom 5118 ‘cfv 5888 LHypclh 35270 LTrncltrn 35387 TEndoctendo 36040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-tendo 36043 |
This theorem is referenced by: tendocnv 36310 tendospcanN 36312 dvalveclem 36314 dvhlveclem 36397 dihjatcclem4 36710 |
Copyright terms: Public domain | W3C validator |