Theorem tendovalco 41639

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 2760	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendof.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendof.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		eqid 2760	. . . . 5 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5		tendof.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	istendo 41634	. . . 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 5837	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
8	7	fveq2d 6883	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑆‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝑔)))
9		fveq2 6879	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑆‘𝑓) = (𝑆‘𝐹))
10	9	coeq1d 5841	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)))
11	8, 10	eqeq12d 2776	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔))))
12		coeq2 5838	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺))
13	12	fveq2d 6883	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑆‘(𝐹 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝐺)))
14		fveq2 6879	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑆‘𝑔) = (𝑆‘𝐺))
15	14	coeq2d 5842	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))
16	13, 15	eqeq12d 2776	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
17	11, 16	rspc2v 3587	. . . . . 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 33	. . . . 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 1152	. . . 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 256	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))))
21	20	3impia 1135	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
22	21	imp 412	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   class class class wbr 5103   ∘ ccom 5659  ⟶wf 6529  ‘cfv 6533  lecple 17350  LHypclh 40858  LTrncltrn 40975  trLctrl 41032  TEndoctendo 41626

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8829  df-tendo 41629

This theorem is used by:  tendoco2  41642  tendococl  41646  tendodi1  41658  tendoicl  41670  cdlemi2  41693  tendospdi1  41894