Theorem tendovalco 40752

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 2729	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendof.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendof.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		eqid 2729	. . . . 5 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5		tendof.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	istendo 40747	. . . 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 5811	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
8	7	fveq2d 6844	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑆‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝑔)))
9		fveq2 6840	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑆‘𝑓) = (𝑆‘𝐹))
10	9	coeq1d 5815	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)))
11	8, 10	eqeq12d 2745	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔))))
12		coeq2 5812	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺))
13	12	fveq2d 6844	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑆‘(𝐹 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝐺)))
14		fveq2 6840	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑆‘𝑔) = (𝑆‘𝐺))
15	14	coeq2d 5816	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))
16	13, 15	eqeq12d 2745	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))
17	11, 16	rspc2v 3596	. . . . . 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 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5102   ∘ ccom 5635  ⟶wf 6495  ‘cfv 6499  lecple 17203  LHypclh 39971  LTrncltrn 40088  trLctrl 40145  TEndoctendo 40739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-tendo 40742

This theorem is referenced by:  tendoco2  40755  tendococl  40759  tendodi1  40771  tendoicl  40783  cdlemi2  40806  tendospdi1  41007