Theorem tendotp 40763

Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Hypotheses

Ref	Expression
tendoset.l	⊢ ≤ = (le‘𝐾)
tendoset.h	⊢ 𝐻 = (LHyp‘𝐾)
tendoset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoset.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
tendoset.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendotp	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))

Proof of Theorem tendotp

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

Step	Hyp	Ref	Expression
1		tendoset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		tendoset.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendoset.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		tendoset.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
5		tendoset.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	istendo 40762	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))))
7		2fveq3 6911	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘(𝑆‘𝑓)) = (𝑅‘(𝑆‘𝐹)))
8		fveq2 6906	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹))
9	7, 8	breq12d 5156	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) ↔ (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))
10	9	rspccv 3619	. . . 4 ⊢ (∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))
11	10	3ad2ant3 1136	. . 3 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))
12	6, 11	biimtrdi 253	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))))
13	12	3imp 1111	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   class class class wbr 5143   ∘ ccom 5689  ⟶wf 6557  ‘cfv 6561  lecple 17304  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  TEndoctendo 40754

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-tendo 40757

This theorem is referenced by:  tendococl  40774  tendoid  40775  tendopltp  40782  tendoicl  40798  cdlemi1  40820  tendotr  40832  cdleml1N  40978  dva1dim  40987  dialss  41048  diblss  41172