Theorem tendovalco 40464

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 2726	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendof.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendof.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		eqid 2726	. . . . 5 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5		tendof.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	istendo 40459	. . . 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 5864	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
8	7	fveq2d 6905	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑆‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝑔)))
9		fveq2 6901	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑆‘𝑓) = (𝑆‘𝐹))
10	9	coeq1d 5868	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)))
11	8, 10	eqeq12d 2742	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔))))
12		coeq2 5865	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺))
13	12	fveq2d 6905	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑆‘(𝐹 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝐺)))
14		fveq2 6901	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑆‘𝑔) = (𝑆‘𝐺))
15	14	coeq2d 5869	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))
16	13, 15	eqeq12d 2742	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
17	11, 16	rspc2v 3619	. . . . . 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 1131	. . . 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 252	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))))
21	20	3impia 1114	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
22	21	imp 405	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∀wral 3051   class class class wbr 5153   ∘ ccom 5686  ⟶wf 6550  ‘cfv 6554  lecple 17273  LHypclh 39683  LTrncltrn 39800  trLctrl 39857  TEndoctendo 40451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-tendo 40454

This theorem is referenced by:  tendoco2  40467  tendococl  40471  tendodi1  40483  tendoicl  40495  cdlemi2  40518  tendospdi1  40719