Theorem tendovalco 40964

Description: Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Hypotheses

Ref	Expression
tendof.h	⊢ 𝐻 = (LHyp‘𝐾)
tendof.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendof.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendovalco	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))

Proof of Theorem tendovalco

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendof.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendof.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		eqid 2734	. . . . 5 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5		tendof.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	istendo 40959	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7		coeq1 5804	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
8	7	fveq2d 6836	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑆‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝑔)))
9		fveq2 6832	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑆‘𝑓) = (𝑆‘𝐹))
10	9	coeq1d 5808	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)))
11	8, 10	eqeq12d 2750	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔))))
12		coeq2 5805	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺))
13	12	fveq2d 6836	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑆‘(𝐹 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝐺)))
14		fveq2 6832	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑆‘𝑔) = (𝑆‘𝐺))
15	14	coeq2d 5809	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))
16	13, 15	eqeq12d 2750	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
17	11, 16	rspc2v 3585	. . . . . 6 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
18	17	com12 32	. . . . 5 ⊢ (∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
19	18	3ad2ant2 1134	. . . 4 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓)) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
20	6, 19	biimtrdi 253	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))))
21	20	3impia 1117	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
22	21	imp 406	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049   class class class wbr 5096   ∘ ccom 5626  ⟶wf 6486  ‘cfv 6490  lecple 17182  LHypclh 40183  LTrncltrn 40300  trLctrl 40357  TEndoctendo 40951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8763  df-tendo 40954

This theorem is referenced by:  tendoco2  40967  tendococl  40971  tendodi1  40983  tendoicl  40995  cdlemi2  41018  tendospdi1  41219