Metamath Proof Explorer

Theorem tendodi2

Description: Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013)

Ref Expression
Hypotheses tendopl.h ⊢H=LHyp⁡K
tendopl.t ⊢T=LTrn⁡K⁡W
tendopl.e ⊢E=TEndo⁡K⁡W
tendopl.p ⊢P=s∈E,t∈E⟼f∈T⟼s⁡f∘t⁡f
Assertion tendodi2 ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→SPU∘V=S∘VPU∘V

Proof

Step Hyp Ref Expression
1 tendopl.h ⊢H=LHyp⁡K
2 tendopl.t ⊢T=LTrn⁡K⁡W
3 tendopl.e ⊢E=TEndo⁡K⁡W
4 tendopl.p ⊢P=s∈E,t∈E⟼f∈T⟼s⁡f∘t⁡f
5 simpl ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→K∈HL∧W∈H
6 simpr1 ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→S∈E
7 simpr2 ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→U∈E
8 1 2 3 4 tendoplcl ⊢K∈HL∧W∈H∧S∈E∧U∈E→SPU∈E
9 5 6 7 8 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→SPU∈E
10 simpr3 ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→V∈E
11 1 3 tendococl ⊢K∈HL∧W∈H∧SPU∈E∧V∈E→SPU∘V∈E
12 5 9 10 11 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→SPU∘V∈E
13 1 3 tendococl ⊢K∈HL∧W∈H∧S∈E∧V∈E→S∘V∈E
14 5 6 10 13 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→S∘V∈E
15 1 3 tendococl ⊢K∈HL∧W∈H∧U∈E∧V∈E→U∘V∈E
16 5 7 10 15 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→U∘V∈E
17 1 2 3 4 tendoplcl ⊢K∈HL∧W∈H∧S∘V∈E∧U∘V∈E→S∘VPU∘V∈E
18 5 14 16 17 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→S∘VPU∘V∈E
19 simpll ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→K∈HL∧W∈H
20 simplr1 ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→S∈E
21 simplr2 ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→U∈E
22 19 20 21 8 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→SPU∈E
23 simplr3 ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→V∈E
24 simpr ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→g∈T
25 1 2 3 tendocoval ⊢K∈HL∧W∈H∧SPU∈E∧V∈E∧g∈T→SPU∘V⁡g=SPU⁡V⁡g
26 19 22 23 24 25 syl121anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→SPU∘V⁡g=SPU⁡V⁡g
27 simplll ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→K∈HL
28 simpllr ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→W∈H
29 1 2 3 tendocoval ⊢K∈HL∧W∈H∧S∈E∧V∈E∧g∈T→S∘V⁡g=S⁡V⁡g
30 27 28 20 23 24 29 syl221anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→S∘V⁡g=S⁡V⁡g
31 1 2 3 tendocoval ⊢K∈HL∧W∈H∧U∈E∧V∈E∧g∈T→U∘V⁡g=U⁡V⁡g
32 27 28 21 23 24 31 syl221anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→U∘V⁡g=U⁡V⁡g
33 30 32 coeq12d ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→S∘V⁡g∘U∘V⁡g=S⁡V⁡g∘U⁡V⁡g
34 19 20 23 13 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→S∘V∈E
35 19 21 23 15 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→U∘V∈E
36 4 2 tendopl2 ⊢S∘V∈E∧U∘V∈E∧g∈T→S∘VPU∘V⁡g=S∘V⁡g∘U∘V⁡g
37 34 35 24 36 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→S∘VPU∘V⁡g=S∘V⁡g∘U∘V⁡g
38 1 2 3 tendocl ⊢K∈HL∧W∈H∧V∈E∧g∈T→V⁡g∈T
39 19 23 24 38 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→V⁡g∈T
40 4 2 tendopl2 ⊢S∈E∧U∈E∧V⁡g∈T→SPU⁡V⁡g=S⁡V⁡g∘U⁡V⁡g
41 20 21 39 40 syl3anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→SPU⁡V⁡g=S⁡V⁡g∘U⁡V⁡g
42 33 37 41 3eqtr4rd ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→SPU⁡V⁡g=S∘VPU∘V⁡g
43 26 42 eqtrd ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E∧g∈T→SPU∘V⁡g=S∘VPU∘V⁡g
44 43 ralrimiva ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→∀g∈TSPU∘V⁡g=S∘VPU∘V⁡g
45 1 2 3 tendoeq1 ⊢K∈HL∧W∈H∧SPU∘V∈E∧S∘VPU∘V∈E∧∀g∈TSPU∘V⁡g=S∘VPU∘V⁡g→SPU∘V=S∘VPU∘V
46 5 12 18 44 45 syl121anc ⊢K∈HL∧W∈H∧S∈E∧U∈E∧V∈E→SPU∘V=S∘VPU∘V