Metamath Proof Explorer

Theorem tendodi1

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 tendodi1 ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → S ∘ U P V = S ∘ U P S ∘ 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 simpr3 ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → V ∈ E
9 1 2 3 4 tendoplcl ⊢ K ∈ HL ∧ W ∈ H ∧ U ∈ E ∧ V ∈ E → U P V ∈ E
10 5 7 8 9 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → U P V ∈ E
11 1 3 tendococl ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U P V ∈ E → S ∘ U P V ∈ E
12 5 6 10 11 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → S ∘ U P V ∈ E
13 1 3 tendococl ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E → S ∘ U ∈ E
14 5 6 7 13 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → S ∘ U ∈ E
15 1 3 tendococl ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ V ∈ E → S ∘ V ∈ E
16 5 6 8 15 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → S ∘ V ∈ E
17 1 2 3 4 tendoplcl ⊢ K ∈ HL ∧ W ∈ H ∧ S ∘ U ∈ E ∧ S ∘ V ∈ E → S ∘ U P S ∘ V ∈ E
18 5 14 16 17 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → S ∘ U P S ∘ V ∈ E
19 simplll ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → K ∈ HL
20 simpllr ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → W ∈ H
21 simplr1 ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∈ E
22 simpll ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → K ∈ HL ∧ W ∈ H
23 simplr2 ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → U ∈ E
24 simpr ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → g ∈ T
25 1 2 3 tendocl ⊢ K ∈ HL ∧ W ∈ H ∧ U ∈ E ∧ g ∈ T → U ⁡ g ∈ T
26 22 23 24 25 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → U ⁡ g ∈ T
27 simplr3 ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → V ∈ E
28 1 2 3 tendocl ⊢ K ∈ HL ∧ W ∈ H ∧ V ∈ E ∧ g ∈ T → V ⁡ g ∈ T
29 22 27 24 28 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → V ⁡ g ∈ T
30 1 2 3 tendovalco ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ⁡ g ∈ T ∧ V ⁡ g ∈ T → S ⁡ U ⁡ g ∘ V ⁡ g = S ⁡ U ⁡ g ∘ S ⁡ V ⁡ g
31 19 20 21 26 29 30 syl32anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ⁡ U ⁡ g ∘ V ⁡ g = S ⁡ U ⁡ g ∘ S ⁡ V ⁡ g
32 4 2 tendopl2 ⊢ U ∈ E ∧ V ∈ E ∧ g ∈ T → U P V ⁡ g = U ⁡ g ∘ V ⁡ g
33 23 27 24 32 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → U P V ⁡ g = U ⁡ g ∘ V ⁡ g
34 33 fveq2d ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ⁡ U P V ⁡ g = S ⁡ U ⁡ g ∘ V ⁡ g
35 1 2 3 tendocoval ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ g ∈ T → S ∘ U ⁡ g = S ⁡ U ⁡ g
36 19 20 21 23 24 35 syl221anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ U ⁡ g = S ⁡ U ⁡ g
37 1 2 3 tendocoval ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ V ⁡ g = S ⁡ V ⁡ g
38 19 20 21 27 24 37 syl221anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ V ⁡ g = S ⁡ V ⁡ g
39 36 38 coeq12d ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ U ⁡ g ∘ S ∘ V ⁡ g = S ⁡ U ⁡ g ∘ S ⁡ V ⁡ g
40 31 34 39 3eqtr4rd ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ U ⁡ g ∘ S ∘ V ⁡ g = S ⁡ U P V ⁡ g
41 22 21 23 13 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ U ∈ E
42 22 21 27 15 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ V ∈ E
43 4 2 tendopl2 ⊢ S ∘ U ∈ E ∧ S ∘ V ∈ E ∧ g ∈ T → S ∘ U P S ∘ V ⁡ g = S ∘ U ⁡ g ∘ S ∘ V ⁡ g
44 41 42 24 43 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ U P S ∘ V ⁡ g = S ∘ U ⁡ g ∘ S ∘ V ⁡ g
45 22 23 27 9 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → U P V ∈ E
46 1 2 3 tendocoval ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U P V ∈ E ∧ g ∈ T → S ∘ U P V ⁡ g = S ⁡ U P V ⁡ g
47 22 21 45 24 46 syl121anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ U P V ⁡ g = S ⁡ U P V ⁡ g
48 40 44 47 3eqtr4rd ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E ∧ g ∈ T → S ∘ U P V ⁡ g = S ∘ U P S ∘ V ⁡ g
49 48 ralrimiva ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → ∀ g ∈ T S ∘ U P V ⁡ g = S ∘ U P S ∘ V ⁡ g
50 1 2 3 tendoeq1 ⊢ K ∈ HL ∧ W ∈ H ∧ S ∘ U P V ∈ E ∧ S ∘ U P S ∘ V ∈ E ∧ ∀ g ∈ T S ∘ U P V ⁡ g = S ∘ U P S ∘ V ⁡ g → S ∘ U P V = S ∘ U P S ∘ V
51 5 12 18 49 50 syl121anc ⊢ K ∈ HL ∧ W ∈ H ∧ S ∈ E ∧ U ∈ E ∧ V ∈ E → S ∘ U P V = S ∘ U P S ∘ V